MAT3633

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	0.2 and 1.1	Loss of significant digits Bisection Method Brief introduction to matlab	binary number system; Taylor's theorem; intermediate value theorem	Nested multiplication for evaluating polynomials Machine representation of real numbers Loss of significant digits in numerical computing Review of Taylor's Theorem Bisection method and implementation
Week.2	1.2 and 1.3	Fixed-Point Iteration Limits of Accuracy: Conditioning of problems	limit of sequences multiplicity of solution of equations.	Geometric interpretation Convergence of fixed point iterations Order of convergence of iterative methods Wilkinson polynomial and other examples Sensitivity analysis of root-finding Error magnification factor for solution of equations
Week.3	1.4 and 1.5	Newton's Method Root-Finding without Derivatives	Remainder of Taylor's series intermediate value theorem.	Algebraic and geometric interpretation of Newton's method Error analysis for Newton's method based on Taylor's theorem Newton's method as a fixed point iteration Modified Newton's method and its rate of convergence Secant Method and its convergence, Method of False Position, Muller's Method: Stopping criteria for iterative methods
Week.4	2.1 and 2.2	Solve Systems of Linear Equations: Gaussian Elmination Solve System of Linear Equations: LU Decomposition	Matrix-matrix products and matrix-vector products inverse matrix elementary row operations product and inverse of matrices for elementary row operations.	Gaussian elimination and its operation counts Gaussian elimination with pivoting Implementation of Gauss elimination Matrices for elementary row operations Gauss elimination as matrix products Advantages of solutions by LU decomposition
Week.5	2.3 and 2.4	Error Analysis for Solution of Ax=b Iterative Methods for Solving Ax=b	Length of vectors eigenvalue and eigenvectors of matrix	various norms for vectors and matrices: compatibility of vector and matrix norms. Error Analysis for the solution of Ax=b Error magnification factor and condition number of matrix Jacobi method, Gauss-Seidel method, Successive-Over-Relaxation (SOR) method Convergence of Jacobi Method, GS method and SOR method. spectral radius of matrix convergence of general iterative method for solving system of linear equations, Sparse Matrix Comparison of Gauss Elimination and iterative methods
Week.6	2.6 and 2.7	Conjugate Gradient Method Nonlinear System of Equations	scalar product of vectors determinant and eigenvalues of matrix quadratic polynomials of n-variables partial derivatives and gradients chain rule for partial derivatives.	Symmetric positive definite matrix and properties Construction of Conjugate Gradient (CG) Method Propertise of CG Method Preconditioning for CG method Taylor's Theorem for multi-variate vector valued functions: Newton's Method: Broyden's Method
Week.7	3.1 and 3.2	Data and Interpolating Functions Interpolation Error and Runge Phenomenon Chebyshev interpolation	Fundamental theorem of algebra Rolle's theorem.	Lagrange Basis Functions: Properties of Lagrange basis functions: Lagrange form of the interpolation polynomials Newton's Divided Differences: Properties of Newton's divided differences: Newton's Form of the interpolation polynomials Interpolation error analysis Runge phenomenon Chebyshev Polynomial Error estimates for Chebyshev interpolation
Week.8	3.4, 3.5 and 4.1	Cubic Splines Bezier Curves Least Square Method	one-sided limits continuity of functions indefinite integrals extremum values of multivariate quadratic functions.	Cubic splines construction of cubic splines for interpolation end conditions properties of cubic spline interpolation Bezier Curve and fonts Least square method for solving inconsistent system of linear equations. Basic properties of least square solutions:
Week.9	4.2 and 4.5	Mathematical Models and Data Fitting Nonlinear Least Square Fitting	linear spaces, basis functions product rule and chain rule for vector valued multivariate functions.	Least square method for curve fitting and statistical modeling. Survey of Models: linear model, periodic model, exponential models, logistic model, etc Taylor's theorem for vector valued multivariate functions. Gauss-Newton Method Levenberg-Marquardt Method
Week.10	5.1, 5.2 and 5.3	Numerical Differentiation Numerical Integration: Newton-Cotes Formulas Numerical Integration: Romberg's Technique	Taylor's theorem interpolation error estimates properties of definite inetgrals	Finite difference (FD) approximations of 1st order derivative and their error analysis FD approximations of 2nd order derivatives and their error analysis Undetermined coefficient method for FD approximation Extropolation technique for improving the order of approximation Midpoint rule, trapezoid rule and Simpson's rule; Error analysis based on Taylor's Theorem and interpolation errors Degree of precision of quadrature rules Composite quadrature rules Motivation, construction and implementation of Romberg's technique.
Week.11	5.4 and 5.5	Adaptive Numerical Integration Gauss Quadrature Formulas	long divisions changing variables for definite integrals	How to estimate the error on a subinterval How to mark subintervals to be further refinement? Implementation of adaptive numerical integration techniques. Motivation and difficulties with straightforward approach. Orthogonal polynomials, Legendre polynomials and their basic properties; Gauss quadrature rule based on Legendre polynomials Degree of precision of Gauss Quadrature Gauss quadrature formula on general interval and composite Gauss rules
Week.12	10.1 and 11.1	Discrete Fourier Transform and FFT Discrete Cosine Transform (optional) Image Compression (optional)	complex numbers and complex variables integration by parts convergence of sequences and series.	Fourier Series, Discrete Fourier Transform Matrix Form of Discrete Fourier Transform: Inverse Discrete Fourier Transform: DFT and Trigonometric interpolation Algorithm for computing DFT: Fast Fourier Transform (FFT) Discrete Cosine Transform (DCT), DCT and Interpolation by Cosine Functions Relation between DFT and DCT: Fourier Transform of 2-Dimensional Functions DCT of 2-Dimensional Functions: Interpolation Theorem for 2-Dimensional DCT Digital Gray scale images and color color images: RGB format: YCbCr (or YUV) format: Convertion between RGB and YUV formats: Quantization, Image Compression and Decompression
Week.13	12.1 and 12.2	Power Iteration Methods QR Algorithm for Computing Eigenvalues	properties of eigen values and eigenvectors Gram-Schmidt orthogonalization	Power iteration and its rate of convergence. Inverse Power Iteration, Inverse Power Iteration with Shift Rayleigh Quotient Iteration Definition and basic properties of orthogonal matrices: QR-Factorization based on Gram-Schmidt Orthogonalization: Normalized Simultaneous Iteration (NSI). Unshifted QR Algorithm: Shifted QR Algorithm:
Week.14	12.2	Algorithm for Computing Eigenvalues: Speed up of QR-algorithm:	matrices for orthogonal projection and reflection block matrices and their products similar matrices.	Upper Hessenberg form (UHF) Householder Reflector Convert a matrix into UHF by Householder reflectors Topics List B Wiki Format 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