Difference between revisions of "MAT3633"

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* [[Block Matrices]]
 
* [[Block Matrices]]
 
* [[Similar 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
 
* Convert a matrix into UHF by Householder reflectors
 
|}
 
|}

Revision as of 09:11, 3 August 2020

Course Catalog

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

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week 1

Section 0.2: Loss of significant digits

  • 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 and Implementation
  • Brief Introduction to Matlab
Week 2

Section 1.2: Fixed-Point Iteration

  • 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

  • Sensitivity Analysis of Root-Finding
  • Error Magnification Factor for Solution of Equations
Week 3

Section 1.4: Newton's Method

  • 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 and its Convergence
  • Stopping Criteria for Iterative Methods
Week 4

Section 2.1 Solve Systems of Linear Equations: Gaussian Elimination

  • 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

  • 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

  • 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

  • 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

  • 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

  • (TBD)
Week 7

Sections 3.1: Data and Interpolating Functions

  • 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

  • (TBD)
Week 8

Section 3.4: Cubic Splines

  • Construction of Cubic Splines for Interpolation
  • End Conditions
  • Properties of Cubic Spline Interpolation
Week 8

Section 3.5: Bezier Curves

  • Bezier Curve and Fonts
Week 8

Section 4.1: Least Square Method

  • 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

  • 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

  • (TBD)
Week 10

Section 5.1: Numerical Differentiation

  • 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

  • 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

  • Motivation, construction and implementation of Romberg's Technique.
Week 11

Section 5.4: Adaptive Numerical Integration

  • 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

  • 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)

  • Matrix Form of Discrete Fourier Transform
  • DFT and Trigonometric Interpolation
Week 12

Section 11.1: Discrete Cosine Transform (optional)

  • 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)

  • 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

  • Convergence of Power Iteration Methods
Week 13

Section 12.2: QR Algorithm for Computing Eigenvalues

  • Definition and basic properties of orthogonal matrices
  • QR-Factorization based on Gram-Schmidt Orthogonalization
Week 14

Section 12.2: QR Algorithm for Computing Eigenvalues

  • Convert a matrix into UHF by Householder reflectors

Topics List B Wiki Format

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week 1
  • Section 0.2: 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
  • 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
  • Intermediate Value Theorem
  • Bisection Method and Implementation
  • Brief Introduction to Matlab
Week 2
  • Section 1.2: 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • Fundamental Theorem of Algebra
  • Rolle's Theorem
  • (TBD)
Week 7
  • Section 3.2: Interpolation Error and Runge Phenomenon
  • Fundamental Theorem of Algebra
  • Rolle's Theorem
  • (TBD)
Week 7
  • Section 3.2: Interpolation Error and Runge Phenomenon
  • Fundamental Theorem of Algebra
  • Rolle's Theorem
  • (TBD)
Week 7
  • Section 3.2: Interpolation Error and Runge Phenomenon
  • Fundamental Theorem of Algebra
  • Rolle's Theorem
  • (TBD)
Week 7
  • Section 3.2: Interpolation Error and Runge Phenomenon
  • Fundamental Theorem of Algebra
  • Rolle's Theorem
  • (TBD)
Week 8
  • Section 3.4: 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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)
  • 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)
  • 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)
  • 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)
  • 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)
  • 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)
  • 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)
  • 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)
  • 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)
  • 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
  • Eigenvalues
  • Eigenvectors
  • Orthonormal Bases and the Gram-Schmidt Process
  • Convergence of Power Iteration Methods
Week 13
  • Section 12.1: 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
  • Eigenvalues
  • Eigenvectors
  • Orthonormal Bases and the Gram-Schmidt Process
  • Convergence of Power Iteration Methods
Week 13
  • Section 12.1: Power Iteration Methods
  • Eigenvalues
  • Eigenvectors
  • Orthonormal Bases and the Gram-Schmidt Process
  • Convergence of Power Iteration Methods
Week 13
  • Section 12.2: QR Algorithm for Computing Eigenvalues
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • Matrices for Orthogonal Projection
  • Matrices for Reflection
  • Block Matrices
  • Similar Matrices
  • Convert a matrix into UHF by Householder reflectors
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