# Math 151B: General Course Outline

## Catalog Description

151B. Applied Numerical Methods. (4) Lecture, three hours; discussion, one hour. Requisite: course 151A. Introduction to numerical methods with emphasis on algorithms, analysis of algorithms, and computer implementation. Numerical solution of ordinary differential equations. Iterative solution of linear systems. Computation of least squares approximations. Discrete Fourier approximation and the fast Fourier transform. Matlab programming. Letter grading.

Assignments Homework assignments in the course consist of both theoretical and computational work. The computational work is completed using matlab.

General Information. Math 151AB is the main course sequence in numerical analysis, important for all of the applied mathematics majors. Mathematics majors who graduate and go into industry often find Math 151AB to be the most useful course for their work.

Math 151A is offered each term, and Math 151B is offered Winter and Spring.

## Textbook

R. Burden and J. Faires, Numerical Analysis, 10th Ed., Brooks/Cole.

Homework assignments in the course consist of both theoretical and computational work. The computational work is completed using matlab.

AS: The topics of stiffness and of absolute stability are not well presented in Burden and Faires. Other textbooks should be consulted.

DLS: The matrix form of the discrete least squares problem is not presented in Burden and Faires. Other textbooks should be consulted.

Outline update: J. Qin, 06/2015

NOTE: While this outline includes only one midterm, it is strongly recommended that the instructor considers giving two. It is difficult to schedule a second midterm late in the quarter if it was not announced at the beginning of the course.

## Schedule of Lectures

Lecture Section Topics

1

5.1

Initial value problem

2

5.2

Euler's method

3

5.3, 5.10

Higher-order Taylor methods. Error analysis of one-step methods

4

5.10, 5.4

Stability of one-step methods. Taylor Theorem in two variables

5

5.4

Runge-Kutta methods

6

5.4

Butcher tableau. Design a Runge-Kutta method

7

5.5

Runge-Kutta-Fehlberg method.

8

5.6

9

5.6, 5.10

Predictor-corrector methods. Analysis of general multistep methods

10

5.10, 5.11

Stability of multistep methods. Stiff differential equations

11

5.11

Region of absolute stability

12

5.9

High-order differential equations. Systems of differential equations

13

11.1

Boundary value problems. Linear shooting method

14

11.2, 11.3

Nonlinear shooting method. Finite difference methods for linear BVP

15

Midterm

16

11.4

Finite-difference methods for nonlinear BVP

17

10.1, 10.2

Solving nonlinear systems of equations. Newton's method

18

10.3

Quasi-Newton method - Broyden's method

19

10.4

Steepest descent method

20

10.5

Homotopy and continuation methods

21

9.1, 9.2

Linear algebra, Eigenvalues, orthogonal matrices and similarity transformations

22

9.3

Power method. Inverse Power method

23

9.4

Householder's transformation. Householder's method

24

9.5

QR factorization. QR algorithm

25

8.1

Discrete least squares approximation. Linearly independent functions

26

8.2

Orthogonal polynomials and least squares approximation

27

8.5

Continuous and discrete trigonometric polynomial approximation.

28

8.6

Fast Fourier transform I

29

8.6

Fast Fourier transform II