## Spring 2007: Math 151B

### APPLIED NUMERICAL METHODS II

 Teaching Assistant: Igor Yanovsky Office Hours: Wednesday, 1:00-2:00 pm, Thursday, 12:00-1:00 pm, open office policy, and by appointment. Office: IPAM 1129D Professor J. Xia Class Page

HOMEWORK SOLUTIONS

Homework 1
Theoretical Problems
Computational Problem 11a
 Euler's method, main routine ODE specification
Computational Problem 12
 Euler's method, main routine ODE specification
Computational Problem 15, Euler's Method
 Euler's method, main routine ODE specification
Homework 2
Theoretical Problems
Computational Problem 3
 Runge-Kutta second order (Midpoint) method ODE specification exact solution
Computational Problem 4b
 Euler's method ODE specification exact solution
Computational Problem 4c
 RK-2 (Midpoint) method
Computational Problem 4d
 RK-2 (Midpoint) method
Homework 3
Theoretical Problems
Computational Problem 5.11, 7b
 Implicit Trapezoidal method function definition function derivative definition exact solution
Computational Problem 5.6, 6a
 Adams Predictor-Corrector method function definition exact solution
Homework 4
Theoretical Problems
Computational Problem 5.9, 2a
 RK-4 method for systems function1, function2 exact solution1, exact solution2
Computational Problem 5.9, 3b
 RK-4 method for systems function1, function2 exact solution1, exact solution2
Homework 5
 Solutions
Homework 6
 Solutions
Homework 7
 Solutions
Homework 8
Theoretical Problems
Computational Problem 1
 Normal equations Least squares with QR
Computational Problem 2

HANDOUTS

Matlab Environment
Euler Method implementation
 main routine ODE definition exact solution
Implicit Trapezoidal method implementation
 main routine function definition function derivative definition exact solution
Midterm Sample Problems
Boundary Value Problem Example
Gram-Schmidt and QR factorization
Driver for gsqr routine
Final Exam Sample Problems

DISCUSSION CONTENTS

April 5
 Matlab Environment page. Introduction to Matlab. See the links in the Resources section below. Euler's method for numerical solutions of ordinary differential equations.

April 12
 Euler's method. Discussion of Homework 1.

April 19
 Truncation error analysis, order of convergence. Runge-Kutta second order method. Implicit Trapezoidal method. Discussion of Homework 2.

April 26
 Stability, regions of absolute stability. Runge-Kutta fourth order method. Adams-Bashforth and Adams-Moulton methods. Adams predictor-corrector algorithm. Discussion of Homework 3.

May 3
 Runge-Kutta methods for systems. Stability for multistep methods, consistency, convergence. Discussion of Homework 4. Midterm Review.

May 10
 Two points boundary value problems. Finite difference approximation. Midterm Discussion. Discussion of Homework 5.

May 17
 Vector and matrix norms. Iterative methods for the solution of linear systems of equations. Gauss-Jacobi and Gauss Seidel. Discussion of Homework 6.

May 24
 Error analysis for iterative methods. Convergence results for iterative methods. Condition number. QR Decomposition and Gram-Schmidt process. Discussion of Homework 7.

May 31
 Discrete least squares approximation. Construction of the normal equations. Matrix/vector formulation of the discrete least squares problem. Using the QR decomposition to solve normal equations. Discussion of Homework 8.

June 7
 Fourier Series. Least squares trigonometric polynomials. Discussion of Homework 8. Final Review.

June 8
 Final Review.

RESOURCES

 Matlab Short Guide Quick Matlab Documentation (created by Professor C. Anderson, UCLA)
 Igor Yanovsky