Difference between revisions of "MAT3633"
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|Week 1 | |Week 1 | ||
|| | || | ||
− | Section 0.2 | + | Section 0.2: Loss of significant digits |
|| | || | ||
* [[Loss of Significant Digits]] | * [[Loss of Significant Digits]] | ||
− | * [[Nested Multiplication | + | * [[Nested Multiplication]] |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
|| | || | ||
* [[Binary Number System]] | * [[Binary Number System]] | ||
* [[Taylor's Theorem]] | * [[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]] | * [[Intermediate Value Theorem]] | ||
|| | || | ||
− | * | + | * Bisection Method and Implementation |
+ | * Brief Introduction to Matlab | ||
|- | |- | ||
|Week 2 | |Week 2 | ||
|| | || | ||
− | + | Section 1.2: Fixed-Point Iteration | |
|| | || | ||
* [[Fixed-Point Iteration]] | * [[Fixed-Point Iteration]] | ||
− | * [[ | + | * [[Order of Convergence]] of Iterative Methods |
− | * [[Convergence of Fixed Point Iterations | + | || |
− | * | + | * [[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]] | * [[Wilkinson Polynomial]] | ||
− | |||
− | |||
|| | || | ||
* [[Limit of Sequences]] | * [[Limit of Sequences]] | ||
* [[Solution Multiplicity of Equations]] | * [[Solution Multiplicity of Equations]] | ||
|| | || | ||
− | * | + | * Sensitivity Analysis of Root-Finding |
+ | * Error Magnification Factor for Solution of Equations | ||
|- | |- | ||
|Week 3 | |Week 3 | ||
|| | || | ||
− | + | Section 1.4: Newton's Method | |
|| | || | ||
* [[Newton's Method]] | * [[Newton's Method]] | ||
− | + | * [[Error Analysis]] for Newton's Method | |
− | * [[Error Analysis | + | * [[Modified Newton's Method]] |
− | |||
− | * [[Modified Newton's Method | ||
* [[Root-Finding Without Derivatives]] | * [[Root-Finding Without Derivatives]] | ||
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* [[Remainder of Taylor's Series]] | * [[Remainder of Taylor's Series]] | ||
* [[Intermediate Value Theorem]] | * [[Intermediate Value Theorem]] | ||
+ | * [[Fixed-Point Iteration]] | ||
+ | || | ||
+ | * Algebraic and Geometric Interpretation of Newton's method | ||
+ | * Error Analysis for Newton's Method Based on Taylor's Theorem | ||
+ | * Newton's Method as a Fixed Point Iteration | ||
+ | * Modified Newton's Method and its Rate of Convergence | ||
+ | |- | ||
+ | |Week 3 | ||
+ | || | ||
+ | Section 1.5 Root-Finding Without Derivatives | ||
+ | || | ||
+ | * [[Secant Method]] | ||
+ | * [[Method of False Position]] | ||
+ | * [[Muller's Method]] | ||
+ | * [[Stopping Criteria]] | ||
+ | || | ||
+ | * [[Remainder of Taylor's Series]] | ||
+ | * [[Intermediate Value Theorem]] | ||
+ | || | ||
+ | * Secant Method and its Convergence | ||
+ | * Stopping Criteria for Iterative Methods | ||
+ | |- | ||
+ | |Week 4 | ||
+ | || | ||
+ | Section 2.1 Solve Systems of Linear Equations: Gaussian Elimination | ||
+ | || | ||
+ | * [[Gaussian Elimination]] | ||
+ | || | ||
+ | * [[Elementary Row Operations]] | ||
|| | || | ||
− | * | + | * Gaussian Elimination and its Operation Counts |
+ | * Gaussian Elimination with Pivoting | ||
+ | * Implementation of Gauss Elimination | ||
|- | |- | ||
|Week 4 | |Week 4 | ||
|| | || | ||
− | + | Section 2.2 Solve Systems of Linear Equations: LU Decomposition | |
|| | || | ||
− | * [[ | + | * [[LU Decomposition]] |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
|| | || | ||
* [[Matrix-Matrix Products]] | * [[Matrix-Matrix Products]] | ||
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* [[Elementary Row Operations]] | * [[Elementary Row Operations]] | ||
|| | || | ||
− | * | + | * Matrices for Elementary Row Operations |
+ | * Gauss Elimination as Matrix Products | ||
+ | * Advantages of Solutions by LU Decomposition | ||
+ | |- | ||
+ | |Week 5 | ||
+ | || | ||
+ | Section 2.3 Error Analysis for Solution of Ax=b | ||
+ | || | ||
+ | * [[Norms]] | ||
+ | * [[Error Analysis]] for Solution of Ax=b | ||
+ | * [[Error Magnification]] Factor and Condition Number of Matrix | ||
+ | || | ||
+ | * [[Length of Vectors]] | ||
+ | * [[Eigenvalues of a Matrix]] | ||
+ | * [[Eigenvectors of a Matrix]] | ||
+ | || | ||
+ | * Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms | ||
+ | * Error Analysis for Solution of Ax=b | ||
+ | * Error Magnification Factor and Condition Number of Matrix | ||
|- | |- | ||
|Week 5 | |Week 5 | ||
|| | || | ||
− | + | Section 2.5: Iterative Methods for Solving Ax=b | |
|| | || | ||
− | + | * [[Iterative Methods]] | |
− | |||
− | |||
− | |||
− | |||
− | * [[Iterative Methods | ||
* [[Jacobi Method]] | * [[Jacobi Method]] | ||
* [[Gauss-Seidel Method]] | * [[Gauss-Seidel Method]] | ||
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* [[Convergence of Iterative Methods]] | * [[Convergence of Iterative Methods]] | ||
* [[Spectral Radius of Matrix]] | * [[Spectral Radius of Matrix]] | ||
− | |||
* [[Sparse Matrix]] | * [[Sparse Matrix]] | ||
− | |||
|| | || | ||
* [[Length of Vectors]] | * [[Length of Vectors]] | ||
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* [[Eigenvectors 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 | |Week 6 | ||
|| | || | ||
− | + | Section 2.6: Conjugate Gradient (CG) Method | |
|| | || | ||
* [[Conjugate Gradient Method]] | * [[Conjugate Gradient Method]] | ||
− | * [[Symmetric Positive Definite Matrix | + | * [[Symmetric Positive Definite Matrix]] |
− | * [[Construction of 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]] | * [[Nonlinear System of Equations]] | ||
* [[Taylor's Theorem for Multi-Variate Vector Valued Functions]] | * [[Taylor's Theorem for Multi-Variate Vector Valued Functions]] | ||
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|Week 7 | |Week 7 | ||
|| | || | ||
− | Sections 3.1 | + | Sections 3.1: Data and Interpolating Functions |
|| | || | ||
− | |||
* [[Lagrange Basis Functions]] | * [[Lagrange Basis Functions]] | ||
− | |||
− | |||
* [[Newton's Divided Differences]] | * [[Newton's Divided Differences]] | ||
* [[Properties of Newton's Divided Differences]] | * [[Properties of Newton's Divided Differences]] | ||
− | * [[Newton's Form of the Interpolation Polynomials | + | || |
− | + | * [[Fundamental Theorem of Algebra]] | |
− | * [[Interpolation Error | + | * [[Rolle's Theorem]] |
− | * [[ | + | || |
+ | * Properties of Lagrange Basis Functions | ||
+ | * Lagrange Form of the Interpolation Polynomials | ||
+ | * Properties of Newton's Divided Differences | ||
+ | * Newton's Form of the Interpolation Polynomials | ||
+ | |- | ||
+ | |Week 7 | ||
+ | || | ||
+ | Section 3.2: Interpolation Error and Runge Phenomenon | ||
+ | || | ||
+ | * [[Interpolation Error]] | ||
+ | * Interpolation [[Error Analysis]] | ||
* [[Runge Phenomenon]] | * [[Runge Phenomenon]] | ||
* [[Chebyshev Polynomial]] | * [[Chebyshev Polynomial]] | ||
− | * [[Error Estimates for Chebyshev Interpolation | + | * [[Error Estimates]] for Chebyshev Interpolation |
|| | || | ||
* [[Fundamental Theorem of Algebra]] | * [[Fundamental Theorem of Algebra]] | ||
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|Week 8 | |Week 8 | ||
|| | || | ||
− | + | Section 3.4: Cubic Splines | |
|| | || | ||
* [[Cubic Splines]] | * [[Cubic Splines]] | ||
− | * [[ | + | || |
− | * [[ | + | * [[One-Sided Limits]] |
− | * [[ | + | * [[Continuity of Functions]] |
− | * [[Properties of Cubic Spline Interpolation | + | * [[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]] | * [[Bezier Curves]] | ||
− | * [[Bezier Curve and Fonts | + | || |
− | + | * [[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]] | * [[Least Square Method]] | ||
− | |||
− | |||
|| | || | ||
* [[One-Sided Limits]] | * [[One-Sided Limits]] | ||
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* [[Extremum Values of Multivariate Quadratic Functions]] | * [[Extremum Values of Multivariate Quadratic Functions]] | ||
|| | || | ||
− | * | + | * Least Square Method for Solving Inconsistent System of Linear Equations] |
+ | * Basic Properties of Least Square Solutions | ||
+ | |- | ||
+ | |Week 9 | ||
+ | || | ||
+ | Section 4.2: Mathematical Models and Data Fitting | ||
+ | || | ||
+ | * [[Curve Fitting]] | ||
+ | * [[Statistical Modeling]] | ||
+ | || | ||
+ | * [[Linear Spaces]] | ||
+ | * [[Basis Functions]] | ||
+ | * [[Product Rule for Vector Valued Multivariate Functions]] | ||
+ | * [[Chain Rule for Vector Valued Multivariate Functions]] | ||
+ | || | ||
+ | * Least square method for curve fitting and statistical modeling | ||
+ | * Survey of Models: linear model, periodic model, exponential models, logistic model, etc | ||
|- | |- | ||
|Week 9 | |Week 9 | ||
|| | || | ||
− | + | Section 4.5: Nonlinear Least Square Fitting | |
|| | || | ||
− | |||
− | |||
− | |||
− | |||
− | |||
* [[Taylor's Theorem for Vector Valued Multivariate Functions]] | * [[Taylor's Theorem for Vector Valued Multivariate Functions]] | ||
* [[Gauss-Newton Method]] | * [[Gauss-Newton Method]] | ||
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|Week 10 | |Week 10 | ||
|| | || | ||
− | + | Section 5.1: Numerical Differentiation | |
|| | || | ||
* [[Numerical Differentiation]] | * [[Numerical Differentiation]] | ||
− | * [[Finite | + | * [[Finite Difference]] (FD) |
− | * | + | * [[Undetermined Coefficient Method]] |
− | * | + | * [[Extrapolation Technique]] |
− | * | + | || |
− | + | * [[Taylor's Theorem]] | |
− | + | * [[Interpolation Error Estimates]] | |
+ | * [[Properties of Definite Integrals]] | ||
+ | || | ||
+ | * Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis | ||
+ | * FD approximations of 2nd order Derivatives and Their Error Analysis | ||
+ | * Undetermined Coefficient Method for FD Approximation | ||
+ | * Extrapolation Technique for Improving the Order of Approximation | ||
+ | |- | ||
+ | |Week 10 | ||
+ | || | ||
+ | Section 5.2: Numerical Integration: Newton-Cotes Formulas | ||
+ | || | ||
+ | * [[Newton-Cotes]] | ||
* [[Midpoint rule]] | * [[Midpoint rule]] | ||
* [[Trapezoid rule]] | * [[Trapezoid rule]] | ||
* [[Simpson's rule]] | * [[Simpson's rule]] | ||
− | |||
* [[Error Analysis based on Interpolation Errors]] | * [[Error Analysis based on Interpolation Errors]] | ||
− | * [[ | + | * [[Quadrature Rules]] |
* [[Composite Quadrature Rules]] | * [[Composite Quadrature Rules]] | ||
− | |||
− | |||
− | |||
|| | || | ||
* [[Taylor's Theorem]] | * [[Taylor's Theorem]] | ||
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* [[Properties of Definite Integrals]] | * [[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 | |Week 11 | ||
|| | || | ||
− | + | Section 5.4: Adaptive Numerical Integration | |
|| | || | ||
* [[Adaptive Numerical Integration]] | * [[Adaptive Numerical Integration]] | ||
* [[Implementation of Adaptive Numerical Integration Techniques]] | * [[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]] | * [[Gauss Quadrature Formulas]] | ||
* [[Orthogonal Polynomials]] | * [[Orthogonal Polynomials]] | ||
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* [[Substitution Methods]] for definite integrals | * [[Substitution Methods]] for definite integrals | ||
|| | || | ||
− | |||
− | |||
− | |||
* Motivation and difficulties with straightforward approach | * Motivation and difficulties with straightforward approach | ||
* Legendre polynomials and their basic properties | * Legendre polynomials and their basic properties | ||
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|Week 12 | |Week 12 | ||
|| | || | ||
− | + | Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT) | |
|| | || | ||
− | |||
* [[Fourier Series]] | * [[Fourier Series]] | ||
* [[Discrete Fourier Transform]] (DFT) | * [[Discrete Fourier Transform]] (DFT) | ||
− | |||
* [[Inverse Discrete Fourier Transform]] | * [[Inverse Discrete Fourier Transform]] | ||
− | |||
* [[Fast Fourier Transform (FFT)]] | * [[Fast Fourier Transform (FFT)]] | ||
− | + | || | |
− | * [[Discrete Cosine 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 11.1: Discrete Cosine Transform (optional) | ||
+ | || | ||
+ | * [[Discrete Cosine Transform]] | ||
* [[Discrete Cosine Transform]](DCT) | * [[Discrete Cosine Transform]](DCT) | ||
− | |||
− | |||
− | |||
− | |||
− | |||
|| | || | ||
* [[Complex Numbers]] | * [[Complex Numbers]] | ||
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* DCT of 2-Dimensional Functions | * DCT of 2-Dimensional Functions | ||
* Interpolation Theorem for 2-Dimensional DCT | * Interpolation Theorem for 2-Dimensional DCT | ||
− | + | |- | |
+ | |Week 12 | ||
+ | || | ||
+ | Section 11.2: Image Compression (optional) | ||
+ | || | ||
+ | * [[Image Compression] | ||
+ | * [[Quantization]] | ||
+ | * [[Image Compression]] | ||
+ | * [[Image Decompression]] | ||
+ | || | ||
+ | * [[Complex Numbers]] | ||
+ | * [[Complex Variables]] | ||
+ | * [[Integration by Parts]] | ||
+ | * [[Convergence of Sequences]] | ||
+ | * [[Convergence of Series]] | ||
+ | || | ||
* Digital Gray scale images and color color images | * Digital Gray scale images and color color images | ||
* RGB format | * RGB format | ||
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|Week 13 | |Week 13 | ||
|| | || | ||
− | + | Section 12.1: Power Iteration Methods | |
|| | || | ||
* [[Power Iteration Methods]] | * [[Power Iteration Methods]] | ||
− | |||
− | |||
* [[Inverse Power Iteration]] | * [[Inverse Power Iteration]] | ||
* [[Inverse Power Iteration with Shift]] | * [[Inverse Power Iteration with Shift]] | ||
* [[Rayleigh Quotient Iteration]] | * [[Rayleigh Quotient Iteration]] | ||
− | + | || | |
− | * [[QR Algorithm for Computing Eigenvalues | + | * [[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]] | * [[Orthogonal Matrices]] | ||
* [[QR-Factorization]] | * [[QR-Factorization]] | ||
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|Week 14 | |Week 14 | ||
|| | || | ||
− | + | Section 12.2: QR Algorithm for Computing Eigenvalues | |
|| | || | ||
− | |||
* [[Upper Hessenberg Form]] (UHF) | * [[Upper Hessenberg Form]] (UHF) | ||
* [[Householder Reflector]] | * [[Householder Reflector]] |
Revision as of 07:28, 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 |
| ||
Week 1 |
Section 1.1: Fixed-Point Iteration |
| ||
Week 2 |
Section 1.2: Fixed-Point Iteration |
|
| |
Week 2 |
Section 1.3: Limits of Accuracy: Conditioning of Problems |
| ||
Week 3 |
Section 1.4: Newton's Method |
|
| |
Week 3 |
Section 1.5 Root-Finding Without Derivatives |
| ||
Week 4 |
Section 2.1 Solve Systems of Linear Equations: Gaussian Elimination |
| ||
Week 4 |
Section 2.2 Solve Systems of Linear Equations: LU Decomposition |
| ||
Week 5 |
Section 2.3 Error Analysis for Solution of Ax=b |
|
| |
Week 5 |
Section 2.5: Iterative Methods for Solving Ax=b |
| ||
Week 6 |
Section 2.6: Conjugate Gradient (CG) Method |
| ||
Week 6 |
Section 2.7: Nonlinear System of Equations |
| ||
Week 7 |
Sections 3.1: Data and Interpolating Functions |
| ||
Week 7 |
Section 3.2: Interpolation Error and Runge Phenomenon |
|
| |
Week 8 |
Section 3.4: Cubic Splines |
| ||
Week 8 |
Section 3.5: Bezier Curves |
| ||
Week 8 |
Section 4.1: Least Square Method |
| ||
Week 9 |
Section 4.2: Mathematical Models and Data Fitting |
| ||
Week 9 |
Section 4.5: Nonlinear Least Square Fitting |
| ||
Week 10 |
Section 5.1: Numerical Differentiation |
| ||
Week 10 |
Section 5.2: Numerical Integration: Newton-Cotes Formulas |
| ||
Week 10 |
Section 5.3: Numerical Integration: Romberg's Technique |
| ||
Week 11 |
Section 5.4: Adaptive Numerical Integration |
|
| |
Week 11 |
Section 5.5: Gauss Quadrature Formulas |
|
| |
Week 12 |
Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT) |
| ||
Week 12 |
Section 11.1: Discrete Cosine Transform (optional) |
| ||
Week 12 |
Section 11.2: Image Compression (optional) |
|
| |
Week 13 |
Section 12.1: Power Iteration Methods |
| ||
Week 13 |
Section 12.2: QR Algorithm for Computing Eigenvalues |
| ||
Week 14 |
Section 12.2: QR Algorithm for Computing Eigenvalues |
|