Math 441 - Introduction to Numerical Analysis

Class: TuTh 1:00 - 2:30PM SOND 208 (8/28-12/10/2024), Instructor: Bedrich Sousedik

We will learn about methods and analysis of techniques used to resolve continuous mathematical problems on the computer. Topics of this course include numerical linear algebra, interpolation, numerical differentiation and integration, solution of nonlinear equations, acceleration of convergence and numerical treatment of differential equations.

Prerequisites: You must complete CMSC201 and Math225 and Math251 and Math301 with a grade of C or better before you can enroll in this class.

The detailed class syllabus (pdf) is here. However, look below to see the actual progress of the class.

Howto get started with Matlab at UMBC document is here, and some basic software tutorials are here.

How Many Decimals of Pi Do We Really Need?

8/29 Introduction. Sources of error.

9/3 Chapter 0: IEEE floating-point arithmetic.

9/5 Chapter 0: IEEE floating-point arithmetic, Loss of significance (idea). Horner's method. Hw 1 (due 9/12).

9/10 1.1 Bisection method.

9/12 1.2 Fixed-point iteration (FPI), 1.4 Newton's method. Hw 2 (due 9/24).

9/17 2.7.1 Newton's method for systems.

9/19 Matlab tutorial: watch.

9/24 2.7.1 Newton's method for systems. Ch 3: Polynomial interpolation theory. 3.1.1 (and 3.1.3) Lagrange polynomials. Hw 3 (due 10/3).

9/26 3.1.2 Newton's divided differences. 3.2.1 Interpolation error formula. 3.2.3 Runge phenomenon.

10/1 3.3 Chebyshev interpolation. 3.4 Cubic splines. Hw 4 (due 10/10).

10/3 Ch 5: Numerical integration. Newton-Cotes integration formulas.

10/8 Composite formulas. Numerical differentiation.

10/10 5.1.3 Extrapolation. 5.3 Romberg integration.

10/15 Test #1. (Ch. 0, 1 and 3).

10/17 5.5 Gaussian quadrature. Chapter 6: Numerical methods for ODEs. Hw 5 (due 10/31).

10/22 6.1 and 6.2.1 (Explicit) Euler's method and its analysis. Hw 6 (due 11/5).

10/24 6.2.2 - 6.2.3, 6.4 Taylor-series, explicit trapezoid, and midpoint methods. Hw 7 (due 11/7).

10/29 Runge-Kutta method. 6.3.1 Higher-order equations. 6.7.1-6.7.2 Multistep methods. Hw 8 (due 11/14).

10/31 6.6 Implicit methods and stiff equations (via Newton's method + illustration by Matlab code).

11/5 2.1 Gausssian elimination, 2.2 LU factorization. 2.3.2 Swamping. 2.4 The PA=LU factorization.

11/7 2.5 Iterative methods (Jacobi and Gauss-Seidel), preconditioning.

11/12 2.3.1 Error magnification and condition number (for linear systems). An example of the PA=LU factorization. Hw 9 (due 12/3).

11/14 4.1 Least squares and the normal equations.

11/19 4.3.1 Gram-Schmidt orthogonalization.

11/21 Test #2 (Ch. 5, 6, and Sections 2.1-2.2).

11/26 4.3.3 Householder reflectors. Hw 10 (due 12/10).

11/28 Thanksgiving (no class).

12/3 Finite difference methods for PDEs.

12/5 Introduction to finite element method.

12/10 Projects, review.

Tuesday 12/17, 3:30-5:30pm Final.