CSci 5302 - Analysis of Numerical Algorithms
index.php

Last Revised 01/18/23 (provisional schedule, dates subject to change)

Textbooks

-->
 WEEK DATE TOPICS 1 1/18 Scientific Computing - Goals and Fundamentals (H: Chap 1. A&G: 1.1-1.2, 2.1-2.3) floating point arithmetic relative error conditioning vs stability: backward error vs forward error truncation error vs rounding error. 2 1/23 Scalar Nonlinear Equs (H: 5.1 (5.2), 5.4. A&G: 3.1-3.4 ) Existence and Uniqueness Sensitivity and Conditioning. backward error vs forward error Methods: Bisection, Newton, Secant. HW1 due Wed Jan 25 3 1/30 ... Nonlinear equations continued General Fixed Point Methods. Convergence Rates and Stopping Criteria ... ... Systems of Linear Equations (H: 2.1-2.4.7. A&G: Chap 9, 9.1) Existence and Uniqueness Norms. 4 2/06 ... Linear equations continued HW2 due Mon Feb 06. Sensitivity and Conditioning. Methods: Gaussian Elimination and Back Substitution. 5 2/13 Nonlinear Equations -- Systems (H: 5.4-5.6, [A&G: Chap 9]) Partial Derivatives: gradients, Jacobians. Fixed Point Methods. Newton's method. 5 2/13 Enhanced Homework I (due Monday). 6 2/20 Polynomial Interpolation (H: 7.1-7.3. A&G: 7.1-7.3) Existence and Uniqueness Methods: undetermined coefs. Methods: Lagrange polynomials. divided differences. 7 2/27 Splines (H: 7.4. A&G: 11.3, 11.6) HW3 due Mon Feb 27. Piecewise polynomial interpolation: splines. Two-dimensional interpolation/splines (from notes). === 3/06 SPRING BREAK Romp in the virtual sand. 8 3/13 Linear Least Squares (H: 3.1-3.2. A&G: 6.1) Existence and Uniqueness Normal equations. Projections. 9 3/20 Unconstrained Optimization Problems (H: 6.1-6.5. A&G: Chap 9, 3.5,9.2-9.3) Existence and Uniqueness: coercive and convex functions Conditioning First & second order conditions for optimality in 1D, gradients, Hessians. One-dimension: Golden Section Search. HW4 due Mon Mar 20. Line Search 10 3/22 Enhanced Homework II (due Wednesday). 10 3/27 ... Unconstrained Optimization continued (multi-dimensional) First & second order conditions for optimality (multi-dim): gradients, Hessians. Conditioning Multi-dimensional methods: steepest descent, Newton, BFGS. 11 4/03 Constrained Optimization Problems (H: 6.7. A&G: 9.3) Equality constraints, inequality constraints First Order conditions for optimality: KKT conditions (Penalty & barrier methods.) Linear Programs. 12 4/10 Numerical Integration and Quadrature (H: 8.1-8.4, 8.7. A&G: 15.1-15.2,15.4-15.5) Existence and Uniqueness Numerical Quadrature (equally-spaced points). Adaptive Quadrature Richardson Extrapolation: Romberg. 13 4/17 Initial Val. Probs. for Ord. Diff. Equs (H: 9.1-9.3. A&G: 16.1-16.6. HW5 due Mon Apr 17. Existence and Uniqueness, stability of solutions Linear ODEs Methods: Euler, Backward Euler. 14 4/24 ... ODEs continued Accuracy and Stability: local vs global truncation error Runge-Kutta Methods Review 15 4/24 Last day of class XX 5/01 Enhanced Homework III (due Monday).