CSci 5302 - Analysis of Numerical Algorithms
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## Last Revised 03/29/22 (provisional schedule, dates subject to change)

Textbooks

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 WEEK DATE TOPICS 1 1/19 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 1/24 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 26 3 1/31 ... 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 2/07 ... Linear equations continued Sensitivity and Conditioning. Methods: Gaussian Elimination and Back Substitution. 5 2/14 Nonlinear Equations -- Systems (H: 5.4-5.6, [A&G: Chap 9]) HW2 due Mon Feb 14. Partial Derivatives: gradients, Jacobians. Fixed Point Methods. Newton's method. 6 2/21 Polynomial Interpolation (H: 7.1-7.3. A&G: 7.1-7.3) Existence and Uniqueness Methods: undetermined coefs. 6 2/22 Enhanced Homework I (out Monday, due Tuesday). 7 2/28 ... Interpolation continued Methods: Lagrange polynomials. divided differences. ... ... Splines (H: 7.4. A&G: 11.3, 11.6) Piecewise polynomial interpolation: splines. Two-dimensional interpolation/splines (from notes). === 3/07 SPRING BREAK Romp in the virtual sand. 8 3/14 Linear Least Squares (H: 3.1-3.2. A&G: 6.1) HW3 due Fri Mar 04. Existence and Uniqueness Normal equations. 9 3/21 Unconstrained Optimization Problems (H: 6.1-6.5. A&G: Chap 9, 3.5,9.2-9.3) First & second order conditions for optimality in 1D, gradients, Hessians. One-dimension: Golden Section Search. Existence and Uniqueness: coercive and convex functions Conditioning 10 3/28 Unconstrained Optimization Problems (continued) Conditioning First & second order conditions for optimality (multi-dim): gradients, Hessians. Multi-dimensional methods: steepest descent, Newton, BFGS. Line Search 11 4/04 Constrained Optimization Problems (H: 6.7. A&G: 9.3) HW4 due Mon Apr 04. Conditioning First & second order conditions for optimality (multi-dim): gradients, Hessians. Equality constraints, inequality constraints First Order conditions for optimality: KKT conditions 12 4/11 Enhanced Homework II (out Monday, due Tuesday). 12 4/11 Constrained Optimization Problems (continued) Penalty & barrier methods. (maybe dual ascent methods?) Linear Programs. 13 4/18 Numerical Integration and Quadrature (H: 8.1-8.4, 8.7. A&G: 15.1-15.2,15.4-15.5) HW5 due Mon Apr 18 Existence and Uniqueness Numerical Quadrature (equally-spaced points). Adaptive Quadrature Richardson Extrapolation: Romberg. 14 4/25 Initial Val. Probs. for Ord. Diff. Equs (H: 9.1-9.3. A&G: 16.1-16.6.) Existence and Uniqueness, stability of solutions Linear ODEs Methods: Euler, Backward Euler. Runge-Kutta Methods Accuracy and Stability: local vs global truncation error 15 5/02 review HW6 due Mon May 02. Final 5/09 Enhanced Homework III (out Monday, due Tuesday).