Department of Mathematics at UTSA - User contributions [en]

MAT3633

2020-08-17T20:38:28Z

Weiming.cao:

MAT3633

2020-08-17T20:34:44Z

Weiming.cao:

MAT3633

2020-08-17T20:31:30Z

Weiming.cao: /* Topics List */

==Course Catalog==
[https://catalog.utsa.edu/undergraduate/sciences/mathematics/#courseinventory MAT 3633. Numerical Analysis.] (3-0) 3 Credit Hours.

Prerequisites: [[MAT2233]], [[MAT3213]], and one of the following: [[CS1063]], [[CS1714]], or [[CS2073]]. Solution of linear and nonlinear equations, curve-fitting, and eigenvalue problems. Generally offered: Fall, Spring. Differential Tuition: $150.

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
| Week.1
||
0.2 and 1.1
||
* Loss of significant digits
* Bisection Method
* Brief introduction to matlab
||
* binary number system;
* Taylor's theorem;
* intermediate value theorem
||
* Nested multiplication for evaluating polynomials
* Machine representation of real numbers
* Loss of significant digits in numerical computing
* Review of Taylor's Theorem
* Bisection method and implementation
||

|-
| Week.2
||
1.2 and 1.3
||
* Fixed-Point Iteration
* Limits of Accuracy: Conditioning of problems
||
* limit of sequences
* multiplicity of solution of equations.
||
* Geometric interpretation
* Convergence of fixed point iterations
* Order of convergence of iterative methods

* Wilkinson polynomial and other examples
* Sensitivity analysis of root-finding
* Error magnification factor for solution of equations

|-
| Week.3
||
1.4 and 1.5
||
* Newton's Method
* Root-Finding without Derivatives
||
* Remainder of Taylor's series
* intermediate value theorem.
||
* Algebraic and geometric interpretation of Newton's method
* Error analysis for Newton's method based on Taylor's theorem
* Newton's method as a fixed point iteration
* Modified Newton's method and its rate of convergence

* Secant Method and its convergence,
* Method of False Position, Muller's Method:
* Stopping criteria for iterative methods

|-
| Week.4
||
2.1 and 2.2
||
* Solve Systems of Linear Equations: Gaussian Elmination
* Solve System of Linear Equations: LU Decomposition
||
* Matrix-matrix products and matrix-vector products
* inverse matrix
* elementary row operations
* product and inverse of matrices for elementary row operations.
||
* Gaussian elimination and its operation counts
* Gaussian elimination with pivoting
* Implementation of Gauss elimination

* Matrices for elementary row operations
* Gauss elimination as matrix products
* Advantages of solutions by LU decomposition

|-
| Week.5
||
2.3 and 2.4
||
* Error Analysis for Solution of Ax=b
* Iterative Methods for Solving Ax=b
||
* Length of vectors
* eigenvalue and eigenvectors of matrix
||
* various norms for vectors and matrices: compatibility of vector and matrix norms.
* Error Analysis for the solution of Ax=b
* Error magnification factor and condition number of matrix

* Jacobi method, Gauss-Seidel method, Successive-Over-Relaxation (SOR) method
* Convergence of Jacobi Method, GS method and SOR method.
* spectral radius of matrix
* convergence of general iterative method for solving system of linear equations,
* Sparse Matrix
* Comparison of Gauss Elimination and iterative methods

|-
| Week.6
||
2.6 and 2.7
||
* Conjugate Gradient Method
* Nonlinear System of Equations
||
* scalar product of vectors
* determinant and eigenvalues of matrix
* quadratic polynomials of n-variables
* partial derivatives and gradients
* chain rule for partial derivatives.
||
* Symmetric positive definite matrix and properties
* Construction of Conjugate Gradient (CG) Method
* Propertise of CG Method
* Preconditioning for CG method

* Taylor's Theorem for multi-variate vector valued functions:
* Newton's Method:
* Broyden's Method

|-
| Week.7
||
3.1 and 3.2
||
* Data and Interpolating Functions
* Interpolation Error and Runge Phenomenon
* Chebyshev interpolation
||
* Fundamental theorem of algebra
* Rolle's theorem.
||
* Lagrange Basis Functions:
* Properties of Lagrange basis functions:
* Lagrange form of the interpolation polynomials

* Newton's Divided Differences:
* Properties of Newton's divided differences:
* Newton's Form of the interpolation polynomials

* Interpolation error analysis
* Runge phenomenon

* Chebyshev Polynomial
* Error estimates for Chebyshev interpolation

|-
| Week.8
||
3.4, 3.5 and 4.1
||
* Cubic Splines
* Bezier Curves
* Least Square Method
||
* one-sided limits
* continuity of functions
* indefinite integrals
* extremum values of multivariate quadratic functions.
||
* Cubic splines
* construction of cubic splines for interpolation
* end conditions
* properties of cubic spline interpolation

* Bezier Curve and fonts

* Least square method for solving inconsistent system of linear equations.
* Basic properties of least square solutions:

|-
| Week.9
||
4.2 and 4.5
||
* Mathematical Models and Data Fitting
* Nonlinear Least Square Fitting
||
* linear spaces, basis functions
* product rule and chain rule for vector valued multivariate functions.
||
* Least square method for curve fitting and statistical modeling.
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc

* Taylor's theorem for vector valued multivariate functions.
* Gauss-Newton Method
* Levenberg-Marquardt Method

|-
| Week.10
||
5.1, 5.2 and 5.3
||
* Numerical Differentiation
* Numerical Integration: Newton-Cotes Formulas
* Numerical Integration: Romberg's Technique
||
* Taylor's theorem
* interpolation error estimates
* properties of definite inetgrals
||
* Finite difference (FD) approximations of 1st order derivative and their error analysis
* FD approximations of 2nd order derivatives and their error analysis
* Undetermined coefficient method for FD approximation
* Extropolation technique for improving the order of approximation

* Midpoint rule, trapezoid rule and Simpson's rule;
* Error analysis based on Taylor's Theorem and interpolation errors
* Degree of precision of quadrature rules
* Composite quadrature rules

* Motivation, construction and implementation of Romberg's technique.

|-
| Week.11
||
5.4 and 5.5
||
* Adaptive Numerical Integration
* Gauss Quadrature Formulas
||
* long divisions
* changing variables for definite integrals
||
* How to estimate the error on a subinterval
* How to mark subintervals to be further refinement?
* Implementation of adaptive numerical integration techniques.

* Motivation and difficulties with straightforward approach.
* Orthogonal polynomials,
* Legendre polynomials and their basic properties;
* Gauss quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules

|-
| Week.12
||
10.1 and 11.1
||
* Discrete Fourier Transform and FFT
* Discrete Cosine Transform (optional)
* Image Compression (optional)
||
* complex numbers and complex variables
* integration by parts
* convergence of sequences and series.
||
* Fourier Series,
* Discrete Fourier Transform
* Matrix Form of Discrete Fourier Transform:
* Inverse Discrete Fourier Transform:
* DFT and Trigonometric interpolation
* Algorithm for computing DFT: Fast Fourier Transform (FFT)

* Discrete Cosine Transform (DCT),
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT:
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions:
* Interpolation Theorem for 2-Dimensional DCT

* Digital Gray scale images and color color images:
* RGB format:
* YCbCr (or YUV) format:
* Convertion between RGB and YUV formats:
* Quantization, Image Compression and Decompression

|-
| Week.13
||
12.1 and 12.2
||
* Power Iteration Methods
* QR Algorithm for Computing Eigenvalues
||
* properties of eigen values and eigenvectors
* Gram-Schmidt orthogonalization
||
* Power iteration and its rate of convergence.
* Inverse Power Iteration,
* Inverse Power Iteration with Shift
* Rayleigh Quotient Iteration

* Definition and basic properties of orthogonal matrices:
* QR-Factorization based on Gram-Schmidt Orthogonalization:
* Normalized Simultaneous Iteration (NSI).
* Unshifted QR Algorithm:
* Shifted QR Algorithm:

|-
| Week.14
||
12.2
||
* Algorithm for Computing Eigenvalues: Speed up of QR-algorithm:
||
* matrices for orthogonal projection and reflection
* block matrices and their products
* similar matrices.
||
* Upper Hessenberg form (UHF)
* Householder Reflector
* Convert a matrix into UHF by Householder reflectors

==Topics List B Wiki Format ==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
* Section 0.2: Loss of significant digits
||
* [[Loss of Significant Digits]]
||
* Binary Number System
* Taylor's Theorem
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
* Section 0.2: Loss of significant digits
||
* [[Nested Multiplication]]
||
* Binary Number System
* Taylor's Theorem
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
* Section 1.1: Fixed-Point Iteration
||
* [[Bisection Method]]
||
* Intermediate Value Theorem
||
* Bisection Method and Implementation
* Brief Introduction to Matlab
|-
|Week 2
||
* Section 1.2: Fixed-Point Iteration
||
* [[Fixed-Point Iteration]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
* Section 1.2: Fixed-Point Iteration
||
* [[Order of Convergence of Iterative Methods]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
* Section 1.3: Limits of Accuracy: Conditioning of Problems
||
* [[Wilkinson Polynomial]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Sensitivity Analysis of Root-Finding
* Error Magnification Factor for Solution of Equations
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Error Analysis for Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Modified Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Root-Finding Without Derivatives]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Secant Method and its Convergence]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Method of False Position, Muller's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Stopping Criteria for Iterative Methods]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Secant Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Method of False Position]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Muller's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Stopping Criteria]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 4
||
* Section 2.1 Solve Systems of Linear Equations: Gaussian Elimination
||
* [[Gaussian Elimination]]
||
* Elementary Row Operations
||
* Gaussian Elimination and its Operation Counts
* Gaussian Elimination with Pivoting
* Implementation of Gauss Elimination
|-
|Week 4
||
* Section 2.2 Solve Systems of Linear Equations: LU Decomposition
||
* [[LU Decomposition]]
||
* Matrix-Matrix Products
* Matrix-Vector Products
* Inverse Matrix
* Elementary Row Operations
||
* Matrices for Elementary Row Operations
* Gauss Elimination as Matrix Products
* Advantages of Solutions by LU Decomposition
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Norms]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Error Analysis for Solution of Ax=b]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Error Magnification Factor and Condition Number of Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Iterative Methods]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Jacobi Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Gauss-Seidel Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Successive-Over-Relaxation (SOR) Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Convergence of Iterative Methods]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Spectral Radius of Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Sparse Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[Conjugate Gradient Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[Symmetric Positive Definite Matrix]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[CG Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Nonlinear System of Equations]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Taylor's Theorem for Multi-Variate Vector Valued Functions]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Newton's Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Broyden's Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Lagrange Basis Functions]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Newton's Divided Differences]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Properties of Newton's Divided Differences]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error Analysis]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Runge Phenomenon]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Chebyshev Polynomial]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Error Estimates for Chebyshev Interpolation]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 8
||
* Section 3.4: Cubic Splines
||
* [[Cubic Splines]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Construction of Cubic Splines for Interpolation
* End Conditions
* Properties of Cubic Spline Interpolation
|-
|Week 8
||
* Section 3.5: Bezier Curves
||
* [[Bezier Curves]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Bezier Curve and Fonts
|-
|Week 8
||
* Section 4.1: Least Square Method
||
* [[Least Square Method]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Least Square Method for Solving Inconsistent System of Linear Equations]
* Basic Properties of Least Square Solutions
|-
|Week 9
||
* Section 4.2: Mathematical Models and Data Fitting
||
* [[Curve Fitting]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
* Section 4.2: Mathematical Models and Data Fitting
||
* [[Statistical Modeling]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Taylor's Theorem for Vector Valued Multivariate Functions]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Gauss-Newton Method]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Levenberg-Marquardt Method]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Numerical Differentiation]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Finite Difference (FD)]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Undetermined Coefficient Method]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Extrapolation Technique]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Newton-Cotes]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Midpoint rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Trapezoid rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Simpson's rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Error Analysis based on Interpolation Errors]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Quadrature Rules]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Composite Quadrature Rules]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.3: Numerical Integration: Romberg's Technique
||
* [[Romberg's Technique]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Motivation, construction and implementation of Romberg's Technique.
|-
|Week 11
||
* Section 5.4: Adaptive Numerical Integration
||
* [[Adaptive Numerical Integration]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
* Section 5.4: Adaptive Numerical Integration
||
* [[Implementation of Adaptive Numerical Integration Techniques]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Formulas]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Orthogonal Polynomials]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Legendre polynomials]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Rule]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fourier Series]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Discrete Fourier Transform (DFT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Inverse Discrete Fourier Transform]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fast Fourier Transform (FFT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
* Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform(DCT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Quantization]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Image Compression]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Image Decompression]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Power Iteration Methods]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Inverse Power Iteration]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Inverse Power Iteration with Shift]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Rayleigh Quotient Iteration]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Orthogonal Matrices]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[QR-Factorization]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Normalized Simultaneous Iteration(NSI)]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Unshifted QR Algorithm]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Shifted QR Algorithm]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 14
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Upper Hessenberg Form (UHF)]]
||
* Matrices for Orthogonal Projection
* Matrices for Reflection
* Block Matrices
* Similar Matrices
||
* Convert a matrix into UHF by Householder reflectors
|-
|Week 14
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Householder Reflector]]
||
* Matrices for Orthogonal Projection
* Matrices for Reflection
* Block Matrices
* Similar Matrices
||
* Convert a matrix into UHF by Householder reflectors
|}

MAT3633

2020-08-17T20:05:40Z

Weiming.cao:

==Course Catalog==
[https://catalog.utsa.edu/undergraduate/sciences/mathematics/#courseinventory MAT 3633. Numerical Analysis.] (3-0) 3 Credit Hours.

Prerequisites: [[MAT2233]], [[MAT3213]], and one of the following: [[CS1063]], [[CS1714]], or [[CS2073]]. Solution of linear and nonlinear equations, curve-fitting, and eigenvalue problems. Generally offered: Fall, Spring. Differential Tuition: $150.

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
| Week.1
||
0.2 and 1.1
||
* Loss of significant digits
* Bisection Method
* Brief introduction to matlab
||
* binary number system;
* Taylor's theorem;
* intermediate value theorem
||
* Nested multiplication for evaluating polynomials
* Machine representation of real numbers
* Loss of significant digits in numerical computing
* Review of Taylor's Theorem
* Bisection method and implementation

|-
| Week.2
||
1.2 and 1.3
||
* Fixed-Point Iteration
* Limits of Accuracy: Conditioning of problems
||
* limit of sequences
* multiplicity of solution of equations.
||
* Geometric interpretation
* Convergence of fixed point iterations
* Order of convergence of iterative methods
* Wilkinson polynomial and other examples
* Sensitivity analysis of root-finding
* Error magnification factor for solution of equations
|-
|Week 1
||
Section 1.1: Fixed-Point Iteration
||
* [[Bisection Method]]
||
* [[Intermediate Value Theorem]]
||
* Bisection Method and Implementation
* Brief Introduction to Matlab
|-
|Week 2
||
Section 1.2: Fixed-Point Iteration
||
* [[Fixed-Point Iteration]]
* [[Order of Convergence]] of Iterative Methods
||
* [[Limit of Sequences]]
* [[Solution Multiplicity of Equations]]
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
Section 1.3: Limits of Accuracy: Conditioning of Problems
||
* [[Wilkinson Polynomial]]
||
* [[Limit of Sequences]]
* [[Solution Multiplicity of Equations]]
||
* Sensitivity Analysis of Root-Finding
* Error Magnification Factor for Solution of Equations
|-
|Week 3
||
Section 1.4: Newton's Method
||
* [[Newton's Method]]
* [[Error Analysis]] for Newton's Method
* [[Modified Newton's Method]]

* [[Root-Finding Without Derivatives]]
* [[Secant Method and its Convergence]]
* [[Method of False Position, Muller's Method]]
* [[Stopping Criteria for Iterative Methods]]
||
* [[Remainder of Taylor's Series]]
* [[Intermediate Value Theorem]]
* [[Fixed-Point Iteration]]
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
Section 1.5 Root-Finding Without Derivatives
||
* [[Secant Method]]
* [[Method of False Position]]
* [[Muller's Method]]
* [[Stopping Criteria]]
||
* [[Remainder of Taylor's Series]]
* [[Intermediate Value Theorem]]
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 4
||
Section 2.1 Solve Systems of Linear Equations: Gaussian Elimination
||
* [[Gaussian Elimination]]
||
* [[Elementary Row Operations]]
||
* Gaussian Elimination and its Operation Counts
* Gaussian Elimination with Pivoting
* Implementation of Gauss Elimination
|-
|Week 4
||
Section 2.2 Solve Systems of Linear Equations: LU Decomposition
||
* [[LU Decomposition]]
||
* [[Matrix-Matrix Products]]
* [[Matrix-Vector Products]]
* [[Inverse Matrix]]
* [[Elementary Row Operations]]
||
* Matrices for Elementary Row Operations
* Gauss Elimination as Matrix Products
* Advantages of Solutions by LU Decomposition
|-
|Week 5
||
Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Norms]]
* [[Error Analysis]] for Solution of Ax=b
* [[Error Magnification]] Factor and Condition Number of Matrix
||
* [[Length of Vectors]]
* [[Eigenvalues of a Matrix]]
* [[Eigenvectors of a Matrix]]
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Iterative Methods]]
* [[Jacobi Method]]
* [[Gauss-Seidel Method]]
* [[Successive-Over-Relaxation (SOR) Method]]
* [[Convergence of Iterative Methods]]
* [[Spectral Radius of Matrix]]
* [[Sparse Matrix]]
||
* [[Length of Vectors]]
* [[Eigenvalues of a Matrix]]
* [[Eigenvectors of a Matrix]]
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 6
||
Section 2.6: Conjugate Gradient (CG) Method
||
* [[Conjugate Gradient Method]]
* [[Symmetric Positive Definite Matrix]]
* [[CG Method]]
||
* [[Scalar Product of Vectors]]
* [[Determinant of a Matrix]]
* [[Eigenvalues of a Matrix]]
* [[Quadratic Polynomials of n-variables]]
* [[Partial Derivatives]]
* [[Gradients]]
* [[Chain Rule for Partial Derivatives]]
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
Section 2.7: Nonlinear System of Equations
||
* [[Nonlinear System of Equations]]
* [[Taylor's Theorem for Multi-Variate Vector Valued Functions]]
* [[Newton's Method]]
* [[Broyden's Method]]
||
* [[Scalar Product of Vectors]]
* [[Determinant of a Matrix]]
* [[Eigenvalues of a Matrix]]
* [[Quadratic Polynomials of n-variables]]
* [[Partial Derivatives]]
* [[Gradients]]
* [[Chain Rule for Partial Derivatives]]
||
* (TBD)
|-
|Week 7
||
Sections 3.1: Data and Interpolating Functions
||
* [[Lagrange Basis Functions]]
* [[Newton's Divided Differences]]
* [[Properties of Newton's Divided Differences]]
||
* [[Fundamental Theorem of Algebra]]
* [[Rolle's Theorem]]
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error]]
* Interpolation [[Error Analysis]]
* [[Runge Phenomenon]]
* [[Chebyshev Polynomial]]
* [[Error Estimates]] for Chebyshev Interpolation
||
* [[Fundamental Theorem of Algebra]]
* [[Rolle's Theorem]]
||
* (TBD)
|-
|Week 8
||
Section 3.4: Cubic Splines
||
* [[Cubic Splines]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Construction of Cubic Splines for Interpolation
* End Conditions
* Properties of Cubic Spline Interpolation
|-
|Week 8
||
Section 3.5: Bezier Curves
||
* [[Bezier Curves]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Bezier Curve and Fonts
|-
|Week 8
||
Section 4.1: Least Square Method
||
* [[Least Square Method]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Least Square Method for Solving Inconsistent System of Linear Equations]
* Basic Properties of Least Square Solutions
|-
|Week 9
||
Section 4.2: Mathematical Models and Data Fitting
||
* [[Curve Fitting]]
* [[Statistical Modeling]]
||
* [[Linear Spaces]]
* [[Basis Functions]]
* [[Product Rule for Vector Valued Multivariate Functions]]
* [[Chain Rule for Vector Valued Multivariate Functions]]
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
Section 4.5: Nonlinear Least Square Fitting
||
* [[Taylor's Theorem for Vector Valued Multivariate Functions]]
* [[Gauss-Newton Method]]
* [[Levenberg-Marquardt Method]]
||
* [[Linear Spaces]]
* [[Basis Functions]]
* [[Product Rule for Vector Valued Multivariate Functions]]
* [[Chain Rule for Vector Valued Multivariate Functions]]
||
* (TBD)
|-
|Week 10
||
Section 5.1: Numerical Differentiation
||
* [[Numerical Differentiation]]
* [[Finite Difference]] (FD)
* [[Undetermined Coefficient Method]]
* [[Extrapolation Technique]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Newton-Cotes]]
* [[Midpoint rule]]
* [[Trapezoid rule]]
* [[Simpson's rule]]
* [[Error Analysis based on Interpolation Errors]]
* [[Quadrature Rules]]
* [[Composite Quadrature Rules]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
Section 5.3: Numerical Integration: Romberg's Technique
||
* [[Romberg's Technique]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Motivation, construction and implementation of Romberg's Technique.
|-
|Week 11
||
Section 5.4: Adaptive Numerical Integration
||
* [[Adaptive Numerical Integration]]
* [[Implementation of Adaptive Numerical Integration Techniques]]
||
* [[Long Divisions]]
* [[Substitution Methods]] for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Formulas]]
* [[Orthogonal Polynomials]]
* [[Legendre polynomials]]
* [[Gauss Quadrature Rule]]
||
* [[Long Divisions]]
* [[Substitution Methods]] for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 12
||
Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fourier Series]]
* [[Discrete Fourier Transform]] (DFT)
* [[Inverse Discrete Fourier Transform]]
* [[Fast Fourier Transform (FFT)]]
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform]]
* [[Discrete Cosine Transform]](DCT)
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
Section 11.2: Image Compression (optional)
||
* [[Quantization]]
* [[Image Compression]]
* [[Image Decompression]]
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 13
||
Section 12.1: Power Iteration Methods
||
* [[Power Iteration Methods]]
* [[Inverse Power Iteration]]
* [[Inverse Power Iteration with Shift]]
* [[Rayleigh Quotient Iteration]]
||
* [[Eigenvalues]]
* [[Eigenvectors]]
* [[Orthonormal Bases and the Gram-Schmidt Process]]
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Orthogonal Matrices]]
* [[QR-Factorization]]
* [[Normalized Simultaneous Iteration]](NSI)
* [[Unshifted QR Algorithm]]
* [[Shifted QR Algorithm]]
||
* [[Eigenvalues]]
* [[Eigenvectors]]
* [[Orthonormal Bases and the Gram-Schmidt Process]]
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 14
||
Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Upper Hessenberg Form]] (UHF)
* [[Householder Reflector]]
||
* [[Matrices for Orthogonal Projection]]
* [[Matrices for Reflection]]
* [[Block Matrices]]
* [[Similar Matrices]]
||
* Convert a matrix into UHF by Householder reflectors
|}

==Topics List B Wiki Format ==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
* Section 0.2: Loss of significant digits
||
* [[Loss of Significant Digits]]
||
* Binary Number System
* Taylor's Theorem
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
* Section 0.2: Loss of significant digits
||
* [[Nested Multiplication]]
||
* Binary Number System
* Taylor's Theorem
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
* Section 1.1: Fixed-Point Iteration
||
* [[Bisection Method]]
||
* Intermediate Value Theorem
||
* Bisection Method and Implementation
* Brief Introduction to Matlab
|-
|Week 2
||
* Section 1.2: Fixed-Point Iteration
||
* [[Fixed-Point Iteration]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
* Section 1.2: Fixed-Point Iteration
||
* [[Order of Convergence of Iterative Methods]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
* Section 1.3: Limits of Accuracy: Conditioning of Problems
||
* [[Wilkinson Polynomial]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Sensitivity Analysis of Root-Finding
* Error Magnification Factor for Solution of Equations
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Error Analysis for Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Modified Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Root-Finding Without Derivatives]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Secant Method and its Convergence]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Method of False Position, Muller's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Stopping Criteria for Iterative Methods]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Secant Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Method of False Position]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Muller's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Stopping Criteria]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 4
||
* Section 2.1 Solve Systems of Linear Equations: Gaussian Elimination
||
* [[Gaussian Elimination]]
||
* Elementary Row Operations
||
* Gaussian Elimination and its Operation Counts
* Gaussian Elimination with Pivoting
* Implementation of Gauss Elimination
|-
|Week 4
||
* Section 2.2 Solve Systems of Linear Equations: LU Decomposition
||
* [[LU Decomposition]]
||
* Matrix-Matrix Products
* Matrix-Vector Products
* Inverse Matrix
* Elementary Row Operations
||
* Matrices for Elementary Row Operations
* Gauss Elimination as Matrix Products
* Advantages of Solutions by LU Decomposition
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Norms]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Error Analysis for Solution of Ax=b]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Error Magnification Factor and Condition Number of Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Iterative Methods]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Jacobi Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Gauss-Seidel Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Successive-Over-Relaxation (SOR) Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Convergence of Iterative Methods]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Spectral Radius of Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Sparse Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[Conjugate Gradient Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[Symmetric Positive Definite Matrix]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[CG Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Nonlinear System of Equations]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Taylor's Theorem for Multi-Variate Vector Valued Functions]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Newton's Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Broyden's Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Lagrange Basis Functions]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Newton's Divided Differences]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Properties of Newton's Divided Differences]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error Analysis]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Runge Phenomenon]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Chebyshev Polynomial]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Error Estimates for Chebyshev Interpolation]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 8
||
* Section 3.4: Cubic Splines
||
* [[Cubic Splines]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Construction of Cubic Splines for Interpolation
* End Conditions
* Properties of Cubic Spline Interpolation
|-
|Week 8
||
* Section 3.5: Bezier Curves
||
* [[Bezier Curves]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Bezier Curve and Fonts
|-
|Week 8
||
* Section 4.1: Least Square Method
||
* [[Least Square Method]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Least Square Method for Solving Inconsistent System of Linear Equations]
* Basic Properties of Least Square Solutions
|-
|Week 9
||
* Section 4.2: Mathematical Models and Data Fitting
||
* [[Curve Fitting]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
* Section 4.2: Mathematical Models and Data Fitting
||
* [[Statistical Modeling]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Taylor's Theorem for Vector Valued Multivariate Functions]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Gauss-Newton Method]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Levenberg-Marquardt Method]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Numerical Differentiation]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Finite Difference (FD)]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Undetermined Coefficient Method]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Extrapolation Technique]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Newton-Cotes]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Midpoint rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Trapezoid rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Simpson's rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Error Analysis based on Interpolation Errors]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Quadrature Rules]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Composite Quadrature Rules]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.3: Numerical Integration: Romberg's Technique
||
* [[Romberg's Technique]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Motivation, construction and implementation of Romberg's Technique.
|-
|Week 11
||
* Section 5.4: Adaptive Numerical Integration
||
* [[Adaptive Numerical Integration]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
* Section 5.4: Adaptive Numerical Integration
||
* [[Implementation of Adaptive Numerical Integration Techniques]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Formulas]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Orthogonal Polynomials]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Legendre polynomials]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Rule]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fourier Series]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Discrete Fourier Transform (DFT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Inverse Discrete Fourier Transform]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fast Fourier Transform (FFT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
* Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform(DCT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Quantization]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Image Compression]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Image Decompression]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Power Iteration Methods]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Inverse Power Iteration]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Inverse Power Iteration with Shift]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Rayleigh Quotient Iteration]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Orthogonal Matrices]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[QR-Factorization]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Normalized Simultaneous Iteration(NSI)]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Unshifted QR Algorithm]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Shifted QR Algorithm]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 14
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Upper Hessenberg Form (UHF)]]
||
* Matrices for Orthogonal Projection
* Matrices for Reflection
* Block Matrices
* Similar Matrices
||
* Convert a matrix into UHF by Householder reflectors
|-
|Week 14
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Householder Reflector]]
||
* Matrices for Orthogonal Projection
* Matrices for Reflection
* Block Matrices
* Similar Matrices
||
* Convert a matrix into UHF by Householder reflectors
|}

MAT3633

2020-08-17T20:04:45Z

Weiming.cao: /* Topics List */

==Course Catalog==
[https://catalog.utsa.edu/undergraduate/sciences/mathematics/#courseinventory MAT 3633. Numerical Analysis.] (3-0) 3 Credit Hours.

Prerequisites: [[MAT2233]], [[MAT3213]], and one of the following: [[CS1063]], [[CS1714]], or [[CS2073]]. Solution of linear and nonlinear equations, curve-fitting, and eigenvalue problems. Generally offered: Fall, Spring. Differential Tuition: $150.

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
| Week.1
||
0.2 and 1.1
||
* Loss of significant digits
* Bisection Method
* Brief introduction to matlab
||
* binary number system;
* Taylor's theorem;
* intermediate value theorem
||
* Nested multiplication for evaluating polynomials
* Machine representation of real numbers
* Loss of significant digits in numerical computing
* Review of Taylor's Theorem
* Bisection method and implementation

|-
|Week 1
|||-
| Week.2
||
1.2 and 1.3
||
* Fixed-Point Iteration
* Limits of Accuracy: Conditioning of problems
||
* limit of sequences
* multiplicity of solution of equations.
||
* Geometric interpretation
* Convergence of fixed point iterations
* Order of convergence of iterative methods
* Wilkinson polynomial and other examples
* Sensitivity analysis of root-finding
* Error magnification factor for solution of equations
|-
|Week 1
||
Section 1.1: Fixed-Point Iteration
||
* [[Bisection Method]]
||
* [[Intermediate Value Theorem]]
||
* Bisection Method and Implementation
* Brief Introduction to Matlab
|-
|Week 2
||
Section 1.2: Fixed-Point Iteration
||
* [[Fixed-Point Iteration]]
* [[Order of Convergence]] of Iterative Methods
||
* [[Limit of Sequences]]
* [[Solution Multiplicity of Equations]]
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
Section 1.3: Limits of Accuracy: Conditioning of Problems
||
* [[Wilkinson Polynomial]]
||
* [[Limit of Sequences]]
* [[Solution Multiplicity of Equations]]
||
* Sensitivity Analysis of Root-Finding
* Error Magnification Factor for Solution of Equations
|-
|Week 3
||
Section 1.4: Newton's Method
||
* [[Newton's Method]]
* [[Error Analysis]] for Newton's Method
* [[Modified Newton's Method]]

* [[Root-Finding Without Derivatives]]
* [[Secant Method and its Convergence]]
* [[Method of False Position, Muller's Method]]
* [[Stopping Criteria for Iterative Methods]]
||
* [[Remainder of Taylor's Series]]
* [[Intermediate Value Theorem]]
* [[Fixed-Point Iteration]]
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
Section 1.5 Root-Finding Without Derivatives
||
* [[Secant Method]]
* [[Method of False Position]]
* [[Muller's Method]]
* [[Stopping Criteria]]
||
* [[Remainder of Taylor's Series]]
* [[Intermediate Value Theorem]]
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 4
||
Section 2.1 Solve Systems of Linear Equations: Gaussian Elimination
||
* [[Gaussian Elimination]]
||
* [[Elementary Row Operations]]
||
* Gaussian Elimination and its Operation Counts
* Gaussian Elimination with Pivoting
* Implementation of Gauss Elimination
|-
|Week 4
||
Section 2.2 Solve Systems of Linear Equations: LU Decomposition
||
* [[LU Decomposition]]
||
* [[Matrix-Matrix Products]]
* [[Matrix-Vector Products]]
* [[Inverse Matrix]]
* [[Elementary Row Operations]]
||
* Matrices for Elementary Row Operations
* Gauss Elimination as Matrix Products
* Advantages of Solutions by LU Decomposition
|-
|Week 5
||
Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Norms]]
* [[Error Analysis]] for Solution of Ax=b
* [[Error Magnification]] Factor and Condition Number of Matrix
||
* [[Length of Vectors]]
* [[Eigenvalues of a Matrix]]
* [[Eigenvectors of a Matrix]]
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Iterative Methods]]
* [[Jacobi Method]]
* [[Gauss-Seidel Method]]
* [[Successive-Over-Relaxation (SOR) Method]]
* [[Convergence of Iterative Methods]]
* [[Spectral Radius of Matrix]]
* [[Sparse Matrix]]
||
* [[Length of Vectors]]
* [[Eigenvalues of a Matrix]]
* [[Eigenvectors of a Matrix]]
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 6
||
Section 2.6: Conjugate Gradient (CG) Method
||
* [[Conjugate Gradient Method]]
* [[Symmetric Positive Definite Matrix]]
* [[CG Method]]
||
* [[Scalar Product of Vectors]]
* [[Determinant of a Matrix]]
* [[Eigenvalues of a Matrix]]
* [[Quadratic Polynomials of n-variables]]
* [[Partial Derivatives]]
* [[Gradients]]
* [[Chain Rule for Partial Derivatives]]
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
Section 2.7: Nonlinear System of Equations
||
* [[Nonlinear System of Equations]]
* [[Taylor's Theorem for Multi-Variate Vector Valued Functions]]
* [[Newton's Method]]
* [[Broyden's Method]]
||
* [[Scalar Product of Vectors]]
* [[Determinant of a Matrix]]
* [[Eigenvalues of a Matrix]]
* [[Quadratic Polynomials of n-variables]]
* [[Partial Derivatives]]
* [[Gradients]]
* [[Chain Rule for Partial Derivatives]]
||
* (TBD)
|-
|Week 7
||
Sections 3.1: Data and Interpolating Functions
||
* [[Lagrange Basis Functions]]
* [[Newton's Divided Differences]]
* [[Properties of Newton's Divided Differences]]
||
* [[Fundamental Theorem of Algebra]]
* [[Rolle's Theorem]]
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error]]
* Interpolation [[Error Analysis]]
* [[Runge Phenomenon]]
* [[Chebyshev Polynomial]]
* [[Error Estimates]] for Chebyshev Interpolation
||
* [[Fundamental Theorem of Algebra]]
* [[Rolle's Theorem]]
||
* (TBD)
|-
|Week 8
||
Section 3.4: Cubic Splines
||
* [[Cubic Splines]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Construction of Cubic Splines for Interpolation
* End Conditions
* Properties of Cubic Spline Interpolation
|-
|Week 8
||
Section 3.5: Bezier Curves
||
* [[Bezier Curves]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Bezier Curve and Fonts
|-
|Week 8
||
Section 4.1: Least Square Method
||
* [[Least Square Method]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Least Square Method for Solving Inconsistent System of Linear Equations]
* Basic Properties of Least Square Solutions
|-
|Week 9
||
Section 4.2: Mathematical Models and Data Fitting
||
* [[Curve Fitting]]
* [[Statistical Modeling]]
||
* [[Linear Spaces]]
* [[Basis Functions]]
* [[Product Rule for Vector Valued Multivariate Functions]]
* [[Chain Rule for Vector Valued Multivariate Functions]]
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
Section 4.5: Nonlinear Least Square Fitting
||
* [[Taylor's Theorem for Vector Valued Multivariate Functions]]
* [[Gauss-Newton Method]]
* [[Levenberg-Marquardt Method]]
||
* [[Linear Spaces]]
* [[Basis Functions]]
* [[Product Rule for Vector Valued Multivariate Functions]]
* [[Chain Rule for Vector Valued Multivariate Functions]]
||
* (TBD)
|-
|Week 10
||
Section 5.1: Numerical Differentiation
||
* [[Numerical Differentiation]]
* [[Finite Difference]] (FD)
* [[Undetermined Coefficient Method]]
* [[Extrapolation Technique]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Newton-Cotes]]
* [[Midpoint rule]]
* [[Trapezoid rule]]
* [[Simpson's rule]]
* [[Error Analysis based on Interpolation Errors]]
* [[Quadrature Rules]]
* [[Composite Quadrature Rules]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
Section 5.3: Numerical Integration: Romberg's Technique
||
* [[Romberg's Technique]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Motivation, construction and implementation of Romberg's Technique.
|-
|Week 11
||
Section 5.4: Adaptive Numerical Integration
||
* [[Adaptive Numerical Integration]]
* [[Implementation of Adaptive Numerical Integration Techniques]]
||
* [[Long Divisions]]
* [[Substitution Methods]] for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Formulas]]
* [[Orthogonal Polynomials]]
* [[Legendre polynomials]]
* [[Gauss Quadrature Rule]]
||
* [[Long Divisions]]
* [[Substitution Methods]] for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 12
||
Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fourier Series]]
* [[Discrete Fourier Transform]] (DFT)
* [[Inverse Discrete Fourier Transform]]
* [[Fast Fourier Transform (FFT)]]
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform]]
* [[Discrete Cosine Transform]](DCT)
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
Section 11.2: Image Compression (optional)
||
* [[Quantization]]
* [[Image Compression]]
* [[Image Decompression]]
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 13
||
Section 12.1: Power Iteration Methods
||
* [[Power Iteration Methods]]
* [[Inverse Power Iteration]]
* [[Inverse Power Iteration with Shift]]
* [[Rayleigh Quotient Iteration]]
||
* [[Eigenvalues]]
* [[Eigenvectors]]
* [[Orthonormal Bases and the Gram-Schmidt Process]]
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Orthogonal Matrices]]
* [[QR-Factorization]]
* [[Normalized Simultaneous Iteration]](NSI)
* [[Unshifted QR Algorithm]]
* [[Shifted QR Algorithm]]
||
* [[Eigenvalues]]
* [[Eigenvectors]]
* [[Orthonormal Bases and the Gram-Schmidt Process]]
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 14
||
Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Upper Hessenberg Form]] (UHF)
* [[Householder Reflector]]
||
* [[Matrices for Orthogonal Projection]]
* [[Matrices for Reflection]]
* [[Block Matrices]]
* [[Similar Matrices]]
||
* Convert a matrix into UHF by Householder reflectors
|}

==Topics List B Wiki Format ==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
* Section 0.2: Loss of significant digits
||
* [[Loss of Significant Digits]]
||
* Binary Number System
* Taylor's Theorem
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
* Section 0.2: Loss of significant digits
||
* [[Nested Multiplication]]
||
* Binary Number System
* Taylor's Theorem
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
* Section 1.1: Fixed-Point Iteration
||
* [[Bisection Method]]
||
* Intermediate Value Theorem
||
* Bisection Method and Implementation
* Brief Introduction to Matlab
|-
|Week 2
||
* Section 1.2: Fixed-Point Iteration
||
* [[Fixed-Point Iteration]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
* Section 1.2: Fixed-Point Iteration
||
* [[Order of Convergence of Iterative Methods]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
* Section 1.3: Limits of Accuracy: Conditioning of Problems
||
* [[Wilkinson Polynomial]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Sensitivity Analysis of Root-Finding
* Error Magnification Factor for Solution of Equations
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Error Analysis for Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Modified Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Root-Finding Without Derivatives]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Secant Method and its Convergence]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Method of False Position, Muller's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Stopping Criteria for Iterative Methods]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Secant Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Method of False Position]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Muller's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Stopping Criteria]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 4
||
* Section 2.1 Solve Systems of Linear Equations: Gaussian Elimination
||
* [[Gaussian Elimination]]
||
* Elementary Row Operations
||
* Gaussian Elimination and its Operation Counts
* Gaussian Elimination with Pivoting
* Implementation of Gauss Elimination
|-
|Week 4
||
* Section 2.2 Solve Systems of Linear Equations: LU Decomposition
||
* [[LU Decomposition]]
||
* Matrix-Matrix Products
* Matrix-Vector Products
* Inverse Matrix
* Elementary Row Operations
||
* Matrices for Elementary Row Operations
* Gauss Elimination as Matrix Products
* Advantages of Solutions by LU Decomposition
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Norms]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Error Analysis for Solution of Ax=b]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Error Magnification Factor and Condition Number of Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Iterative Methods]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Jacobi Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Gauss-Seidel Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Successive-Over-Relaxation (SOR) Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Convergence of Iterative Methods]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Spectral Radius of Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Sparse Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[Conjugate Gradient Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[Symmetric Positive Definite Matrix]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[CG Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Nonlinear System of Equations]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Taylor's Theorem for Multi-Variate Vector Valued Functions]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Newton's Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Broyden's Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Lagrange Basis Functions]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Newton's Divided Differences]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Properties of Newton's Divided Differences]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error Analysis]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Runge Phenomenon]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Chebyshev Polynomial]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Error Estimates for Chebyshev Interpolation]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 8
||
* Section 3.4: Cubic Splines
||
* [[Cubic Splines]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Construction of Cubic Splines for Interpolation
* End Conditions
* Properties of Cubic Spline Interpolation
|-
|Week 8
||
* Section 3.5: Bezier Curves
||
* [[Bezier Curves]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Bezier Curve and Fonts
|-
|Week 8
||
* Section 4.1: Least Square Method
||
* [[Least Square Method]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Least Square Method for Solving Inconsistent System of Linear Equations]
* Basic Properties of Least Square Solutions
|-
|Week 9
||
* Section 4.2: Mathematical Models and Data Fitting
||
* [[Curve Fitting]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
* Section 4.2: Mathematical Models and Data Fitting
||
* [[Statistical Modeling]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Taylor's Theorem for Vector Valued Multivariate Functions]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Gauss-Newton Method]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Levenberg-Marquardt Method]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Numerical Differentiation]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Finite Difference (FD)]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Undetermined Coefficient Method]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Extrapolation Technique]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Newton-Cotes]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Midpoint rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Trapezoid rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Simpson's rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Error Analysis based on Interpolation Errors]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Quadrature Rules]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Composite Quadrature Rules]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.3: Numerical Integration: Romberg's Technique
||
* [[Romberg's Technique]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Motivation, construction and implementation of Romberg's Technique.
|-
|Week 11
||
* Section 5.4: Adaptive Numerical Integration
||
* [[Adaptive Numerical Integration]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
* Section 5.4: Adaptive Numerical Integration
||
* [[Implementation of Adaptive Numerical Integration Techniques]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Formulas]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Orthogonal Polynomials]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Legendre polynomials]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Rule]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fourier Series]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Discrete Fourier Transform (DFT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Inverse Discrete Fourier Transform]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fast Fourier Transform (FFT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
* Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform(DCT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Quantization]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Image Compression]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Image Decompression]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Power Iteration Methods]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Inverse Power Iteration]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Inverse Power Iteration with Shift]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Rayleigh Quotient Iteration]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Orthogonal Matrices]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[QR-Factorization]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Normalized Simultaneous Iteration(NSI)]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Unshifted QR Algorithm]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Shifted QR Algorithm]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 14
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Upper Hessenberg Form (UHF)]]
||
* Matrices for Orthogonal Projection
* Matrices for Reflection
* Block Matrices
* Similar Matrices
||
* Convert a matrix into UHF by Householder reflectors
|-
|Week 14
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Householder Reflector]]
||
* Matrices for Orthogonal Projection
* Matrices for Reflection
* Block Matrices
* Similar Matrices
||
* Convert a matrix into UHF by Householder reflectors
|}

MAT3633

2020-08-17T19:58:53Z

Weiming.cao: /* Topics List */

==Course Catalog==
[https://catalog.utsa.edu/undergraduate/sciences/mathematics/#courseinventory MAT 3633. Numerical Analysis.] (3-0) 3 Credit Hours.

Prerequisites: [[MAT2233]], [[MAT3213]], and one of the following: [[CS1063]], [[CS1714]], or [[CS2073]]. Solution of linear and nonlinear equations, curve-fitting, and eigenvalue problems. Generally offered: Fall, Spring. Differential Tuition: $150.

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
| Week.1
||
0.2 and 1.1
||
* Loss of significant digits
* Bisection Method
* Brief introduction to matlab
||
* binary number system;
* Taylor's theorem;
* intermediate value theorem
||
* Nested multiplication for evaluating polynomials
* Machine representation of real numbers
* Loss of significant digits in numerical computing
* Review of Taylor's Theorem
* Bisection method and implementation
|-
|Week 1
||
Section 0.2: Loss of significant digits
||
* [[Loss of Significant Digits]]
* [[Nested Multiplication]]
* [[my addition]]
||
* [[Binary Number System]]
* [[Taylor's Theorem]]
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
Section 1.1: Fixed-Point Iteration
||
* [[Bisection Method]]
||
* [[Intermediate Value Theorem]]
||
* Bisection Method and Implementation
* Brief Introduction to Matlab
|-
|Week 2
||
Section 1.2: Fixed-Point Iteration
||
* [[Fixed-Point Iteration]]
* [[Order of Convergence]] of Iterative Methods
||
* [[Limit of Sequences]]
* [[Solution Multiplicity of Equations]]
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
Section 1.3: Limits of Accuracy: Conditioning of Problems
||
* [[Wilkinson Polynomial]]
||
* [[Limit of Sequences]]
* [[Solution Multiplicity of Equations]]
||
* Sensitivity Analysis of Root-Finding
* Error Magnification Factor for Solution of Equations
|-
|Week 3
||
Section 1.4: Newton's Method
||
* [[Newton's Method]]
* [[Error Analysis]] for Newton's Method
* [[Modified Newton's Method]]

* [[Root-Finding Without Derivatives]]
* [[Secant Method and its Convergence]]
* [[Method of False Position, Muller's Method]]
* [[Stopping Criteria for Iterative Methods]]
||
* [[Remainder of Taylor's Series]]
* [[Intermediate Value Theorem]]
* [[Fixed-Point Iteration]]
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
Section 1.5 Root-Finding Without Derivatives
||
* [[Secant Method]]
* [[Method of False Position]]
* [[Muller's Method]]
* [[Stopping Criteria]]
||
* [[Remainder of Taylor's Series]]
* [[Intermediate Value Theorem]]
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 4
||
Section 2.1 Solve Systems of Linear Equations: Gaussian Elimination
||
* [[Gaussian Elimination]]
||
* [[Elementary Row Operations]]
||
* Gaussian Elimination and its Operation Counts
* Gaussian Elimination with Pivoting
* Implementation of Gauss Elimination
|-
|Week 4
||
Section 2.2 Solve Systems of Linear Equations: LU Decomposition
||
* [[LU Decomposition]]
||
* [[Matrix-Matrix Products]]
* [[Matrix-Vector Products]]
* [[Inverse Matrix]]
* [[Elementary Row Operations]]
||
* Matrices for Elementary Row Operations
* Gauss Elimination as Matrix Products
* Advantages of Solutions by LU Decomposition
|-
|Week 5
||
Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Norms]]
* [[Error Analysis]] for Solution of Ax=b
* [[Error Magnification]] Factor and Condition Number of Matrix
||
* [[Length of Vectors]]
* [[Eigenvalues of a Matrix]]
* [[Eigenvectors of a Matrix]]
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Iterative Methods]]
* [[Jacobi Method]]
* [[Gauss-Seidel Method]]
* [[Successive-Over-Relaxation (SOR) Method]]
* [[Convergence of Iterative Methods]]
* [[Spectral Radius of Matrix]]
* [[Sparse Matrix]]
||
* [[Length of Vectors]]
* [[Eigenvalues of a Matrix]]
* [[Eigenvectors of a Matrix]]
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 6
||
Section 2.6: Conjugate Gradient (CG) Method
||
* [[Conjugate Gradient Method]]
* [[Symmetric Positive Definite Matrix]]
* [[CG Method]]
||
* [[Scalar Product of Vectors]]
* [[Determinant of a Matrix]]
* [[Eigenvalues of a Matrix]]
* [[Quadratic Polynomials of n-variables]]
* [[Partial Derivatives]]
* [[Gradients]]
* [[Chain Rule for Partial Derivatives]]
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
Section 2.7: Nonlinear System of Equations
||
* [[Nonlinear System of Equations]]
* [[Taylor's Theorem for Multi-Variate Vector Valued Functions]]
* [[Newton's Method]]
* [[Broyden's Method]]
||
* [[Scalar Product of Vectors]]
* [[Determinant of a Matrix]]
* [[Eigenvalues of a Matrix]]
* [[Quadratic Polynomials of n-variables]]
* [[Partial Derivatives]]
* [[Gradients]]
* [[Chain Rule for Partial Derivatives]]
||
* (TBD)
|-
|Week 7
||
Sections 3.1: Data and Interpolating Functions
||
* [[Lagrange Basis Functions]]
* [[Newton's Divided Differences]]
* [[Properties of Newton's Divided Differences]]
||
* [[Fundamental Theorem of Algebra]]
* [[Rolle's Theorem]]
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error]]
* Interpolation [[Error Analysis]]
* [[Runge Phenomenon]]
* [[Chebyshev Polynomial]]
* [[Error Estimates]] for Chebyshev Interpolation
||
* [[Fundamental Theorem of Algebra]]
* [[Rolle's Theorem]]
||
* (TBD)
|-
|Week 8
||
Section 3.4: Cubic Splines
||
* [[Cubic Splines]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Construction of Cubic Splines for Interpolation
* End Conditions
* Properties of Cubic Spline Interpolation
|-
|Week 8
||
Section 3.5: Bezier Curves
||
* [[Bezier Curves]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Bezier Curve and Fonts
|-
|Week 8
||
Section 4.1: Least Square Method
||
* [[Least Square Method]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Least Square Method for Solving Inconsistent System of Linear Equations]
* Basic Properties of Least Square Solutions
|-
|Week 9
||
Section 4.2: Mathematical Models and Data Fitting
||
* [[Curve Fitting]]
* [[Statistical Modeling]]
||
* [[Linear Spaces]]
* [[Basis Functions]]
* [[Product Rule for Vector Valued Multivariate Functions]]
* [[Chain Rule for Vector Valued Multivariate Functions]]
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
Section 4.5: Nonlinear Least Square Fitting
||
* [[Taylor's Theorem for Vector Valued Multivariate Functions]]
* [[Gauss-Newton Method]]
* [[Levenberg-Marquardt Method]]
||
* [[Linear Spaces]]
* [[Basis Functions]]
* [[Product Rule for Vector Valued Multivariate Functions]]
* [[Chain Rule for Vector Valued Multivariate Functions]]
||
* (TBD)
|-
|Week 10
||
Section 5.1: Numerical Differentiation
||
* [[Numerical Differentiation]]
* [[Finite Difference]] (FD)
* [[Undetermined Coefficient Method]]
* [[Extrapolation Technique]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Newton-Cotes]]
* [[Midpoint rule]]
* [[Trapezoid rule]]
* [[Simpson's rule]]
* [[Error Analysis based on Interpolation Errors]]
* [[Quadrature Rules]]
* [[Composite Quadrature Rules]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
Section 5.3: Numerical Integration: Romberg's Technique
||
* [[Romberg's Technique]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Motivation, construction and implementation of Romberg's Technique.
|-
|Week 11
||
Section 5.4: Adaptive Numerical Integration
||
* [[Adaptive Numerical Integration]]
* [[Implementation of Adaptive Numerical Integration Techniques]]
||
* [[Long Divisions]]
* [[Substitution Methods]] for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Formulas]]
* [[Orthogonal Polynomials]]
* [[Legendre polynomials]]
* [[Gauss Quadrature Rule]]
||
* [[Long Divisions]]
* [[Substitution Methods]] for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 12
||
Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fourier Series]]
* [[Discrete Fourier Transform]] (DFT)
* [[Inverse Discrete Fourier Transform]]
* [[Fast Fourier Transform (FFT)]]
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform]]
* [[Discrete Cosine Transform]](DCT)
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
Section 11.2: Image Compression (optional)
||
* [[Quantization]]
* [[Image Compression]]
* [[Image Decompression]]
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 13
||
Section 12.1: Power Iteration Methods
||
* [[Power Iteration Methods]]
* [[Inverse Power Iteration]]
* [[Inverse Power Iteration with Shift]]
* [[Rayleigh Quotient Iteration]]
||
* [[Eigenvalues]]
* [[Eigenvectors]]
* [[Orthonormal Bases and the Gram-Schmidt Process]]
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Orthogonal Matrices]]
* [[QR-Factorization]]
* [[Normalized Simultaneous Iteration]](NSI)
* [[Unshifted QR Algorithm]]
* [[Shifted QR Algorithm]]
||
* [[Eigenvalues]]
* [[Eigenvectors]]
* [[Orthonormal Bases and the Gram-Schmidt Process]]
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 14
||
Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Upper Hessenberg Form]] (UHF)
* [[Householder Reflector]]
||
* [[Matrices for Orthogonal Projection]]
* [[Matrices for Reflection]]
* [[Block Matrices]]
* [[Similar Matrices]]
||
* Convert a matrix into UHF by Householder reflectors
|}

==Topics List B Wiki Format ==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
* Section 0.2: Loss of significant digits
||
* [[Loss of Significant Digits]]
||
* Binary Number System
* Taylor's Theorem
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
* Section 0.2: Loss of significant digits
||
* [[Nested Multiplication]]
||
* Binary Number System
* Taylor's Theorem
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
* Section 1.1: Fixed-Point Iteration
||
* [[Bisection Method]]
||
* Intermediate Value Theorem
||
* Bisection Method and Implementation
* Brief Introduction to Matlab
|-
|Week 2
||
* Section 1.2: Fixed-Point Iteration
||
* [[Fixed-Point Iteration]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
* Section 1.2: Fixed-Point Iteration
||
* [[Order of Convergence of Iterative Methods]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
* Section 1.3: Limits of Accuracy: Conditioning of Problems
||
* [[Wilkinson Polynomial]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Sensitivity Analysis of Root-Finding
* Error Magnification Factor for Solution of Equations
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Error Analysis for Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Modified Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Root-Finding Without Derivatives]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Secant Method and its Convergence]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Method of False Position, Muller's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Stopping Criteria for Iterative Methods]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Secant Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Method of False Position]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Muller's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Stopping Criteria]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 4
||
* Section 2.1 Solve Systems of Linear Equations: Gaussian Elimination
||
* [[Gaussian Elimination]]
||
* Elementary Row Operations
||
* Gaussian Elimination and its Operation Counts
* Gaussian Elimination with Pivoting
* Implementation of Gauss Elimination
|-
|Week 4
||
* Section 2.2 Solve Systems of Linear Equations: LU Decomposition
||
* [[LU Decomposition]]
||
* Matrix-Matrix Products
* Matrix-Vector Products
* Inverse Matrix
* Elementary Row Operations
||
* Matrices for Elementary Row Operations
* Gauss Elimination as Matrix Products
* Advantages of Solutions by LU Decomposition
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Norms]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Error Analysis for Solution of Ax=b]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Error Magnification Factor and Condition Number of Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Iterative Methods]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Jacobi Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Gauss-Seidel Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Successive-Over-Relaxation (SOR) Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Convergence of Iterative Methods]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Spectral Radius of Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Sparse Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[Conjugate Gradient Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[Symmetric Positive Definite Matrix]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[CG Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Nonlinear System of Equations]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Taylor's Theorem for Multi-Variate Vector Valued Functions]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Newton's Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Broyden's Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Lagrange Basis Functions]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Newton's Divided Differences]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Properties of Newton's Divided Differences]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error Analysis]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Runge Phenomenon]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Chebyshev Polynomial]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Error Estimates for Chebyshev Interpolation]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 8
||
* Section 3.4: Cubic Splines
||
* [[Cubic Splines]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Construction of Cubic Splines for Interpolation
* End Conditions
* Properties of Cubic Spline Interpolation
|-
|Week 8
||
* Section 3.5: Bezier Curves
||
* [[Bezier Curves]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Bezier Curve and Fonts
|-
|Week 8
||
* Section 4.1: Least Square Method
||
* [[Least Square Method]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Least Square Method for Solving Inconsistent System of Linear Equations]
* Basic Properties of Least Square Solutions
|-
|Week 9
||
* Section 4.2: Mathematical Models and Data Fitting
||
* [[Curve Fitting]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
* Section 4.2: Mathematical Models and Data Fitting
||
* [[Statistical Modeling]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Taylor's Theorem for Vector Valued Multivariate Functions]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Gauss-Newton Method]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Levenberg-Marquardt Method]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Numerical Differentiation]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Finite Difference (FD)]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Undetermined Coefficient Method]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Extrapolation Technique]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Newton-Cotes]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Midpoint rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Trapezoid rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Simpson's rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Error Analysis based on Interpolation Errors]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Quadrature Rules]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Composite Quadrature Rules]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.3: Numerical Integration: Romberg's Technique
||
* [[Romberg's Technique]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Motivation, construction and implementation of Romberg's Technique.
|-
|Week 11
||
* Section 5.4: Adaptive Numerical Integration
||
* [[Adaptive Numerical Integration]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
* Section 5.4: Adaptive Numerical Integration
||
* [[Implementation of Adaptive Numerical Integration Techniques]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Formulas]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Orthogonal Polynomials]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Legendre polynomials]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Rule]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fourier Series]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Discrete Fourier Transform (DFT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Inverse Discrete Fourier Transform]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fast Fourier Transform (FFT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
* Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform(DCT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Quantization]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Image Compression]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Image Decompression]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Power Iteration Methods]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Inverse Power Iteration]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Inverse Power Iteration with Shift]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Rayleigh Quotient Iteration]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Orthogonal Matrices]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[QR-Factorization]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Normalized Simultaneous Iteration(NSI)]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Unshifted QR Algorithm]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Shifted QR Algorithm]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 14
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Upper Hessenberg Form (UHF)]]
||
* Matrices for Orthogonal Projection
* Matrices for Reflection
* Block Matrices
* Similar Matrices
||
* Convert a matrix into UHF by Householder reflectors
|-
|Week 14
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Householder Reflector]]
||
* Matrices for Orthogonal Projection
* Matrices for Reflection
* Block Matrices
* Similar Matrices
||
* Convert a matrix into UHF by Householder reflectors
|}

MAT3633

2020-08-17T19:57:59Z

Weiming.cao: /* Topics List */

==Course Catalog==
[https://catalog.utsa.edu/undergraduate/sciences/mathematics/#courseinventory MAT 3633. Numerical Analysis.] (3-0) 3 Credit Hours.

Prerequisites: [[MAT2233]], [[MAT3213]], and one of the following: [[CS1063]], [[CS1714]], or [[CS2073]]. Solution of linear and nonlinear equations, curve-fitting, and eigenvalue problems. Generally offered: Fall, Spring. Differential Tuition: $150.

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
| Week 1
||
0.2 and 1.1
||
* Loss of significant digits
* Bisection Method
* Brief introduction to matlab
||
* binary number system;
* Taylor's theorem;
* intermediate value theorem
||
* Nested multiplication for evaluating polynomials
* Machine representation of real numbers
* Loss of significant digits in numerical computing
* Review of Taylor's Theorem
* Bisection method and implementation
|-
|Week 1
||
Section 0.2: Loss of significant digits
||
* [[Loss of Significant Digits]]
* [[Nested Multiplication]]
* [[my addition]]
||
* [[Binary Number System]]
* [[Taylor's Theorem]]
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
Section 1.1: Fixed-Point Iteration
||
* [[Bisection Method]]
||
* [[Intermediate Value Theorem]]
||
* Bisection Method and Implementation
* Brief Introduction to Matlab
|-
|Week 2
||
Section 1.2: Fixed-Point Iteration
||
* [[Fixed-Point Iteration]]
* [[Order of Convergence]] of Iterative Methods
||
* [[Limit of Sequences]]
* [[Solution Multiplicity of Equations]]
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
Section 1.3: Limits of Accuracy: Conditioning of Problems
||
* [[Wilkinson Polynomial]]
||
* [[Limit of Sequences]]
* [[Solution Multiplicity of Equations]]
||
* Sensitivity Analysis of Root-Finding
* Error Magnification Factor for Solution of Equations
|-
|Week 3
||
Section 1.4: Newton's Method
||
* [[Newton's Method]]
* [[Error Analysis]] for Newton's Method
* [[Modified Newton's Method]]

* [[Root-Finding Without Derivatives]]
* [[Secant Method and its Convergence]]
* [[Method of False Position, Muller's Method]]
* [[Stopping Criteria for Iterative Methods]]
||
* [[Remainder of Taylor's Series]]
* [[Intermediate Value Theorem]]
* [[Fixed-Point Iteration]]
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
Section 1.5 Root-Finding Without Derivatives
||
* [[Secant Method]]
* [[Method of False Position]]
* [[Muller's Method]]
* [[Stopping Criteria]]
||
* [[Remainder of Taylor's Series]]
* [[Intermediate Value Theorem]]
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 4
||
Section 2.1 Solve Systems of Linear Equations: Gaussian Elimination
||
* [[Gaussian Elimination]]
||
* [[Elementary Row Operations]]
||
* Gaussian Elimination and its Operation Counts
* Gaussian Elimination with Pivoting
* Implementation of Gauss Elimination
|-
|Week 4
||
Section 2.2 Solve Systems of Linear Equations: LU Decomposition
||
* [[LU Decomposition]]
||
* [[Matrix-Matrix Products]]
* [[Matrix-Vector Products]]
* [[Inverse Matrix]]
* [[Elementary Row Operations]]
||
* Matrices for Elementary Row Operations
* Gauss Elimination as Matrix Products
* Advantages of Solutions by LU Decomposition
|-
|Week 5
||
Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Norms]]
* [[Error Analysis]] for Solution of Ax=b
* [[Error Magnification]] Factor and Condition Number of Matrix
||
* [[Length of Vectors]]
* [[Eigenvalues of a Matrix]]
* [[Eigenvectors of a Matrix]]
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Iterative Methods]]
* [[Jacobi Method]]
* [[Gauss-Seidel Method]]
* [[Successive-Over-Relaxation (SOR) Method]]
* [[Convergence of Iterative Methods]]
* [[Spectral Radius of Matrix]]
* [[Sparse Matrix]]
||
* [[Length of Vectors]]
* [[Eigenvalues of a Matrix]]
* [[Eigenvectors of a Matrix]]
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 6
||
Section 2.6: Conjugate Gradient (CG) Method
||
* [[Conjugate Gradient Method]]
* [[Symmetric Positive Definite Matrix]]
* [[CG Method]]
||
* [[Scalar Product of Vectors]]
* [[Determinant of a Matrix]]
* [[Eigenvalues of a Matrix]]
* [[Quadratic Polynomials of n-variables]]
* [[Partial Derivatives]]
* [[Gradients]]
* [[Chain Rule for Partial Derivatives]]
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
Section 2.7: Nonlinear System of Equations
||
* [[Nonlinear System of Equations]]
* [[Taylor's Theorem for Multi-Variate Vector Valued Functions]]
* [[Newton's Method]]
* [[Broyden's Method]]
||
* [[Scalar Product of Vectors]]
* [[Determinant of a Matrix]]
* [[Eigenvalues of a Matrix]]
* [[Quadratic Polynomials of n-variables]]
* [[Partial Derivatives]]
* [[Gradients]]
* [[Chain Rule for Partial Derivatives]]
||
* (TBD)
|-
|Week 7
||
Sections 3.1: Data and Interpolating Functions
||
* [[Lagrange Basis Functions]]
* [[Newton's Divided Differences]]
* [[Properties of Newton's Divided Differences]]
||
* [[Fundamental Theorem of Algebra]]
* [[Rolle's Theorem]]
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error]]
* Interpolation [[Error Analysis]]
* [[Runge Phenomenon]]
* [[Chebyshev Polynomial]]
* [[Error Estimates]] for Chebyshev Interpolation
||
* [[Fundamental Theorem of Algebra]]
* [[Rolle's Theorem]]
||
* (TBD)
|-
|Week 8
||
Section 3.4: Cubic Splines
||
* [[Cubic Splines]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Construction of Cubic Splines for Interpolation
* End Conditions
* Properties of Cubic Spline Interpolation
|-
|Week 8
||
Section 3.5: Bezier Curves
||
* [[Bezier Curves]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Bezier Curve and Fonts
|-
|Week 8
||
Section 4.1: Least Square Method
||
* [[Least Square Method]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Least Square Method for Solving Inconsistent System of Linear Equations]
* Basic Properties of Least Square Solutions
|-
|Week 9
||
Section 4.2: Mathematical Models and Data Fitting
||
* [[Curve Fitting]]
* [[Statistical Modeling]]
||
* [[Linear Spaces]]
* [[Basis Functions]]
* [[Product Rule for Vector Valued Multivariate Functions]]
* [[Chain Rule for Vector Valued Multivariate Functions]]
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
Section 4.5: Nonlinear Least Square Fitting
||
* [[Taylor's Theorem for Vector Valued Multivariate Functions]]
* [[Gauss-Newton Method]]
* [[Levenberg-Marquardt Method]]
||
* [[Linear Spaces]]
* [[Basis Functions]]
* [[Product Rule for Vector Valued Multivariate Functions]]
* [[Chain Rule for Vector Valued Multivariate Functions]]
||
* (TBD)
|-
|Week 10
||
Section 5.1: Numerical Differentiation
||
* [[Numerical Differentiation]]
* [[Finite Difference]] (FD)
* [[Undetermined Coefficient Method]]
* [[Extrapolation Technique]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Newton-Cotes]]
* [[Midpoint rule]]
* [[Trapezoid rule]]
* [[Simpson's rule]]
* [[Error Analysis based on Interpolation Errors]]
* [[Quadrature Rules]]
* [[Composite Quadrature Rules]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
Section 5.3: Numerical Integration: Romberg's Technique
||
* [[Romberg's Technique]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Motivation, construction and implementation of Romberg's Technique.
|-
|Week 11
||
Section 5.4: Adaptive Numerical Integration
||
* [[Adaptive Numerical Integration]]
* [[Implementation of Adaptive Numerical Integration Techniques]]
||
* [[Long Divisions]]
* [[Substitution Methods]] for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Formulas]]
* [[Orthogonal Polynomials]]
* [[Legendre polynomials]]
* [[Gauss Quadrature Rule]]
||
* [[Long Divisions]]
* [[Substitution Methods]] for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 12
||
Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fourier Series]]
* [[Discrete Fourier Transform]] (DFT)
* [[Inverse Discrete Fourier Transform]]
* [[Fast Fourier Transform (FFT)]]
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform]]
* [[Discrete Cosine Transform]](DCT)
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
Section 11.2: Image Compression (optional)
||
* [[Quantization]]
* [[Image Compression]]
* [[Image Decompression]]
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 13
||
Section 12.1: Power Iteration Methods
||
* [[Power Iteration Methods]]
* [[Inverse Power Iteration]]
* [[Inverse Power Iteration with Shift]]
* [[Rayleigh Quotient Iteration]]
||
* [[Eigenvalues]]
* [[Eigenvectors]]
* [[Orthonormal Bases and the Gram-Schmidt Process]]
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Orthogonal Matrices]]
* [[QR-Factorization]]
* [[Normalized Simultaneous Iteration]](NSI)
* [[Unshifted QR Algorithm]]
* [[Shifted QR Algorithm]]
||
* [[Eigenvalues]]
* [[Eigenvectors]]
* [[Orthonormal Bases and the Gram-Schmidt Process]]
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 14
||
Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Upper Hessenberg Form]] (UHF)
* [[Householder Reflector]]
||
* [[Matrices for Orthogonal Projection]]
* [[Matrices for Reflection]]
* [[Block Matrices]]
* [[Similar Matrices]]
||
* Convert a matrix into UHF by Householder reflectors
|}

==Topics List B Wiki Format ==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
* Section 0.2: Loss of significant digits
||
* [[Loss of Significant Digits]]
||
* Binary Number System
* Taylor's Theorem
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
* Section 0.2: Loss of significant digits
||
* [[Nested Multiplication]]
||
* Binary Number System
* Taylor's Theorem
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
* Section 1.1: Fixed-Point Iteration
||
* [[Bisection Method]]
||
* Intermediate Value Theorem
||
* Bisection Method and Implementation
* Brief Introduction to Matlab
|-
|Week 2
||
* Section 1.2: Fixed-Point Iteration
||
* [[Fixed-Point Iteration]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
* Section 1.2: Fixed-Point Iteration
||
* [[Order of Convergence of Iterative Methods]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
* Section 1.3: Limits of Accuracy: Conditioning of Problems
||
* [[Wilkinson Polynomial]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Sensitivity Analysis of Root-Finding
* Error Magnification Factor for Solution of Equations
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Error Analysis for Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Modified Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Root-Finding Without Derivatives]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Secant Method and its Convergence]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Method of False Position, Muller's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Stopping Criteria for Iterative Methods]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Secant Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Method of False Position]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Muller's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Stopping Criteria]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 4
||
* Section 2.1 Solve Systems of Linear Equations: Gaussian Elimination
||
* [[Gaussian Elimination]]
||
* Elementary Row Operations
||
* Gaussian Elimination and its Operation Counts
* Gaussian Elimination with Pivoting
* Implementation of Gauss Elimination
|-
|Week 4
||
* Section 2.2 Solve Systems of Linear Equations: LU Decomposition
||
* [[LU Decomposition]]
||
* Matrix-Matrix Products
* Matrix-Vector Products
* Inverse Matrix
* Elementary Row Operations
||
* Matrices for Elementary Row Operations
* Gauss Elimination as Matrix Products
* Advantages of Solutions by LU Decomposition
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Norms]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Error Analysis for Solution of Ax=b]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Error Magnification Factor and Condition Number of Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Iterative Methods]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Jacobi Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Gauss-Seidel Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Successive-Over-Relaxation (SOR) Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Convergence of Iterative Methods]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Spectral Radius of Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Sparse Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[Conjugate Gradient Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[Symmetric Positive Definite Matrix]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[CG Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Nonlinear System of Equations]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Taylor's Theorem for Multi-Variate Vector Valued Functions]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Newton's Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Broyden's Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Lagrange Basis Functions]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Newton's Divided Differences]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Properties of Newton's Divided Differences]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error Analysis]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Runge Phenomenon]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Chebyshev Polynomial]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Error Estimates for Chebyshev Interpolation]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 8
||
* Section 3.4: Cubic Splines
||
* [[Cubic Splines]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Construction of Cubic Splines for Interpolation
* End Conditions
* Properties of Cubic Spline Interpolation
|-
|Week 8
||
* Section 3.5: Bezier Curves
||
* [[Bezier Curves]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Bezier Curve and Fonts
|-
|Week 8
||
* Section 4.1: Least Square Method
||
* [[Least Square Method]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Least Square Method for Solving Inconsistent System of Linear Equations]
* Basic Properties of Least Square Solutions
|-
|Week 9
||
* Section 4.2: Mathematical Models and Data Fitting
||
* [[Curve Fitting]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
* Section 4.2: Mathematical Models and Data Fitting
||
* [[Statistical Modeling]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Taylor's Theorem for Vector Valued Multivariate Functions]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Gauss-Newton Method]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Levenberg-Marquardt Method]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Numerical Differentiation]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Finite Difference (FD)]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Undetermined Coefficient Method]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Extrapolation Technique]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Newton-Cotes]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Midpoint rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Trapezoid rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Simpson's rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Error Analysis based on Interpolation Errors]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Quadrature Rules]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Composite Quadrature Rules]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.3: Numerical Integration: Romberg's Technique
||
* [[Romberg's Technique]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Motivation, construction and implementation of Romberg's Technique.
|-
|Week 11
||
* Section 5.4: Adaptive Numerical Integration
||
* [[Adaptive Numerical Integration]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
* Section 5.4: Adaptive Numerical Integration
||
* [[Implementation of Adaptive Numerical Integration Techniques]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Formulas]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Orthogonal Polynomials]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Legendre polynomials]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Rule]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fourier Series]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Discrete Fourier Transform (DFT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Inverse Discrete Fourier Transform]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fast Fourier Transform (FFT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
* Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform(DCT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Quantization]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Image Compression]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Image Decompression]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Power Iteration Methods]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Inverse Power Iteration]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Inverse Power Iteration with Shift]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Rayleigh Quotient Iteration]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Orthogonal Matrices]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[QR-Factorization]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Normalized Simultaneous Iteration(NSI)]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Unshifted QR Algorithm]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Shifted QR Algorithm]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 14
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Upper Hessenberg Form (UHF)]]
||
* Matrices for Orthogonal Projection
* Matrices for Reflection
* Block Matrices
* Similar Matrices
||
* Convert a matrix into UHF by Householder reflectors
|-
|Week 14
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Householder Reflector]]
||
* Matrices for Orthogonal Projection
* Matrices for Reflection
* Block Matrices
* Similar Matrices
||
* Convert a matrix into UHF by Householder reflectors
|}

MAT3633

2020-08-17T19:56:21Z

Weiming.cao: /* Topics List */

==Course Catalog==
[https://catalog.utsa.edu/undergraduate/sciences/mathematics/#courseinventory MAT 3633. Numerical Analysis.] (3-0) 3 Credit Hours.

Prerequisites: [[MAT2233]], [[MAT3213]], and one of the following: [[CS1063]], [[CS1714]], or [[CS2073]]. Solution of linear and nonlinear equations, curve-fitting, and eigenvalue problems. Generally offered: Fall, Spring. Differential Tuition: $150.

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
Week 1
||
0.2 and 1.1
||
Sec.0.2:
||
* Loss of significant digits
* Bisection Method
* Brief introduction to matlab
||
* binary number system;
* Taylor's theorem;
* intermediate value theorem
||
* Nested multiplication for evaluating polynomials
* Machine representation of real numbers
* Loss of significant digits in numerical computing
* Review of Taylor's Theorem
* Bisection method and implementation
|-
|Week 1
||
Section 0.2: Loss of significant digits
||
* [[Loss of Significant Digits]]
* [[Nested Multiplication]]
* [[my addition]]
||
* [[Binary Number System]]
* [[Taylor's Theorem]]
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
Section 1.1: Fixed-Point Iteration
||
* [[Bisection Method]]
||
* [[Intermediate Value Theorem]]
||
* Bisection Method and Implementation
* Brief Introduction to Matlab
|-
|Week 2
||
Section 1.2: Fixed-Point Iteration
||
* [[Fixed-Point Iteration]]
* [[Order of Convergence]] of Iterative Methods
||
* [[Limit of Sequences]]
* [[Solution Multiplicity of Equations]]
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
Section 1.3: Limits of Accuracy: Conditioning of Problems
||
* [[Wilkinson Polynomial]]
||
* [[Limit of Sequences]]
* [[Solution Multiplicity of Equations]]
||
* Sensitivity Analysis of Root-Finding
* Error Magnification Factor for Solution of Equations
|-
|Week 3
||
Section 1.4: Newton's Method
||
* [[Newton's Method]]
* [[Error Analysis]] for Newton's Method
* [[Modified Newton's Method]]

* [[Root-Finding Without Derivatives]]
* [[Secant Method and its Convergence]]
* [[Method of False Position, Muller's Method]]
* [[Stopping Criteria for Iterative Methods]]
||
* [[Remainder of Taylor's Series]]
* [[Intermediate Value Theorem]]
* [[Fixed-Point Iteration]]
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
Section 1.5 Root-Finding Without Derivatives
||
* [[Secant Method]]
* [[Method of False Position]]
* [[Muller's Method]]
* [[Stopping Criteria]]
||
* [[Remainder of Taylor's Series]]
* [[Intermediate Value Theorem]]
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 4
||
Section 2.1 Solve Systems of Linear Equations: Gaussian Elimination
||
* [[Gaussian Elimination]]
||
* [[Elementary Row Operations]]
||
* Gaussian Elimination and its Operation Counts
* Gaussian Elimination with Pivoting
* Implementation of Gauss Elimination
|-
|Week 4
||
Section 2.2 Solve Systems of Linear Equations: LU Decomposition
||
* [[LU Decomposition]]
||
* [[Matrix-Matrix Products]]
* [[Matrix-Vector Products]]
* [[Inverse Matrix]]
* [[Elementary Row Operations]]
||
* Matrices for Elementary Row Operations
* Gauss Elimination as Matrix Products
* Advantages of Solutions by LU Decomposition
|-
|Week 5
||
Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Norms]]
* [[Error Analysis]] for Solution of Ax=b
* [[Error Magnification]] Factor and Condition Number of Matrix
||
* [[Length of Vectors]]
* [[Eigenvalues of a Matrix]]
* [[Eigenvectors of a Matrix]]
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Iterative Methods]]
* [[Jacobi Method]]
* [[Gauss-Seidel Method]]
* [[Successive-Over-Relaxation (SOR) Method]]
* [[Convergence of Iterative Methods]]
* [[Spectral Radius of Matrix]]
* [[Sparse Matrix]]
||
* [[Length of Vectors]]
* [[Eigenvalues of a Matrix]]
* [[Eigenvectors of a Matrix]]
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 6
||
Section 2.6: Conjugate Gradient (CG) Method
||
* [[Conjugate Gradient Method]]
* [[Symmetric Positive Definite Matrix]]
* [[CG Method]]
||
* [[Scalar Product of Vectors]]
* [[Determinant of a Matrix]]
* [[Eigenvalues of a Matrix]]
* [[Quadratic Polynomials of n-variables]]
* [[Partial Derivatives]]
* [[Gradients]]
* [[Chain Rule for Partial Derivatives]]
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
Section 2.7: Nonlinear System of Equations
||
* [[Nonlinear System of Equations]]
* [[Taylor's Theorem for Multi-Variate Vector Valued Functions]]
* [[Newton's Method]]
* [[Broyden's Method]]
||
* [[Scalar Product of Vectors]]
* [[Determinant of a Matrix]]
* [[Eigenvalues of a Matrix]]
* [[Quadratic Polynomials of n-variables]]
* [[Partial Derivatives]]
* [[Gradients]]
* [[Chain Rule for Partial Derivatives]]
||
* (TBD)
|-
|Week 7
||
Sections 3.1: Data and Interpolating Functions
||
* [[Lagrange Basis Functions]]
* [[Newton's Divided Differences]]
* [[Properties of Newton's Divided Differences]]
||
* [[Fundamental Theorem of Algebra]]
* [[Rolle's Theorem]]
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error]]
* Interpolation [[Error Analysis]]
* [[Runge Phenomenon]]
* [[Chebyshev Polynomial]]
* [[Error Estimates]] for Chebyshev Interpolation
||
* [[Fundamental Theorem of Algebra]]
* [[Rolle's Theorem]]
||
* (TBD)
|-
|Week 8
||
Section 3.4: Cubic Splines
||
* [[Cubic Splines]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Construction of Cubic Splines for Interpolation
* End Conditions
* Properties of Cubic Spline Interpolation
|-
|Week 8
||
Section 3.5: Bezier Curves
||
* [[Bezier Curves]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Bezier Curve and Fonts
|-
|Week 8
||
Section 4.1: Least Square Method
||
* [[Least Square Method]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Least Square Method for Solving Inconsistent System of Linear Equations]
* Basic Properties of Least Square Solutions
|-
|Week 9
||
Section 4.2: Mathematical Models and Data Fitting
||
* [[Curve Fitting]]
* [[Statistical Modeling]]
||
* [[Linear Spaces]]
* [[Basis Functions]]
* [[Product Rule for Vector Valued Multivariate Functions]]
* [[Chain Rule for Vector Valued Multivariate Functions]]
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
Section 4.5: Nonlinear Least Square Fitting
||
* [[Taylor's Theorem for Vector Valued Multivariate Functions]]
* [[Gauss-Newton Method]]
* [[Levenberg-Marquardt Method]]
||
* [[Linear Spaces]]
* [[Basis Functions]]
* [[Product Rule for Vector Valued Multivariate Functions]]
* [[Chain Rule for Vector Valued Multivariate Functions]]
||
* (TBD)
|-
|Week 10
||
Section 5.1: Numerical Differentiation
||
* [[Numerical Differentiation]]
* [[Finite Difference]] (FD)
* [[Undetermined Coefficient Method]]
* [[Extrapolation Technique]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Newton-Cotes]]
* [[Midpoint rule]]
* [[Trapezoid rule]]
* [[Simpson's rule]]
* [[Error Analysis based on Interpolation Errors]]
* [[Quadrature Rules]]
* [[Composite Quadrature Rules]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
Section 5.3: Numerical Integration: Romberg's Technique
||
* [[Romberg's Technique]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Motivation, construction and implementation of Romberg's Technique.
|-
|Week 11
||
Section 5.4: Adaptive Numerical Integration
||
* [[Adaptive Numerical Integration]]
* [[Implementation of Adaptive Numerical Integration Techniques]]
||
* [[Long Divisions]]
* [[Substitution Methods]] for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Formulas]]
* [[Orthogonal Polynomials]]
* [[Legendre polynomials]]
* [[Gauss Quadrature Rule]]
||
* [[Long Divisions]]
* [[Substitution Methods]] for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 12
||
Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fourier Series]]
* [[Discrete Fourier Transform]] (DFT)
* [[Inverse Discrete Fourier Transform]]
* [[Fast Fourier Transform (FFT)]]
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform]]
* [[Discrete Cosine Transform]](DCT)
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
Section 11.2: Image Compression (optional)
||
* [[Quantization]]
* [[Image Compression]]
* [[Image Decompression]]
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 13
||
Section 12.1: Power Iteration Methods
||
* [[Power Iteration Methods]]
* [[Inverse Power Iteration]]
* [[Inverse Power Iteration with Shift]]
* [[Rayleigh Quotient Iteration]]
||
* [[Eigenvalues]]
* [[Eigenvectors]]
* [[Orthonormal Bases and the Gram-Schmidt Process]]
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Orthogonal Matrices]]
* [[QR-Factorization]]
* [[Normalized Simultaneous Iteration]](NSI)
* [[Unshifted QR Algorithm]]
* [[Shifted QR Algorithm]]
||
* [[Eigenvalues]]
* [[Eigenvectors]]
* [[Orthonormal Bases and the Gram-Schmidt Process]]
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 14
||
Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Upper Hessenberg Form]] (UHF)
* [[Householder Reflector]]
||
* [[Matrices for Orthogonal Projection]]
* [[Matrices for Reflection]]
* [[Block Matrices]]
* [[Similar Matrices]]
||
* Convert a matrix into UHF by Householder reflectors
|}

==Topics List B Wiki Format ==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
* Section 0.2: Loss of significant digits
||
* [[Loss of Significant Digits]]
||
* Binary Number System
* Taylor's Theorem
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
* Section 0.2: Loss of significant digits
||
* [[Nested Multiplication]]
||
* Binary Number System
* Taylor's Theorem
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
* Section 1.1: Fixed-Point Iteration
||
* [[Bisection Method]]
||
* Intermediate Value Theorem
||
* Bisection Method and Implementation
* Brief Introduction to Matlab
|-
|Week 2
||
* Section 1.2: Fixed-Point Iteration
||
* [[Fixed-Point Iteration]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
* Section 1.2: Fixed-Point Iteration
||
* [[Order of Convergence of Iterative Methods]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
* Section 1.3: Limits of Accuracy: Conditioning of Problems
||
* [[Wilkinson Polynomial]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Sensitivity Analysis of Root-Finding
* Error Magnification Factor for Solution of Equations
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Error Analysis for Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Modified Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Root-Finding Without Derivatives]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Secant Method and its Convergence]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Method of False Position, Muller's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Stopping Criteria for Iterative Methods]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Secant Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Method of False Position]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Muller's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Stopping Criteria]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 4
||
* Section 2.1 Solve Systems of Linear Equations: Gaussian Elimination
||
* [[Gaussian Elimination]]
||
* Elementary Row Operations
||
* Gaussian Elimination and its Operation Counts
* Gaussian Elimination with Pivoting
* Implementation of Gauss Elimination
|-
|Week 4
||
* Section 2.2 Solve Systems of Linear Equations: LU Decomposition
||
* [[LU Decomposition]]
||
* Matrix-Matrix Products
* Matrix-Vector Products
* Inverse Matrix
* Elementary Row Operations
||
* Matrices for Elementary Row Operations
* Gauss Elimination as Matrix Products
* Advantages of Solutions by LU Decomposition
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Norms]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Error Analysis for Solution of Ax=b]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Error Magnification Factor and Condition Number of Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Iterative Methods]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Jacobi Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Gauss-Seidel Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Successive-Over-Relaxation (SOR) Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Convergence of Iterative Methods]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Spectral Radius of Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Sparse Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[Conjugate Gradient Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[Symmetric Positive Definite Matrix]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[CG Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Nonlinear System of Equations]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Taylor's Theorem for Multi-Variate Vector Valued Functions]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Newton's Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Broyden's Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Lagrange Basis Functions]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Newton's Divided Differences]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Properties of Newton's Divided Differences]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error Analysis]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Runge Phenomenon]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Chebyshev Polynomial]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Error Estimates for Chebyshev Interpolation]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 8
||
* Section 3.4: Cubic Splines
||
* [[Cubic Splines]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Construction of Cubic Splines for Interpolation
* End Conditions
* Properties of Cubic Spline Interpolation
|-
|Week 8
||
* Section 3.5: Bezier Curves
||
* [[Bezier Curves]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Bezier Curve and Fonts
|-
|Week 8
||
* Section 4.1: Least Square Method
||
* [[Least Square Method]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Least Square Method for Solving Inconsistent System of Linear Equations]
* Basic Properties of Least Square Solutions
|-
|Week 9
||
* Section 4.2: Mathematical Models and Data Fitting
||
* [[Curve Fitting]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
* Section 4.2: Mathematical Models and Data Fitting
||
* [[Statistical Modeling]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Taylor's Theorem for Vector Valued Multivariate Functions]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Gauss-Newton Method]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Levenberg-Marquardt Method]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Numerical Differentiation]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Finite Difference (FD)]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Undetermined Coefficient Method]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Extrapolation Technique]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Newton-Cotes]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Midpoint rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Trapezoid rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Simpson's rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Error Analysis based on Interpolation Errors]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Quadrature Rules]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Composite Quadrature Rules]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.3: Numerical Integration: Romberg's Technique
||
* [[Romberg's Technique]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Motivation, construction and implementation of Romberg's Technique.
|-
|Week 11
||
* Section 5.4: Adaptive Numerical Integration
||
* [[Adaptive Numerical Integration]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
* Section 5.4: Adaptive Numerical Integration
||
* [[Implementation of Adaptive Numerical Integration Techniques]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Formulas]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Orthogonal Polynomials]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Legendre polynomials]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Rule]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fourier Series]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Discrete Fourier Transform (DFT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Inverse Discrete Fourier Transform]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fast Fourier Transform (FFT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
* Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform(DCT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Quantization]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Image Compression]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Image Decompression]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Power Iteration Methods]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Inverse Power Iteration]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Inverse Power Iteration with Shift]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Rayleigh Quotient Iteration]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Orthogonal Matrices]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[QR-Factorization]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Normalized Simultaneous Iteration(NSI)]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Unshifted QR Algorithm]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Shifted QR Algorithm]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 14
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Upper Hessenberg Form (UHF)]]
||
* Matrices for Orthogonal Projection
* Matrices for Reflection
* Block Matrices
* Similar Matrices
||
* Convert a matrix into UHF by Householder reflectors
|-
|Week 14
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Householder Reflector]]
||
* Matrices for Orthogonal Projection
* Matrices for Reflection
* Block Matrices
* Similar Matrices
||
* Convert a matrix into UHF by Householder reflectors
|}

MAT3633

2020-08-17T19:52:58Z

Weiming.cao:

==Course Catalog==
[https://catalog.utsa.edu/undergraduate/sciences/mathematics/#courseinventory MAT 3633. Numerical Analysis.] (3-0) 3 Credit Hours.

Prerequisites: [[MAT2233]], [[MAT3213]], and one of the following: [[CS1063]], [[CS1714]], or [[CS2073]]. Solution of linear and nonlinear equations, curve-fitting, and eigenvalue problems. Generally offered: Fall, Spring. Differential Tuition: $150.

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
0.2 and 1.1
||
* Loss of significant digits
* Bisection Method
||
* [[Nested multiplication for evaluating polynomials]]
* [[Machine representation of real numbers]]
* [[Loss of significant digits in numerical computing]]
* [[Review of Taylor's Theorem]]
* [[]]
* [[Bisection method and implementation]]
* [[Brief introduction to matlab]]
||
* [[binary number system; Taylor's theorem; intermediate value theorem]]
|-
|Week 1
||
Section 0.2: Loss of significant digits
||
* [[Loss of Significant Digits]]
* [[Nested Multiplication]]
* [[my addition]]
||
* [[Binary Number System]]
* [[Taylor's Theorem]]
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
Section 1.1: Fixed-Point Iteration
||
* [[Bisection Method]]
||
* [[Intermediate Value Theorem]]
||
* Bisection Method and Implementation
* Brief Introduction to Matlab
|-
|Week 2
||
Section 1.2: Fixed-Point Iteration
||
* [[Fixed-Point Iteration]]
* [[Order of Convergence]] of Iterative Methods
||
* [[Limit of Sequences]]
* [[Solution Multiplicity of Equations]]
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
Section 1.3: Limits of Accuracy: Conditioning of Problems
||
* [[Wilkinson Polynomial]]
||
* [[Limit of Sequences]]
* [[Solution Multiplicity of Equations]]
||
* Sensitivity Analysis of Root-Finding
* Error Magnification Factor for Solution of Equations
|-
|Week 3
||
Section 1.4: Newton's Method
||
* [[Newton's Method]]
* [[Error Analysis]] for Newton's Method
* [[Modified Newton's Method]]

* [[Root-Finding Without Derivatives]]
* [[Secant Method and its Convergence]]
* [[Method of False Position, Muller's Method]]
* [[Stopping Criteria for Iterative Methods]]
||
* [[Remainder of Taylor's Series]]
* [[Intermediate Value Theorem]]
* [[Fixed-Point Iteration]]
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
Section 1.5 Root-Finding Without Derivatives
||
* [[Secant Method]]
* [[Method of False Position]]
* [[Muller's Method]]
* [[Stopping Criteria]]
||
* [[Remainder of Taylor's Series]]
* [[Intermediate Value Theorem]]
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 4
||
Section 2.1 Solve Systems of Linear Equations: Gaussian Elimination
||
* [[Gaussian Elimination]]
||
* [[Elementary Row Operations]]
||
* Gaussian Elimination and its Operation Counts
* Gaussian Elimination with Pivoting
* Implementation of Gauss Elimination
|-
|Week 4
||
Section 2.2 Solve Systems of Linear Equations: LU Decomposition
||
* [[LU Decomposition]]
||
* [[Matrix-Matrix Products]]
* [[Matrix-Vector Products]]
* [[Inverse Matrix]]
* [[Elementary Row Operations]]
||
* Matrices for Elementary Row Operations
* Gauss Elimination as Matrix Products
* Advantages of Solutions by LU Decomposition
|-
|Week 5
||
Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Norms]]
* [[Error Analysis]] for Solution of Ax=b
* [[Error Magnification]] Factor and Condition Number of Matrix
||
* [[Length of Vectors]]
* [[Eigenvalues of a Matrix]]
* [[Eigenvectors of a Matrix]]
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Iterative Methods]]
* [[Jacobi Method]]
* [[Gauss-Seidel Method]]
* [[Successive-Over-Relaxation (SOR) Method]]
* [[Convergence of Iterative Methods]]
* [[Spectral Radius of Matrix]]
* [[Sparse Matrix]]
||
* [[Length of Vectors]]
* [[Eigenvalues of a Matrix]]
* [[Eigenvectors of a Matrix]]
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 6
||
Section 2.6: Conjugate Gradient (CG) Method
||
* [[Conjugate Gradient Method]]
* [[Symmetric Positive Definite Matrix]]
* [[CG Method]]
||
* [[Scalar Product of Vectors]]
* [[Determinant of a Matrix]]
* [[Eigenvalues of a Matrix]]
* [[Quadratic Polynomials of n-variables]]
* [[Partial Derivatives]]
* [[Gradients]]
* [[Chain Rule for Partial Derivatives]]
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
Section 2.7: Nonlinear System of Equations
||
* [[Nonlinear System of Equations]]
* [[Taylor's Theorem for Multi-Variate Vector Valued Functions]]
* [[Newton's Method]]
* [[Broyden's Method]]
||
* [[Scalar Product of Vectors]]
* [[Determinant of a Matrix]]
* [[Eigenvalues of a Matrix]]
* [[Quadratic Polynomials of n-variables]]
* [[Partial Derivatives]]
* [[Gradients]]
* [[Chain Rule for Partial Derivatives]]
||
* (TBD)
|-
|Week 7
||
Sections 3.1: Data and Interpolating Functions
||
* [[Lagrange Basis Functions]]
* [[Newton's Divided Differences]]
* [[Properties of Newton's Divided Differences]]
||
* [[Fundamental Theorem of Algebra]]
* [[Rolle's Theorem]]
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error]]
* Interpolation [[Error Analysis]]
* [[Runge Phenomenon]]
* [[Chebyshev Polynomial]]
* [[Error Estimates]] for Chebyshev Interpolation
||
* [[Fundamental Theorem of Algebra]]
* [[Rolle's Theorem]]
||
* (TBD)
|-
|Week 8
||
Section 3.4: Cubic Splines
||
* [[Cubic Splines]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Construction of Cubic Splines for Interpolation
* End Conditions
* Properties of Cubic Spline Interpolation
|-
|Week 8
||
Section 3.5: Bezier Curves
||
* [[Bezier Curves]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Bezier Curve and Fonts
|-
|Week 8
||
Section 4.1: Least Square Method
||
* [[Least Square Method]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Least Square Method for Solving Inconsistent System of Linear Equations]
* Basic Properties of Least Square Solutions
|-
|Week 9
||
Section 4.2: Mathematical Models and Data Fitting
||
* [[Curve Fitting]]
* [[Statistical Modeling]]
||
* [[Linear Spaces]]
* [[Basis Functions]]
* [[Product Rule for Vector Valued Multivariate Functions]]
* [[Chain Rule for Vector Valued Multivariate Functions]]
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
Section 4.5: Nonlinear Least Square Fitting
||
* [[Taylor's Theorem for Vector Valued Multivariate Functions]]
* [[Gauss-Newton Method]]
* [[Levenberg-Marquardt Method]]
||
* [[Linear Spaces]]
* [[Basis Functions]]
* [[Product Rule for Vector Valued Multivariate Functions]]
* [[Chain Rule for Vector Valued Multivariate Functions]]
||
* (TBD)
|-
|Week 10
||
Section 5.1: Numerical Differentiation
||
* [[Numerical Differentiation]]
* [[Finite Difference]] (FD)
* [[Undetermined Coefficient Method]]
* [[Extrapolation Technique]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Newton-Cotes]]
* [[Midpoint rule]]
* [[Trapezoid rule]]
* [[Simpson's rule]]
* [[Error Analysis based on Interpolation Errors]]
* [[Quadrature Rules]]
* [[Composite Quadrature Rules]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
Section 5.3: Numerical Integration: Romberg's Technique
||
* [[Romberg's Technique]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Motivation, construction and implementation of Romberg's Technique.
|-
|Week 11
||
Section 5.4: Adaptive Numerical Integration
||
* [[Adaptive Numerical Integration]]
* [[Implementation of Adaptive Numerical Integration Techniques]]
||
* [[Long Divisions]]
* [[Substitution Methods]] for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Formulas]]
* [[Orthogonal Polynomials]]
* [[Legendre polynomials]]
* [[Gauss Quadrature Rule]]
||
* [[Long Divisions]]
* [[Substitution Methods]] for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 12
||
Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fourier Series]]
* [[Discrete Fourier Transform]] (DFT)
* [[Inverse Discrete Fourier Transform]]
* [[Fast Fourier Transform (FFT)]]
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform]]
* [[Discrete Cosine Transform]](DCT)
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
Section 11.2: Image Compression (optional)
||
* [[Quantization]]
* [[Image Compression]]
* [[Image Decompression]]
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 13
||
Section 12.1: Power Iteration Methods
||
* [[Power Iteration Methods]]
* [[Inverse Power Iteration]]
* [[Inverse Power Iteration with Shift]]
* [[Rayleigh Quotient Iteration]]
||
* [[Eigenvalues]]
* [[Eigenvectors]]
* [[Orthonormal Bases and the Gram-Schmidt Process]]
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Orthogonal Matrices]]
* [[QR-Factorization]]
* [[Normalized Simultaneous Iteration]](NSI)
* [[Unshifted QR Algorithm]]
* [[Shifted QR Algorithm]]
||
* [[Eigenvalues]]
* [[Eigenvectors]]
* [[Orthonormal Bases and the Gram-Schmidt Process]]
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 14
||
Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Upper Hessenberg Form]] (UHF)
* [[Householder Reflector]]
||
* [[Matrices for Orthogonal Projection]]
* [[Matrices for Reflection]]
* [[Block Matrices]]
* [[Similar Matrices]]
||
* Convert a matrix into UHF by Householder reflectors
|}

==Topics List B Wiki Format ==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
* Section 0.2: Loss of significant digits
||
* [[Loss of Significant Digits]]
||
* Binary Number System
* Taylor's Theorem
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
* Section 0.2: Loss of significant digits
||
* [[Nested Multiplication]]
||
* Binary Number System
* Taylor's Theorem
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
* Section 1.1: Fixed-Point Iteration
||
* [[Bisection Method]]
||
* Intermediate Value Theorem
||
* Bisection Method and Implementation
* Brief Introduction to Matlab
|-
|Week 2
||
* Section 1.2: Fixed-Point Iteration
||
* [[Fixed-Point Iteration]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
* Section 1.2: Fixed-Point Iteration
||
* [[Order of Convergence of Iterative Methods]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
* Section 1.3: Limits of Accuracy: Conditioning of Problems
||
* [[Wilkinson Polynomial]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Sensitivity Analysis of Root-Finding
* Error Magnification Factor for Solution of Equations
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Error Analysis for Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Modified Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Root-Finding Without Derivatives]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Secant Method and its Convergence]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Method of False Position, Muller's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Stopping Criteria for Iterative Methods]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Secant Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Method of False Position]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Muller's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Stopping Criteria]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 4
||
* Section 2.1 Solve Systems of Linear Equations: Gaussian Elimination
||
* [[Gaussian Elimination]]
||
* Elementary Row Operations
||
* Gaussian Elimination and its Operation Counts
* Gaussian Elimination with Pivoting
* Implementation of Gauss Elimination
|-
|Week 4
||
* Section 2.2 Solve Systems of Linear Equations: LU Decomposition
||
* [[LU Decomposition]]
||
* Matrix-Matrix Products
* Matrix-Vector Products
* Inverse Matrix
* Elementary Row Operations
||
* Matrices for Elementary Row Operations
* Gauss Elimination as Matrix Products
* Advantages of Solutions by LU Decomposition
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Norms]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Error Analysis for Solution of Ax=b]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Error Magnification Factor and Condition Number of Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Iterative Methods]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Jacobi Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Gauss-Seidel Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Successive-Over-Relaxation (SOR) Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Convergence of Iterative Methods]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Spectral Radius of Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Sparse Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[Conjugate Gradient Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[Symmetric Positive Definite Matrix]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[CG Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Nonlinear System of Equations]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Taylor's Theorem for Multi-Variate Vector Valued Functions]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Newton's Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Broyden's Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Lagrange Basis Functions]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Newton's Divided Differences]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Properties of Newton's Divided Differences]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error Analysis]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Runge Phenomenon]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Chebyshev Polynomial]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Error Estimates for Chebyshev Interpolation]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 8
||
* Section 3.4: Cubic Splines
||
* [[Cubic Splines]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Construction of Cubic Splines for Interpolation
* End Conditions
* Properties of Cubic Spline Interpolation
|-
|Week 8
||
* Section 3.5: Bezier Curves
||
* [[Bezier Curves]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Bezier Curve and Fonts
|-
|Week 8
||
* Section 4.1: Least Square Method
||
* [[Least Square Method]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Least Square Method for Solving Inconsistent System of Linear Equations]
* Basic Properties of Least Square Solutions
|-
|Week 9
||
* Section 4.2: Mathematical Models and Data Fitting
||
* [[Curve Fitting]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
* Section 4.2: Mathematical Models and Data Fitting
||
* [[Statistical Modeling]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Taylor's Theorem for Vector Valued Multivariate Functions]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Gauss-Newton Method]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Levenberg-Marquardt Method]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Numerical Differentiation]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Finite Difference (FD)]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Undetermined Coefficient Method]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Extrapolation Technique]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Newton-Cotes]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Midpoint rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Trapezoid rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Simpson's rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Error Analysis based on Interpolation Errors]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Quadrature Rules]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Composite Quadrature Rules]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.3: Numerical Integration: Romberg's Technique
||
* [[Romberg's Technique]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Motivation, construction and implementation of Romberg's Technique.
|-
|Week 11
||
* Section 5.4: Adaptive Numerical Integration
||
* [[Adaptive Numerical Integration]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
* Section 5.4: Adaptive Numerical Integration
||
* [[Implementation of Adaptive Numerical Integration Techniques]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Formulas]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Orthogonal Polynomials]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Legendre polynomials]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Rule]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fourier Series]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Discrete Fourier Transform (DFT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Inverse Discrete Fourier Transform]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fast Fourier Transform (FFT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
* Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform(DCT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Quantization]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Image Compression]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Image Decompression]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Power Iteration Methods]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Inverse Power Iteration]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Inverse Power Iteration with Shift]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Rayleigh Quotient Iteration]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Orthogonal Matrices]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[QR-Factorization]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Normalized Simultaneous Iteration(NSI)]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Unshifted QR Algorithm]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Shifted QR Algorithm]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 14
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Upper Hessenberg Form (UHF)]]
||
* Matrices for Orthogonal Projection
* Matrices for Reflection
* Block Matrices
* Similar Matrices
||
* Convert a matrix into UHF by Householder reflectors
|-
|Week 14
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Householder Reflector]]
||
* Matrices for Orthogonal Projection
* Matrices for Reflection
* Block Matrices
* Similar Matrices
||
* Convert a matrix into UHF by Householder reflectors
|}

MAT3633

2020-08-17T19:52:05Z

Weiming.cao: /* Topics List */

==Course Catalog==
[https://catalog.utsa.edu/undergraduate/sciences/mathematics/#courseinventory MAT 3633. Numerical Analysis.] (3-0) 3 Credit Hours.

Prerequisites: [[MAT2233]], [[MAT3213]], and one of the following: [[CS1063]], [[CS1714]], or [[CS2073]]. Solution of linear and nonlinear equations, curve-fitting, and eigenvalue problems. Generally offered: Fall, Spring. Differential Tuition: $150.

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
0.2 and 1.1
||
Sec.0.2:
||
* Loss of significant digits
* Bisection Method
||
* [[Nested multiplication for evaluating polynomials]]
* [[Machine representation of real numbers]]
* [[Loss of significant digits in numerical computing]]
* [[Review of Taylor's Theorem]]
* [[]]
* [[Bisection method and implementation]]
* [[Brief introduction to matlab]]
||
* [[binary number system; Taylor's theorem; intermediate value theorem]]
|-
|Week 1
||
Section 0.2: Loss of significant digits
||
* [[Loss of Significant Digits]]
* [[Nested Multiplication]]
* [[my addition]]
||
* [[Binary Number System]]
* [[Taylor's Theorem]]
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
Section 1.1: Fixed-Point Iteration
||
* [[Bisection Method]]
||
* [[Intermediate Value Theorem]]
||
* Bisection Method and Implementation
* Brief Introduction to Matlab
|-
|Week 2
||
Section 1.2: Fixed-Point Iteration
||
* [[Fixed-Point Iteration]]
* [[Order of Convergence]] of Iterative Methods
||
* [[Limit of Sequences]]
* [[Solution Multiplicity of Equations]]
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
Section 1.3: Limits of Accuracy: Conditioning of Problems
||
* [[Wilkinson Polynomial]]
||
* [[Limit of Sequences]]
* [[Solution Multiplicity of Equations]]
||
* Sensitivity Analysis of Root-Finding
* Error Magnification Factor for Solution of Equations
|-
|Week 3
||
Section 1.4: Newton's Method
||
* [[Newton's Method]]
* [[Error Analysis]] for Newton's Method
* [[Modified Newton's Method]]

* [[Root-Finding Without Derivatives]]
* [[Secant Method and its Convergence]]
* [[Method of False Position, Muller's Method]]
* [[Stopping Criteria for Iterative Methods]]
||
* [[Remainder of Taylor's Series]]
* [[Intermediate Value Theorem]]
* [[Fixed-Point Iteration]]
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
Section 1.5 Root-Finding Without Derivatives
||
* [[Secant Method]]
* [[Method of False Position]]
* [[Muller's Method]]
* [[Stopping Criteria]]
||
* [[Remainder of Taylor's Series]]
* [[Intermediate Value Theorem]]
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 4
||
Section 2.1 Solve Systems of Linear Equations: Gaussian Elimination
||
* [[Gaussian Elimination]]
||
* [[Elementary Row Operations]]
||
* Gaussian Elimination and its Operation Counts
* Gaussian Elimination with Pivoting
* Implementation of Gauss Elimination
|-
|Week 4
||
Section 2.2 Solve Systems of Linear Equations: LU Decomposition
||
* [[LU Decomposition]]
||
* [[Matrix-Matrix Products]]
* [[Matrix-Vector Products]]
* [[Inverse Matrix]]
* [[Elementary Row Operations]]
||
* Matrices for Elementary Row Operations
* Gauss Elimination as Matrix Products
* Advantages of Solutions by LU Decomposition
|-
|Week 5
||
Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Norms]]
* [[Error Analysis]] for Solution of Ax=b
* [[Error Magnification]] Factor and Condition Number of Matrix
||
* [[Length of Vectors]]
* [[Eigenvalues of a Matrix]]
* [[Eigenvectors of a Matrix]]
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Iterative Methods]]
* [[Jacobi Method]]
* [[Gauss-Seidel Method]]
* [[Successive-Over-Relaxation (SOR) Method]]
* [[Convergence of Iterative Methods]]
* [[Spectral Radius of Matrix]]
* [[Sparse Matrix]]
||
* [[Length of Vectors]]
* [[Eigenvalues of a Matrix]]
* [[Eigenvectors of a Matrix]]
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 6
||
Section 2.6: Conjugate Gradient (CG) Method
||
* [[Conjugate Gradient Method]]
* [[Symmetric Positive Definite Matrix]]
* [[CG Method]]
||
* [[Scalar Product of Vectors]]
* [[Determinant of a Matrix]]
* [[Eigenvalues of a Matrix]]
* [[Quadratic Polynomials of n-variables]]
* [[Partial Derivatives]]
* [[Gradients]]
* [[Chain Rule for Partial Derivatives]]
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
Section 2.7: Nonlinear System of Equations
||
* [[Nonlinear System of Equations]]
* [[Taylor's Theorem for Multi-Variate Vector Valued Functions]]
* [[Newton's Method]]
* [[Broyden's Method]]
||
* [[Scalar Product of Vectors]]
* [[Determinant of a Matrix]]
* [[Eigenvalues of a Matrix]]
* [[Quadratic Polynomials of n-variables]]
* [[Partial Derivatives]]
* [[Gradients]]
* [[Chain Rule for Partial Derivatives]]
||
* (TBD)
|-
|Week 7
||
Sections 3.1: Data and Interpolating Functions
||
* [[Lagrange Basis Functions]]
* [[Newton's Divided Differences]]
* [[Properties of Newton's Divided Differences]]
||
* [[Fundamental Theorem of Algebra]]
* [[Rolle's Theorem]]
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error]]
* Interpolation [[Error Analysis]]
* [[Runge Phenomenon]]
* [[Chebyshev Polynomial]]
* [[Error Estimates]] for Chebyshev Interpolation
||
* [[Fundamental Theorem of Algebra]]
* [[Rolle's Theorem]]
||
* (TBD)
|-
|Week 8
||
Section 3.4: Cubic Splines
||
* [[Cubic Splines]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Construction of Cubic Splines for Interpolation
* End Conditions
* Properties of Cubic Spline Interpolation
|-
|Week 8
||
Section 3.5: Bezier Curves
||
* [[Bezier Curves]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Bezier Curve and Fonts
|-
|Week 8
||
Section 4.1: Least Square Method
||
* [[Least Square Method]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Least Square Method for Solving Inconsistent System of Linear Equations]
* Basic Properties of Least Square Solutions
|-
|Week 9
||
Section 4.2: Mathematical Models and Data Fitting
||
* [[Curve Fitting]]
* [[Statistical Modeling]]
||
* [[Linear Spaces]]
* [[Basis Functions]]
* [[Product Rule for Vector Valued Multivariate Functions]]
* [[Chain Rule for Vector Valued Multivariate Functions]]
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
Section 4.5: Nonlinear Least Square Fitting
||
* [[Taylor's Theorem for Vector Valued Multivariate Functions]]
* [[Gauss-Newton Method]]
* [[Levenberg-Marquardt Method]]
||
* [[Linear Spaces]]
* [[Basis Functions]]
* [[Product Rule for Vector Valued Multivariate Functions]]
* [[Chain Rule for Vector Valued Multivariate Functions]]
||
* (TBD)
|-
|Week 10
||
Section 5.1: Numerical Differentiation
||
* [[Numerical Differentiation]]
* [[Finite Difference]] (FD)
* [[Undetermined Coefficient Method]]
* [[Extrapolation Technique]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Newton-Cotes]]
* [[Midpoint rule]]
* [[Trapezoid rule]]
* [[Simpson's rule]]
* [[Error Analysis based on Interpolation Errors]]
* [[Quadrature Rules]]
* [[Composite Quadrature Rules]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
Section 5.3: Numerical Integration: Romberg's Technique
||
* [[Romberg's Technique]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Motivation, construction and implementation of Romberg's Technique.
|-
|Week 11
||
Section 5.4: Adaptive Numerical Integration
||
* [[Adaptive Numerical Integration]]
* [[Implementation of Adaptive Numerical Integration Techniques]]
||
* [[Long Divisions]]
* [[Substitution Methods]] for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Formulas]]
* [[Orthogonal Polynomials]]
* [[Legendre polynomials]]
* [[Gauss Quadrature Rule]]
||
* [[Long Divisions]]
* [[Substitution Methods]] for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 12
||
Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fourier Series]]
* [[Discrete Fourier Transform]] (DFT)
* [[Inverse Discrete Fourier Transform]]
* [[Fast Fourier Transform (FFT)]]
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform]]
* [[Discrete Cosine Transform]](DCT)
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
Section 11.2: Image Compression (optional)
||
* [[Quantization]]
* [[Image Compression]]
* [[Image Decompression]]
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 13
||
Section 12.1: Power Iteration Methods
||
* [[Power Iteration Methods]]
* [[Inverse Power Iteration]]
* [[Inverse Power Iteration with Shift]]
* [[Rayleigh Quotient Iteration]]
||
* [[Eigenvalues]]
* [[Eigenvectors]]
* [[Orthonormal Bases and the Gram-Schmidt Process]]
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Orthogonal Matrices]]
* [[QR-Factorization]]
* [[Normalized Simultaneous Iteration]](NSI)
* [[Unshifted QR Algorithm]]
* [[Shifted QR Algorithm]]
||
* [[Eigenvalues]]
* [[Eigenvectors]]
* [[Orthonormal Bases and the Gram-Schmidt Process]]
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 14
||
Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Upper Hessenberg Form]] (UHF)
* [[Householder Reflector]]
||
* [[Matrices for Orthogonal Projection]]
* [[Matrices for Reflection]]
* [[Block Matrices]]
* [[Similar Matrices]]
||
* Convert a matrix into UHF by Householder reflectors
|}

==Topics List B Wiki Format ==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
* Section 0.2: Loss of significant digits
||
* [[Loss of Significant Digits]]
||
* Binary Number System
* Taylor's Theorem
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
* Section 0.2: Loss of significant digits
||
* [[Nested Multiplication]]
||
* Binary Number System
* Taylor's Theorem
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
* Section 1.1: Fixed-Point Iteration
||
* [[Bisection Method]]
||
* Intermediate Value Theorem
||
* Bisection Method and Implementation
* Brief Introduction to Matlab
|-
|Week 2
||
* Section 1.2: Fixed-Point Iteration
||
* [[Fixed-Point Iteration]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
* Section 1.2: Fixed-Point Iteration
||
* [[Order of Convergence of Iterative Methods]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
* Section 1.3: Limits of Accuracy: Conditioning of Problems
||
* [[Wilkinson Polynomial]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Sensitivity Analysis of Root-Finding
* Error Magnification Factor for Solution of Equations
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Error Analysis for Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Modified Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Root-Finding Without Derivatives]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Secant Method and its Convergence]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Method of False Position, Muller's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Stopping Criteria for Iterative Methods]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Secant Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Method of False Position]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Muller's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Stopping Criteria]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 4
||
* Section 2.1 Solve Systems of Linear Equations: Gaussian Elimination
||
* [[Gaussian Elimination]]
||
* Elementary Row Operations
||
* Gaussian Elimination and its Operation Counts
* Gaussian Elimination with Pivoting
* Implementation of Gauss Elimination
|-
|Week 4
||
* Section 2.2 Solve Systems of Linear Equations: LU Decomposition
||
* [[LU Decomposition]]
||
* Matrix-Matrix Products
* Matrix-Vector Products
* Inverse Matrix
* Elementary Row Operations
||
* Matrices for Elementary Row Operations
* Gauss Elimination as Matrix Products
* Advantages of Solutions by LU Decomposition
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Norms]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Error Analysis for Solution of Ax=b]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Error Magnification Factor and Condition Number of Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Iterative Methods]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Jacobi Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Gauss-Seidel Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Successive-Over-Relaxation (SOR) Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Convergence of Iterative Methods]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Spectral Radius of Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Sparse Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[Conjugate Gradient Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[Symmetric Positive Definite Matrix]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[CG Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Nonlinear System of Equations]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Taylor's Theorem for Multi-Variate Vector Valued Functions]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Newton's Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Broyden's Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Lagrange Basis Functions]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Newton's Divided Differences]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Properties of Newton's Divided Differences]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error Analysis]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Runge Phenomenon]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Chebyshev Polynomial]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Error Estimates for Chebyshev Interpolation]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 8
||
* Section 3.4: Cubic Splines
||
* [[Cubic Splines]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Construction of Cubic Splines for Interpolation
* End Conditions
* Properties of Cubic Spline Interpolation
|-
|Week 8
||
* Section 3.5: Bezier Curves
||
* [[Bezier Curves]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Bezier Curve and Fonts
|-
|Week 8
||
* Section 4.1: Least Square Method
||
* [[Least Square Method]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Least Square Method for Solving Inconsistent System of Linear Equations]
* Basic Properties of Least Square Solutions
|-
|Week 9
||
* Section 4.2: Mathematical Models and Data Fitting
||
* [[Curve Fitting]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
* Section 4.2: Mathematical Models and Data Fitting
||
* [[Statistical Modeling]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Taylor's Theorem for Vector Valued Multivariate Functions]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Gauss-Newton Method]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Levenberg-Marquardt Method]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Numerical Differentiation]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Finite Difference (FD)]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Undetermined Coefficient Method]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Extrapolation Technique]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Newton-Cotes]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Midpoint rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Trapezoid rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Simpson's rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Error Analysis based on Interpolation Errors]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Quadrature Rules]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Composite Quadrature Rules]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.3: Numerical Integration: Romberg's Technique
||
* [[Romberg's Technique]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Motivation, construction and implementation of Romberg's Technique.
|-
|Week 11
||
* Section 5.4: Adaptive Numerical Integration
||
* [[Adaptive Numerical Integration]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
* Section 5.4: Adaptive Numerical Integration
||
* [[Implementation of Adaptive Numerical Integration Techniques]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Formulas]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Orthogonal Polynomials]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Legendre polynomials]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Rule]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fourier Series]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Discrete Fourier Transform (DFT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Inverse Discrete Fourier Transform]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fast Fourier Transform (FFT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
* Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform(DCT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Quantization]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Image Compression]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Image Decompression]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Power Iteration Methods]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Inverse Power Iteration]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Inverse Power Iteration with Shift]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Rayleigh Quotient Iteration]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Orthogonal Matrices]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[QR-Factorization]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Normalized Simultaneous Iteration(NSI)]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Unshifted QR Algorithm]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Shifted QR Algorithm]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 14
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Upper Hessenberg Form (UHF)]]
||
* Matrices for Orthogonal Projection
* Matrices for Reflection
* Block Matrices
* Similar Matrices
||
* Convert a matrix into UHF by Householder reflectors
|-
|Week 14
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Householder Reflector]]
||
* Matrices for Orthogonal Projection
* Matrices for Reflection
* Block Matrices
* Similar Matrices
||
* Convert a matrix into UHF by Householder reflectors
|}

MAT3633

2020-08-17T19:37:57Z

Weiming.cao: /* Topics List */

==Course Catalog==
[https://catalog.utsa.edu/undergraduate/sciences/mathematics/#courseinventory MAT 3633. Numerical Analysis.] (3-0) 3 Credit Hours.

Prerequisites: [[MAT2233]], [[MAT3213]], and one of the following: [[CS1063]], [[CS1714]], or [[CS2073]]. Solution of linear and nonlinear equations, curve-fitting, and eigenvalue problems. Generally offered: Fall, Spring. Differential Tuition: $150.

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
Section 0.2: Loss of significant digits
||
* [[Loss of Significant Digits]]
* [[Nested Multiplication]]
* [[my addition]]
||
* [[Binary Number System]]
* [[Taylor's Theorem]]
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
Section 1.1: Fixed-Point Iteration
||
* [[Bisection Method]]
||
* [[Intermediate Value Theorem]]
||
* Bisection Method and Implementation
* Brief Introduction to Matlab
|-
|Week 2
||
Section 1.2: Fixed-Point Iteration
||
* [[Fixed-Point Iteration]]
* [[Order of Convergence]] of Iterative Methods
||
* [[Limit of Sequences]]
* [[Solution Multiplicity of Equations]]
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
Section 1.3: Limits of Accuracy: Conditioning of Problems
||
* [[Wilkinson Polynomial]]
||
* [[Limit of Sequences]]
* [[Solution Multiplicity of Equations]]
||
* Sensitivity Analysis of Root-Finding
* Error Magnification Factor for Solution of Equations
|-
|Week 3
||
Section 1.4: Newton's Method
||
* [[Newton's Method]]
* [[Error Analysis]] for Newton's Method
* [[Modified Newton's Method]]

* [[Root-Finding Without Derivatives]]
* [[Secant Method and its Convergence]]
* [[Method of False Position, Muller's Method]]
* [[Stopping Criteria for Iterative Methods]]
||
* [[Remainder of Taylor's Series]]
* [[Intermediate Value Theorem]]
* [[Fixed-Point Iteration]]
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
Section 1.5 Root-Finding Without Derivatives
||
* [[Secant Method]]
* [[Method of False Position]]
* [[Muller's Method]]
* [[Stopping Criteria]]
||
* [[Remainder of Taylor's Series]]
* [[Intermediate Value Theorem]]
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 4
||
Section 2.1 Solve Systems of Linear Equations: Gaussian Elimination
||
* [[Gaussian Elimination]]
||
* [[Elementary Row Operations]]
||
* Gaussian Elimination and its Operation Counts
* Gaussian Elimination with Pivoting
* Implementation of Gauss Elimination
|-
|Week 4
||
Section 2.2 Solve Systems of Linear Equations: LU Decomposition
||
* [[LU Decomposition]]
||
* [[Matrix-Matrix Products]]
* [[Matrix-Vector Products]]
* [[Inverse Matrix]]
* [[Elementary Row Operations]]
||
* Matrices for Elementary Row Operations
* Gauss Elimination as Matrix Products
* Advantages of Solutions by LU Decomposition
|-
|Week 5
||
Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Norms]]
* [[Error Analysis]] for Solution of Ax=b
* [[Error Magnification]] Factor and Condition Number of Matrix
||
* [[Length of Vectors]]
* [[Eigenvalues of a Matrix]]
* [[Eigenvectors of a Matrix]]
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Iterative Methods]]
* [[Jacobi Method]]
* [[Gauss-Seidel Method]]
* [[Successive-Over-Relaxation (SOR) Method]]
* [[Convergence of Iterative Methods]]
* [[Spectral Radius of Matrix]]
* [[Sparse Matrix]]
||
* [[Length of Vectors]]
* [[Eigenvalues of a Matrix]]
* [[Eigenvectors of a Matrix]]
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 6
||
Section 2.6: Conjugate Gradient (CG) Method
||
* [[Conjugate Gradient Method]]
* [[Symmetric Positive Definite Matrix]]
* [[CG Method]]
||
* [[Scalar Product of Vectors]]
* [[Determinant of a Matrix]]
* [[Eigenvalues of a Matrix]]
* [[Quadratic Polynomials of n-variables]]
* [[Partial Derivatives]]
* [[Gradients]]
* [[Chain Rule for Partial Derivatives]]
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
Section 2.7: Nonlinear System of Equations
||
* [[Nonlinear System of Equations]]
* [[Taylor's Theorem for Multi-Variate Vector Valued Functions]]
* [[Newton's Method]]
* [[Broyden's Method]]
||
* [[Scalar Product of Vectors]]
* [[Determinant of a Matrix]]
* [[Eigenvalues of a Matrix]]
* [[Quadratic Polynomials of n-variables]]
* [[Partial Derivatives]]
* [[Gradients]]
* [[Chain Rule for Partial Derivatives]]
||
* (TBD)
|-
|Week 7
||
Sections 3.1: Data and Interpolating Functions
||
* [[Lagrange Basis Functions]]
* [[Newton's Divided Differences]]
* [[Properties of Newton's Divided Differences]]
||
* [[Fundamental Theorem of Algebra]]
* [[Rolle's Theorem]]
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error]]
* Interpolation [[Error Analysis]]
* [[Runge Phenomenon]]
* [[Chebyshev Polynomial]]
* [[Error Estimates]] for Chebyshev Interpolation
||
* [[Fundamental Theorem of Algebra]]
* [[Rolle's Theorem]]
||
* (TBD)
|-
|Week 8
||
Section 3.4: Cubic Splines
||
* [[Cubic Splines]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Construction of Cubic Splines for Interpolation
* End Conditions
* Properties of Cubic Spline Interpolation
|-
|Week 8
||
Section 3.5: Bezier Curves
||
* [[Bezier Curves]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Bezier Curve and Fonts
|-
|Week 8
||
Section 4.1: Least Square Method
||
* [[Least Square Method]]
||
* [[One-Sided Limits]]
* [[Continuity of Functions]]
* [[Indefinite Integrals]]
* [[Extremum Values of Multivariate Quadratic Functions]]
||
* Least Square Method for Solving Inconsistent System of Linear Equations]
* Basic Properties of Least Square Solutions
|-
|Week 9
||
Section 4.2: Mathematical Models and Data Fitting
||
* [[Curve Fitting]]
* [[Statistical Modeling]]
||
* [[Linear Spaces]]
* [[Basis Functions]]
* [[Product Rule for Vector Valued Multivariate Functions]]
* [[Chain Rule for Vector Valued Multivariate Functions]]
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
Section 4.5: Nonlinear Least Square Fitting
||
* [[Taylor's Theorem for Vector Valued Multivariate Functions]]
* [[Gauss-Newton Method]]
* [[Levenberg-Marquardt Method]]
||
* [[Linear Spaces]]
* [[Basis Functions]]
* [[Product Rule for Vector Valued Multivariate Functions]]
* [[Chain Rule for Vector Valued Multivariate Functions]]
||
* (TBD)
|-
|Week 10
||
Section 5.1: Numerical Differentiation
||
* [[Numerical Differentiation]]
* [[Finite Difference]] (FD)
* [[Undetermined Coefficient Method]]
* [[Extrapolation Technique]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Newton-Cotes]]
* [[Midpoint rule]]
* [[Trapezoid rule]]
* [[Simpson's rule]]
* [[Error Analysis based on Interpolation Errors]]
* [[Quadrature Rules]]
* [[Composite Quadrature Rules]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
Section 5.3: Numerical Integration: Romberg's Technique
||
* [[Romberg's Technique]]
||
* [[Taylor's Theorem]]
* [[Interpolation Error Estimates]]
* [[Properties of Definite Integrals]]
||
* Motivation, construction and implementation of Romberg's Technique.
|-
|Week 11
||
Section 5.4: Adaptive Numerical Integration
||
* [[Adaptive Numerical Integration]]
* [[Implementation of Adaptive Numerical Integration Techniques]]
||
* [[Long Divisions]]
* [[Substitution Methods]] for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Formulas]]
* [[Orthogonal Polynomials]]
* [[Legendre polynomials]]
* [[Gauss Quadrature Rule]]
||
* [[Long Divisions]]
* [[Substitution Methods]] for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 12
||
Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fourier Series]]
* [[Discrete Fourier Transform]] (DFT)
* [[Inverse Discrete Fourier Transform]]
* [[Fast Fourier Transform (FFT)]]
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform]]
* [[Discrete Cosine Transform]](DCT)
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
Section 11.2: Image Compression (optional)
||
* [[Quantization]]
* [[Image Compression]]
* [[Image Decompression]]
||
* [[Complex Numbers]]
* [[Complex Variables]]
* [[Integration by Parts]]
* [[Convergence of Sequences]]
* [[Convergence of Series]]
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 13
||
Section 12.1: Power Iteration Methods
||
* [[Power Iteration Methods]]
* [[Inverse Power Iteration]]
* [[Inverse Power Iteration with Shift]]
* [[Rayleigh Quotient Iteration]]
||
* [[Eigenvalues]]
* [[Eigenvectors]]
* [[Orthonormal Bases and the Gram-Schmidt Process]]
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Orthogonal Matrices]]
* [[QR-Factorization]]
* [[Normalized Simultaneous Iteration]](NSI)
* [[Unshifted QR Algorithm]]
* [[Shifted QR Algorithm]]
||
* [[Eigenvalues]]
* [[Eigenvectors]]
* [[Orthonormal Bases and the Gram-Schmidt Process]]
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 14
||
Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Upper Hessenberg Form]] (UHF)
* [[Householder Reflector]]
||
* [[Matrices for Orthogonal Projection]]
* [[Matrices for Reflection]]
* [[Block Matrices]]
* [[Similar Matrices]]
||
* Convert a matrix into UHF by Householder reflectors
|}

==Topics List B Wiki Format ==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week 1
||
* Section 0.2: Loss of significant digits
||
* [[Loss of Significant Digits]]
||
* Binary Number System
* Taylor's Theorem
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
* Section 0.2: Loss of significant digits
||
* [[Nested Multiplication]]
||
* Binary Number System
* Taylor's Theorem
||
* Nested Multiplication for Evaluating Polynomials
* Machine Representation of Real Numbers
* Loss of Significant Digits in Numerical Computing
* Review of Taylor's Theorem
|-
|Week 1
||
* Section 1.1: Fixed-Point Iteration
||
* [[Bisection Method]]
||
* Intermediate Value Theorem
||
* Bisection Method and Implementation
* Brief Introduction to Matlab
|-
|Week 2
||
* Section 1.2: Fixed-Point Iteration
||
* [[Fixed-Point Iteration]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
* Section 1.2: Fixed-Point Iteration
||
* [[Order of Convergence of Iterative Methods]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Geometric Interpretation of Fixed-Point Iteration
* Convergence of Fixed Point Iterations
* Order of Convergence of Iterative Methods
|-
|Week 2
||
* Section 1.3: Limits of Accuracy: Conditioning of Problems
||
* [[Wilkinson Polynomial]]
||
* Limit of Sequences
* Solution Multiplicity of Equations
||
* Sensitivity Analysis of Root-Finding
* Error Magnification Factor for Solution of Equations
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Error Analysis for Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Modified Newton's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Root-Finding Without Derivatives]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Secant Method and its Convergence]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Method of False Position, Muller's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.4: Newton's Method
||
* [[Stopping Criteria for Iterative Methods]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
* Fixed-Point Iteration
||
* Algebraic and Geometric Interpretation of Newton's method
* Error Analysis for Newton's Method Based on Taylor's Theorem
* Newton's Method as a Fixed Point Iteration
* Modified Newton's Method and its Rate of Convergence
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Secant Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Method of False Position]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Muller's Method]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 3
||
* Section 1.5 Root-Finding Without Derivatives
||
* [[Stopping Criteria]]
||
* Remainder of Taylor's Series
* Intermediate Value Theorem
||
* Secant Method and its Convergence
* Stopping Criteria for Iterative Methods
|-
|Week 4
||
* Section 2.1 Solve Systems of Linear Equations: Gaussian Elimination
||
* [[Gaussian Elimination]]
||
* Elementary Row Operations
||
* Gaussian Elimination and its Operation Counts
* Gaussian Elimination with Pivoting
* Implementation of Gauss Elimination
|-
|Week 4
||
* Section 2.2 Solve Systems of Linear Equations: LU Decomposition
||
* [[LU Decomposition]]
||
* Matrix-Matrix Products
* Matrix-Vector Products
* Inverse Matrix
* Elementary Row Operations
||
* Matrices for Elementary Row Operations
* Gauss Elimination as Matrix Products
* Advantages of Solutions by LU Decomposition
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Norms]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Error Analysis for Solution of Ax=b]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.3 Error Analysis for Solution of Ax=b
||
* [[Error Magnification Factor and Condition Number of Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
* Error Analysis for Solution of Ax=b
* Error Magnification Factor and Condition Number of Matrix
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Iterative Methods]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Jacobi Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Gauss-Seidel Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Successive-Over-Relaxation (SOR) Method]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Convergence of Iterative Methods]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Spectral Radius of Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 5
||
* Section 2.5: Iterative Methods for Solving Ax=b
||
* [[Sparse Matrix]]
||
* Length of Vectors
* Eigenvalues of a Matrix
* Eigenvectors of a Matrix
||
* Convergence of General Iterative Method for Solving System of Linear Equations
* Comparison of Gauss Elimination and Iterative Methods
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[Conjugate Gradient Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[Symmetric Positive Definite Matrix]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.6: Conjugate Gradient (CG) Method
||
* [[CG Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* Symmetric Positive Definite Matrix and Properties
* Construction of Conjugate Gradient (CG) Method
* Properties of CG Method
* Preconditioning for CG Method
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Nonlinear System of Equations]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Taylor's Theorem for Multi-Variate Vector Valued Functions]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Newton's Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 6
||
* Section 2.7: Nonlinear System of Equations
||
* [[Broyden's Method]]
||
* Scalar Product of Vectors
* Determinant of a Matrix
* Eigenvalues of a Matrix
* Quadratic Polynomials of n-variables
* Partial Derivatives
* Gradients
* Chain Rule for Partial Derivatives
||
* (TBD)
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Lagrange Basis Functions]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Newton's Divided Differences]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Sections 3.1: Data and Interpolating Functions
||
* [[Properties of Newton's Divided Differences]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* Properties of Lagrange Basis Functions
* Lagrange Form of the Interpolation Polynomials
* Properties of Newton's Divided Differences
* Newton's Form of the Interpolation Polynomials
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Interpolation Error Analysis]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Runge Phenomenon]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Chebyshev Polynomial]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 7
||
* Section 3.2: Interpolation Error and Runge Phenomenon
||
* [[Error Estimates for Chebyshev Interpolation]]
||
* Fundamental Theorem of Algebra
* Rolle's Theorem
||
* (TBD)
|-
|Week 8
||
* Section 3.4: Cubic Splines
||
* [[Cubic Splines]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Construction of Cubic Splines for Interpolation
* End Conditions
* Properties of Cubic Spline Interpolation
|-
|Week 8
||
* Section 3.5: Bezier Curves
||
* [[Bezier Curves]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Bezier Curve and Fonts
|-
|Week 8
||
* Section 4.1: Least Square Method
||
* [[Least Square Method]]
||
* One-Sided Limits
* Continuity of Functions
* Indefinite Integrals
* Extremum Values of Multivariate Quadratic Functions
||
* Least Square Method for Solving Inconsistent System of Linear Equations]
* Basic Properties of Least Square Solutions
|-
|Week 9
||
* Section 4.2: Mathematical Models and Data Fitting
||
* [[Curve Fitting]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
* Section 4.2: Mathematical Models and Data Fitting
||
* [[Statistical Modeling]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* Least square method for curve fitting and statistical modeling
* Survey of Models: linear model, periodic model, exponential models, logistic model, etc
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Taylor's Theorem for Vector Valued Multivariate Functions]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Gauss-Newton Method]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 9
||
* Section 4.5: Nonlinear Least Square Fitting
||
* [[Levenberg-Marquardt Method]]
||
* Linear Spaces
* Basis Functions
* Product Rule for Vector Valued Multivariate Functions
* Chain Rule for Vector Valued Multivariate Functions
||
* (TBD)
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Numerical Differentiation]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Finite Difference (FD)]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Undetermined Coefficient Method]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.1: Numerical Differentiation
||
* [[Extrapolation Technique]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
* FD approximations of 2nd order Derivatives and Their Error Analysis
* Undetermined Coefficient Method for FD Approximation
* Extrapolation Technique for Improving the Order of Approximation
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Newton-Cotes]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Midpoint rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Trapezoid rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Simpson's rule]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Error Analysis based on Interpolation Errors]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Quadrature Rules]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.2: Numerical Integration: Newton-Cotes Formulas
||
* [[Composite Quadrature Rules]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Error Analysis based on Taylor's Theorem
* Error Analysis based on Interpolation Errors
* Degree of Precision of Quadrature Rules
|-
|Week 10
||
* Section 5.3: Numerical Integration: Romberg's Technique
||
* [[Romberg's Technique]]
||
* Taylor's Theorem
* Interpolation Error Estimates
* Properties of Definite Integrals
||
* Motivation, construction and implementation of Romberg's Technique.
|-
|Week 11
||
* Section 5.4: Adaptive Numerical Integration
||
* [[Adaptive Numerical Integration]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
* Section 5.4: Adaptive Numerical Integration
||
* [[Implementation of Adaptive Numerical Integration Techniques]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* How to estimate the error on a sub interval
* How to mark sub intervals to be further refinement?
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Formulas]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Orthogonal Polynomials]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Legendre polynomials]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 11
||
* Section 5.5: Gauss Quadrature Formulas
||
* [[Gauss Quadrature Rule]]
||
* Long Divisions
* Substitution Methods for definite integrals
||
* Motivation and difficulties with straightforward approach
* Legendre polynomials and their basic properties
* Gauss Quadrature rule based on Legendre polynomials
* Degree of precision of Gauss Quadrature
* Gauss quadrature formula on general interval and composite Gauss rules
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fourier Series]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Discrete Fourier Transform (DFT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Inverse Discrete Fourier Transform]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
||
* [[Fast Fourier Transform (FFT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Matrix Form of Discrete Fourier Transform
* DFT and Trigonometric Interpolation
|-
|Week 12
||
* Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
* Section 11.1: Discrete Cosine Transform (optional)
||
* [[Discrete Cosine Transform(DCT)]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* DCT and Interpolation by Cosine Functions
* Relation between DFT and DCT
* Fourier Transform of 2-Dimensional Functions
* DCT of 2-Dimensional Functions
* Interpolation Theorem for 2-Dimensional DCT
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Quantization]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Image Compression]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 12
||
* Section 11.2: Image Compression (optional)
||
* [[Image Decompression]]
||
* Complex Numbers
* Complex Variables
* Integration by Parts
* Convergence of Sequences
* Convergence of Series
||
* Digital Gray scale images and color color images
* RGB format
* YCbCr (or YUV) format
* Convertion between RGB and YUV formats
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Power Iteration Methods]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Inverse Power Iteration]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Inverse Power Iteration with Shift]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.1: Power Iteration Methods
||
* [[Rayleigh Quotient Iteration]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Convergence of Power Iteration Methods
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Orthogonal Matrices]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[QR-Factorization]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Normalized Simultaneous Iteration(NSI)]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Unshifted QR Algorithm]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 13
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Shifted QR Algorithm]]
||
* Eigenvalues
* Eigenvectors
* Orthonormal Bases and the Gram-Schmidt Process
||
* Definition and basic properties of orthogonal matrices
* QR-Factorization based on Gram-Schmidt Orthogonalization
|-
|Week 14
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Upper Hessenberg Form (UHF)]]
||
* Matrices for Orthogonal Projection
* Matrices for Reflection
* Block Matrices
* Similar Matrices
||
* Convert a matrix into UHF by Householder reflectors
|-
|Week 14
||
* Section 12.2: QR Algorithm for Computing Eigenvalues
||
* [[Householder Reflector]]
||
* Matrices for Orthogonal Projection
* Matrices for Reflection
* Block Matrices
* Similar Matrices
||
* Convert a matrix into UHF by Householder reflectors
|}