Math 151B: General Course Outline
Course 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(s)
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 
Higherorder Taylor methods. Error analysis of onestep methods 
4 
5.10, 5.4 
Stability of onestep methods. Taylor Theorem in two variables 
5 
5.4 
RungeKutta methods 
6 
5.4 
Butcher tableau. Design a RungeKutta method 
7 
5.5 
RungeKuttaFehlberg method. 
8 
5.6 
AdamsBashforth/AdamsMoulton multistep methods 
9 
5.6, 5.10 
Predictorcorrector 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 
Highorder 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 
Finitedifference methods for nonlinear BVP 
17 
10.1, 10.2 
Solving nonlinear systems of equations. Newton's method 
18 
10.3 
QuasiNewton 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 