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Math 151A: General Course Outline

Catalog Description

151A. Applied Numerical Methods. (4) Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B, 115A, Program in Computing 10A. Introduction to numerical methods with emphasis on algorithms, analysis of algorithms, and computer implementation issues. Solution of nonlinear equations. Numerical differentiation, integration, and interpolation. Direct methods for solving linear systems. 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.

* This topic is not in Burden and Faires. It can be found in Cheney-Kincaid, Numerical Mathematics and Computing, Brooks/Cole, section 4.2.

Topics in parenthesis are optional and can be included under the discretion of the instructor.

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

1.2

General course overview and machine numbers

2

1.2

Errors

3

1.3

Algorithms and convergence

4

2.1

The bisection method

5

2.2

Fixed-point iteration

6

2.3

Newton's method

7

2.3

Secant method, and method of False Position

8

2.4

Convergence order. Multiple roots

9

2.5

Accelerating convergence

10

2.6

Zeros of polynomials. Horner's method

11

2.6, 3.1

Deflation and Lagrange polynomials

12

3.1, 3.2

Lagrange polynomials and Neville's method

13

3.3

Divided differences

14

3.3

Interpolation nodes and finite difference

15

Midterm

16

3.4

Hermite Interpolation

17

3.5

Cubic spline interpolation

18

4.1

Forward/backward difference

19

4.1

Finite-difference formulas

20

4.2, 4.3.

Richardson's extrapolation. Interpolation based numerical integration

21

4.3, 4.4

Newton-Cotes formulas. Composite integration formulas

22

4.5

Romberg integration

23

4.7

Gaussian quadrature

24

6.1

Solving linear systems

25

6.2

Pivoting

26

6.6

Special types of matrices

27

7.1, 7.3

Review of matrix algebra. Jacobi's method

28

7.3

Gauss-Seidel method