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| | ==Topics List== | | ==Topics List== |
| | {| class="wikitable sortable" | | {| class="wikitable sortable" |
| − | ! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes | + | ! Date !! Sections [Sauer 3rd ed] !! Topics !! Prerequisite Skills !! Student Learning Outcomes |
| | |- | | |- |
| − | |Week 1 | + | | Week.1 |
| | || | | || |
| | 0.2 and 1.1 | | 0.2 and 1.1 |
| Line 14: |
Line 14: |
| | * Loss of significant digits | | * Loss of significant digits |
| | * Bisection Method | | * Bisection Method |
| | + | * Brief introduction to matlab |
| | || | | || |
| − | * [[Nested multiplication for evaluating polynomials]] | + | * binary number system; |
| − | * [[Machine representation of real numbers]] | + | * Taylor's theorem; |
| − | * [[Loss of significant digits in numerical computing]]
| + | * intermediate value theorem |
| − | * [[Review of Taylor's Theorem]]
| |
| − | * [[]]
| |
| − | * [[Bisection method and implementation]]
| |
| − | * [[Brief introduction to matlab]] | |
| | || | | || |
| − | * [[binary number system; Taylor's theorem; intermediate value theorem]]
| + | * Nested multiplication for evaluating polynomials |
| − | |-
| + | * Machine representation of real numbers |
| − | |Week 1
| + | * Loss of significant digits in numerical computing |
| − | ||
| |
| − | 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 | | * Review of Taylor's Theorem |
| − | |-
| + | * Bisection method and implementation |
| − | |Week 1
| |
| | || | | || |
| − | Section 1.1: Fixed-Point Iteration
| + | |
| − | ||
| |
| − | * [[Bisection Method]]
| |
| − | ||
| |
| − | * [[Intermediate Value Theorem]]
| |
| − | ||
| |
| − | * Bisection Method and Implementation
| |
| − | * Brief Introduction to Matlab
| |
| | |- | | |- |
| − | |Week 2 | + | | Week.2 |
| | || | | || |
| − | Section 1.2: Fixed-Point Iteration
| + | 1.2 and 1.3 |
| | || | | || |
| − | * [[Fixed-Point Iteration]] | + | * Fixed-Point Iteration |
| − | * [[Order of Convergence]] of Iterative Methods | + | * Limits of Accuracy: Conditioning of problems |
| | || | | || |
| − | * [[Limit of Sequences]] | + | * limit of sequences |
| − | * [[Solution Multiplicity of Equations]] | + | * multiplicity of solution of equations. |
| | || | | || |
| − | * Geometric Interpretation of Fixed-Point Iteration | + | * Geometric interpretation |
| − | * Convergence of Fixed Point Iterations | + | * Convergence of fixed point iterations |
| − | * Order of Convergence of Iterative Methods | + | * Order of convergence of iterative methods |
| | + | |
| | + | * Wilkinson polynomial and other examples |
| | + | * Sensitivity analysis of root-finding |
| | + | * Error magnification factor for solution of equations |
| | + | |
| | |- | | |- |
| − | |Week 2 | + | | Week.3 |
| | || | | || |
| − | Section 1.3: Limits of Accuracy: Conditioning of Problems
| + | 1.4 and 1.5 |
| | || | | || |
| − | * [[Wilkinson Polynomial]] | + | * Newton's Method |
| | + | * Root-Finding without Derivatives |
| | || | | || |
| − | * [[Limit of Sequences]] | + | * Remainder of Taylor's series |
| − | * [[Solution Multiplicity of Equations]] | + | * intermediate value theorem. |
| | || | | || |
| − | * Sensitivity Analysis of Root-Finding | + | * Algebraic and geometric interpretation of Newton's method |
| − | * Error Magnification Factor for Solution of Equations | + | * Error analysis for Newton's method based on Taylor's theorem |
| − | |-
| + | * Newton's method as a fixed point iteration |
| − | |Week 3
| + | * Modified Newton's method and its rate of convergence |
| − | ||
| + | |
| − | Section 1.4: Newton's Method
| + | * Secant Method and its convergence, |
| − | ||
| + | * Method of False Position, Muller's Method: |
| − | * [[Newton's Method]] | + | * Stopping criteria for iterative methods |
| − | * [[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 | + | | Week.4 |
| | || | | || |
| − | Section 1.5 Root-Finding Without Derivatives
| + | 2.1 and 2.2 |
| | || | | || |
| − | * [[Secant Method]] | + | * Solve Systems of Linear Equations: Gaussian Elmination |
| − | * [[Method of False Position]] | + | * Solve System of Linear Equations: LU Decomposition |
| − | * [[Muller's Method]]
| |
| − | * [[Stopping Criteria]]
| |
| | || | | || |
| − | * [[Remainder of Taylor's Series]] | + | * Matrix-matrix products and matrix-vector products |
| − | * [[Intermediate Value Theorem]] | + | * inverse matrix |
| | + | * elementary row operations |
| | + | * product and inverse of matrices for elementary row operations. |
| | || | | || |
| − | * Secant Method and its Convergence | + | * Gaussian elimination and its operation counts |
| − | * Stopping Criteria for Iterative Methods | + | * Gaussian elimination with pivoting |
| − | |-
| + | * Implementation of Gauss elimination |
| − | |Week 4
| + | |
| − | ||
| + | * Matrices for elementary row operations |
| − | Section 2.1 Solve Systems of Linear Equations: Gaussian Elimination
| + | * Gauss elimination as matrix products |
| − | ||
| + | * Advantages of solutions by LU decomposition |
| − | * [[Gaussian Elimination]] | + | |
| − | ||
| |
| − | * [[Elementary Row Operations]] | |
| − | ||
| |
| − | * Gaussian Elimination and its Operation Counts | |
| − | * Gaussian Elimination with Pivoting | |
| − | * Implementation of Gauss Elimination
| |
| | |- | | |- |
| − | |Week 4 | + | | Week.5 |
| | || | | || |
| − | Section 2.2 Solve Systems of Linear Equations: LU Decomposition
| + | 2.3 and 2.4 |
| | || | | || |
| − | * [[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 Analysis for Solution of Ax=b |
| − | * Error Magnification Factor and Condition Number of Matrix | + | * Iterative Methods for Solving Ax=b |
| − | |-
| |
| − | |Week 5
| |
| | || | | || |
| − | Section 2.5: Iterative Methods for Solving Ax=b
| + | * Length of vectors |
| | + | * eigenvalue and eigenvectors of matrix |
| | || | | || |
| − | * [[Iterative Methods]] | + | * various norms for vectors and matrices: compatibility of vector and matrix norms. |
| − | * [[Jacobi Method]] | + | * Error Analysis for the solution of Ax=b |
| − | * [[Gauss-Seidel Method]] | + | * Error magnification factor and condition number of matrix |
| − | * [[Successive-Over-Relaxation (SOR) Method]]
| + | |
| − | * [[Convergence of Iterative Methods]] | + | * Jacobi method, Gauss-Seidel method, Successive-Over-Relaxation (SOR) method |
| − | * [[Spectral Radius of Matrix]] | + | * Convergence of Jacobi Method, GS method and SOR method. |
| − | * [[Sparse Matrix]]
| + | * spectral radius of matrix |
| − | ||
| + | * convergence of general iterative method for solving system of linear equations, |
| − | * [[Length of Vectors]] | + | * Sparse Matrix |
| − | * [[Eigenvalues of a Matrix]]
| + | * Comparison of Gauss Elimination and iterative methods |
| − | * [[Eigenvectors of a Matrix]] | + | |
| − | ||
| + | |
| − | * Convergence of General Iterative Method for Solving System of Linear Equations
| |
| − | * Comparison of Gauss Elimination and Iterative Methods | |
| | |- | | |- |
| − | |Week 6 | + | | Week.6 |
| | || | | || |
| − | Section 2.6: Conjugate Gradient (CG) Method
| + | 2.6 and 2.7 |
| | || | | || |
| − | * [[Conjugate Gradient Method]] | + | * Conjugate Gradient Method |
| − | * [[Symmetric Positive Definite Matrix]] | + | * Nonlinear System of Equations |
| − | * [[CG Method]]
| |
| | || | | || |
| − | * [[Scalar Product of Vectors]] | + | * scalar product of vectors |
| − | * [[Determinant of a Matrix]] | + | * determinant and eigenvalues of matrix |
| − | * [[Eigenvalues of a Matrix]] | + | * quadratic polynomials of n-variables |
| − | * [[Quadratic Polynomials of n-variables]]
| + | * partial derivatives and gradients |
| − | * [[Partial Derivatives]]
| + | * chain rule for partial derivatives. |
| − | * [[Gradients]] | |
| − | * [[Chain Rule for Partial Derivatives]] | |
| | || | | || |
| − | * Symmetric Positive Definite Matrix and Properties | + | * Symmetric positive definite matrix and properties |
| | * Construction of Conjugate Gradient (CG) Method | | * Construction of Conjugate Gradient (CG) Method |
| − | * Properties of CG Method | + | * Propertise of CG Method |
| − | * Preconditioning for CG Method | + | * Preconditioning for CG method |
| | + | |
| | + | * Taylor's Theorem for multi-variate vector valued functions: |
| | + | * Newton's Method: |
| | + | * Broyden's Method |
| | + | |
| | |- | | |- |
| − | |Week 6 | + | | Week.7 |
| | || | | || |
| − | Section 2.7: Nonlinear System of Equations
| + | 3.1 and 3.2 |
| | || | | || |
| − | * [[Nonlinear System of Equations]] | + | * Data and Interpolating Functions |
| − | * [[Taylor's Theorem for Multi-Variate Vector Valued Functions]]
| + | * Interpolation Error and Runge Phenomenon |
| − | * [[Newton's Method]] | + | * Chebyshev interpolation |
| − | * [[Broyden's Method]] | |
| | || | | || |
| − | * [[Scalar Product of Vectors]] | + | * Fundamental theorem of algebra |
| − | * [[Determinant of a Matrix]] | + | * Rolle's theorem. |
| − | * [[Eigenvalues of a Matrix]]
| |
| − | * [[Quadratic Polynomials of n-variables]]
| |
| − | * [[Partial Derivatives]]
| |
| − | * [[Gradients]]
| |
| − | * [[Chain Rule for Partial Derivatives]]
| |
| | || | | || |
| − | * (TBD) | + | * 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 7 | + | | Week.8 |
| | || | | || |
| − | Sections 3.1: Data and Interpolating Functions
| + | 3.4, 3.5 and 4.1 |
| | || | | || |
| − | * [[Lagrange Basis Functions]] | + | * Cubic Splines |
| − | * [[Newton's Divided Differences]] | + | * Bezier Curves |
| − | * [[Properties of Newton's Divided Differences]] | + | * Least Square Method |
| | || | | || |
| − | * [[Fundamental Theorem of Algebra]] | + | * one-sided limits |
| − | * [[Rolle's Theorem]] | + | * continuity of functions |
| | + | * indefinite integrals |
| | + | * extremum values of multivariate quadratic functions. |
| | || | | || |
| − | * Properties of Lagrange Basis Functions | + | * Cubic splines |
| − | * Lagrange Form of the Interpolation Polynomials | + | * construction of cubic splines for interpolation |
| − | * Properties of Newton's Divided Differences | + | * end conditions |
| − | * Newton's Form of the Interpolation Polynomials | + | * properties of cubic spline interpolation |
| − | |-
| + | |
| − | |Week 7
| + | * Bezier Curve and fonts |
| − | ||
| + | |
| − | Section 3.2: Interpolation Error and Runge Phenomenon
| + | * Least square method for solving inconsistent system of linear equations. |
| − | ||
| + | * Basic properties of least square solutions: |
| − | * [[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 | + | | Week.9 |
| | || | | || |
| − | Section 4.1: Least Square Method
| + | 4.2 and 4.5 |
| | || | | || |
| − | * [[Least Square Method]] | + | * Mathematical Models and Data Fitting |
| | + | * Nonlinear Least Square Fitting |
| | || | | || |
| − | * [[One-Sided Limits]] | + | * linear spaces, basis functions |
| − | * [[Continuity of Functions]] | + | * product rule and chain rule for vector valued multivariate functions. |
| − | * [[Indefinite Integrals]]
| |
| − | * [[Extremum Values of Multivariate Quadratic Functions]]
| |
| | || | | || |
| − | * Least Square Method for Solving Inconsistent System of Linear Equations]
| + | * Least square method for curve fitting and statistical modeling. |
| − | * 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 | | * 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 9 | + | | Week.10 |
| | || | | || |
| − | Section 4.5: Nonlinear Least Square Fitting
| + | 5.1, 5.2 and 5.3 |
| | || | | || |
| − | * [[Taylor's Theorem for Vector Valued Multivariate Functions]] | + | * Numerical Differentiation |
| − | * [[Gauss-Newton Method]] | + | * Numerical Integration: Newton-Cotes Formulas |
| − | * [[Levenberg-Marquardt Method]] | + | * Numerical Integration: Romberg's Technique |
| | || | | || |
| − | * [[Linear Spaces]] | + | * Taylor's theorem |
| − | * [[Basis Functions]] | + | * interpolation error estimates |
| − | * [[Product Rule for Vector Valued Multivariate Functions]] | + | * properties of definite inetgrals |
| − | * [[Chain Rule for Vector Valued Multivariate Functions]]
| |
| | || | | || |
| − | * (TBD) | + | * 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 10 | + | | Week.11 |
| | || | | || |
| − | Section 5.1: Numerical Differentiation
| + | 5.4 and 5.5 |
| | || | | || |
| − | * [[Numerical Differentiation]] | + | * Adaptive Numerical Integration |
| − | * [[Finite Difference]] (FD) | + | * Gauss Quadrature Formulas |
| − | * [[Undetermined Coefficient Method]]
| |
| − | * [[Extrapolation Technique]]
| |
| | || | | || |
| − | * [[Taylor's Theorem]] | + | * long divisions |
| − | * [[Interpolation Error Estimates]] | + | * changing variables for definite integrals |
| − | * [[Properties of Definite Integrals]]
| |
| | || | | || |
| − | * Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
| + | * How to estimate the error on a subinterval |
| − | * FD approximations of 2nd order Derivatives and Their Error Analysis
| + | * How to mark subintervals to be further refinement? |
| − | * Undetermined Coefficient Method for FD Approximation
| + | * Implementation of adaptive numerical integration techniques. |
| − | * Extrapolation Technique for Improving the Order of Approximation
| + | |
| − | |-
| + | * Motivation and difficulties with straightforward approach. |
| − | |Week 10
| + | * Orthogonal polynomials, |
| − | ||
| + | * Legendre polynomials and their basic properties; |
| − | Section 5.2: Numerical Integration: Newton-Cotes Formulas
| + | * Gauss quadrature rule based on Legendre polynomials |
| − | ||
| |
| − | * [[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 | | * Degree of precision of Gauss Quadrature |
| | * Gauss quadrature formula on general interval and composite Gauss rules | | * Gauss quadrature formula on general interval and composite Gauss rules |
| | + | |
| | + | |
| | |- | | |- |
| − | |Week 12 | + | | Week.12 |
| | || | | || |
| − | Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)
| + | 10.1 and 11.1 |
| | || | | || |
| − | * [[Fourier Series]] | + | * Discrete Fourier Transform and FFT |
| − | * [[Discrete Fourier Transform]] (DFT) | + | * Discrete Cosine Transform (optional) |
| − | * [[Inverse Discrete Fourier Transform]] | + | * Image Compression (optional) |
| − | * [[Fast Fourier Transform (FFT)]]
| |
| | || | | || |
| − | * [[Complex Numbers]] | + | * complex numbers and complex variables |
| − | * [[Complex Variables]] | + | * integration by parts |
| − | * [[Integration by Parts]]
| + | * convergence of sequences and series. |
| − | * [[Convergence of Sequences]]
| |
| − | * [[Convergence of Series]] | |
| | || | | || |
| − | * Matrix Form of Discrete Fourier Transform | + | * Fourier Series, |
| − | * DFT and Trigonometric Interpolation | + | * 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 12 | + | | Week.13 |
| | || | | || |
| − | Section 11.1: Discrete Cosine Transform (optional)
| + | 12.1 and 12.2 |
| | || | | || |
| − | * [[Discrete Cosine Transform]] | + | * Power Iteration Methods |
| − | * [[Discrete Cosine Transform]](DCT) | + | * QR Algorithm for Computing Eigenvalues |
| | || | | || |
| − | * [[Complex Numbers]] | + | * properties of eigen values and eigenvectors |
| − | * [[Complex Variables]]
| + | * Gram-Schmidt orthogonalization |
| − | * [[Integration by Parts]]
| |
| − | * [[Convergence of Sequences]]
| |
| − | * [[Convergence of Series]] | |
| | || | | || |
| − | * DCT and Interpolation by Cosine Functions | + | * Power iteration and its rate of convergence. |
| − | * Relation between DFT and DCT | + | * Inverse Power Iteration, |
| − | * Fourier Transform of 2-Dimensional Functions | + | * Inverse Power Iteration with Shift |
| − | * DCT of 2-Dimensional Functions | + | * Rayleigh Quotient Iteration |
| − | * Interpolation Theorem for 2-Dimensional DCT | + | |
| | + | |
| | + | * 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 12 | + | | Week.14 |
| − | ||
| |
| − | Section 11.2: Image Compression (optional)
| |
| | || | | || |
| − | * [[Quantization]]
| + | 12.2 |
| − | * [[Image Compression]]
| |
| − | * [[Image Decompression]]
| |
| | || | | || |
| − | * [[Complex Numbers]] | + | * Algorithm for Computing Eigenvalues: Speed up of QR-algorithm: |
| − | * [[Complex Variables]]
| |
| − | * [[Integration by Parts]]
| |
| − | * [[Convergence of Sequences]]
| |
| − | * [[Convergence of Series]]
| |
| | || | | || |
| − | * Digital Gray scale images and color color images | + | * matrices for orthogonal projection and reflection |
| − | * RGB format | + | * block matrices and their products |
| − | * YCbCr (or YUV) format
| + | * similar matrices. |
| − | * 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]]
| |
| | || | | || |
| | + | * Upper Hessenberg form (UHF) |
| | + | * Householder Reflector |
| | * Convert a matrix into UHF by Householder reflectors | | * 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
| |
| | |} | | |} |
