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	Section 0.2 & 1.1	Loss of Significant Digits Nested Multiplication for Evaluating Polynomials Machine Representation of Real Numbers Loss of Significant Digits in Numerical Computing Review of Taylor's Theorem Bisection Method Bisection Method and Implementation Brief Introduction to Matlab	Binary Number System Taylor's Theorem Intermediate Value Theorem	(To be Entered)
Week 2	Sections 1.2 & 1.3	Fixed-Point Iteration Geometric Interpretation Convergence of Fixed Point Iterations Order of Convergence of Iterative Methods Limits of Accuracy: Conditioning of Problems Wilkinson Polynomial Sensitivity Analysis of Root-Finding Error Magnification Factor for Solution of Equations	Limit of Sequences Solution Multiplicity of Equations	(TBD)
Week 3	Sections 1.4 & 1.5	Newton's Method Algebraic and Geometric Interpretation of Newton's method Error Analysis for Newton's Method Based on Taylor's Theorem Newton's Method as a Fixed Point Iteration Modified Newton's Method and its Rate of Convergence 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	(TBD)
Week 4	Sections 2.1 & 2.2	Solve Systems of Linear Equations: Gaussian Elimination Gaussian Elimination and its Operation Counts Gaussian Elimination with Pivoting Implementation of Gauss Elimination Solve System of Linear Equations: LU Decomposition Matrices for Elementary Row Operations Gauss Elimination as Matrix Products Advantages of Solutions by LU Decomposition	Matrix-Matrix Products Matrix-Vector Products Inverse Matrix Elementary Row Operations	(TBD)
Week 5	Sections 2.3 & 2.5	Error Analysis for Solution of Ax=b Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms Error Analysis for Solution of Ax=b Error Magnification Factor and Condition Number of Matrix Iterative Methods for Solving Ax=b Jacobi Method Gauss-Seidel Method Successive-Over-Relaxation (SOR) Method Convergence of Iterative Methods Spectral Radius of Matrix Convergence of General Iterative Method for Solving System of Linear Equations Sparse Matrix Comparison of Gauss Elimination and Iterative Methods	Length of Vectors Eigenvalues of a Matrix Eigenvectors of a Matrix	(TBD)
Week 6	Sections 2.6 & 2.7	Conjugate Gradient Method Symmetric Positive Definite Matrix and Properties Construction of Conjugate Gradient (CG) Method Properties of CG Method Preconditioning for CG Method Nonlinear System of Equations Taylor's Theorem for Multi-Variate Vector Valued Functions Newton's Method Broyden's Method	Scalar Product of Vectors Determinant of a Matrix Eigenvalues of a Matrix Quadratic Polynomials of n-variables Partial Derivatives Gradients Chain Rule for Partial Derivatives	(TBD)
Week 7	Sections 3.1 & 3.2	Data and Interpolating Functions 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 and Runge Phenomenon Interpolation Error Analysis Runge Phenomenon Chebyshev Polynomial Error Estimates for Chebyshev Interpolation	Fundamental Theorem of Algebra Rolle's Theorem	(TBD)
Week 8	Sections 3.4, 3.5, & 4.1	Cubic Splines Cubic Splines Construction of Cubic Splines for Interpolation End Conditions Properties of Cubic Spline Interpolation Bezier Curves Bezier Curve and Fonts Least Square Method Least Square Method for Solving Inconsistent System of Linear Equations Basic Properties of Least Square Solutions	One-Sided Limits Continuity of Functions Indefinite Integrals Extremum Values of Multivariate Quadratic Functions	(TBD)
Week 9	Sections 4.2 & 4.5	Mathematical Models and Data Fitting Least square method for curve fitting and statistical modeling Survey of Models: linear model, periodic model, exponential models, logistic model, etc Nonlinear Least Square Fitting Taylor's Theorem for Vector Valued Multivariate Functions Gauss-Newton Method Levenberg-Marquardt Method	Linear Spaces Basis Functions Product Rule for Vector Valued Multivariate Functions Chain Rule for Vector Valued Multivariate Functions	(TBD)
Week 10	Sections 5.1, 5.2, & 5.3	Numerical Differentiation Finite difference (FD) Approximations of 1st order Derivative and Their Error Analysis FD approximations of 2nd order Derivatives and Their Error Analysis Undetermined Coefficient Method for FD Approximation Extrapolation Technique for Improving the Order of Approximation Numerical Integration: Newton-Cotes Formulas Midpoint rule Trapezoid rule Simpson's rule Error Analysis based on Taylor's Theorem Error Analysis based on Interpolation Errors Degree of Precision of Quadrature Rules Composite Quadrature Rules Numerical Integration: Romberg's Technique Motivation, construction and implementation of Romberg's Technique.	Taylor's Theorem Interpolation Error Estimates Properties of Definite Integrals	(TBD)
Week 11	Sections 5.4 & 5.5	Adaptive Numerical Integration Implementation of Adaptive Numerical Integration Techniques Gauss Quadrature Formulas Orthogonal Polynomials Legendre polynomials Gauss Quadrature Rule	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? 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	Sections 10.1, 11.1, & 11.2	Discrete Fourier Transform and Fast Fourier Transform (FTT) Fourier Series Discrete Fourier Transform (DFT) Matrix Form of Discrete Fourier Transform Inverse Discrete Fourier Transform DFT and Trigonometric Interpolation Fast Fourier Transform (FFT) Discrete Cosine Transform(optional) Discrete Cosine Transform(DCT) Image Compression(optional) Quantization Image Compression Image Decompression	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 Digital Gray scale images and color color images RGB format YCbCr (or YUV) format Convertion between RGB and YUV formats
Week 13	Sections 12.1 & 12.2	Power Iteration Methods Power Iteration Methods Convergence of Power Iteration Methods Inverse Power Iteration Inverse Power Iteration with Shift Rayleigh Quotient Iteration 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	Sections 12.2	QR Algorithm for Computing Eigenvalues Upper Hessenberg Form (UHF) Householder Reflector	Matrices for Orthogonal Projection Matrices for Reflection Block Matrices Similar Matrices	Convert a matrix into UHF by Householder reflectors