Math 502 Homework@ISU

The assignments include both theoretical problems and computational projects. Here are some guideline for doing the computation project:

(1) The program should be written in a relatively formal fashion, be easy to read, and contain comments of major steps;
(2) Please turn in a disk or email me all your programs and data files that are necessary to run your programs, including a simple README file.
(3) You are encouraged to discuss with other fellow students, but you need to finish and turn in your own project separately.

HW#1 --- Due September 01
Problems 2.4 (pp. 19) / #3, 4; (Floating point Arithmetic)
Problem 3.5 (pp.28)/ #1, #2(b), #4, #5*; (The conditioning & Error Analysis)
Problem 4.7 (pp. 45)/ #3, #4 (a), (d). (Algorithms & Complexity)

Numerical Tests:
(1) Write a program to find the smallest positive integers (p, q, r) so that
i) 1+1/2^p=1;
ii) 1/2^q=0;
iii) 2^r=inf.
(2) Plot the function y=(x-1)^6 near x=1 with increasingly refined scale and check what happens if using
its expansion f(x) = x^6-6x^5 +15x^4 -20x^3 +15x^2 -6x +1.

HW#2 ---due Sept. 15
Problem 1.7 (pp.60)/ #1, #3*, #5. (Gauss Elimination & Pivoting)

Numerical Tests: (here is a sample of matlab code: GEpiv.m; LtriSol.m; UTriSol.m)
(1) Determine the operation count of Gaussian elimination (with partial pivoting) for solving Ax=b.
(2) Write a Gaussian elimination code that does partial pivoting and use it to solve the system of equations Ax=b, where
A=[9 3 2 0 7; 7 6 9 6 4; 2 7 7 8 2; 0 9 7 2 2; 7 3 6 4 3] and b=[35 58 53 37 39]' . The correct answer is x=[0 1 2 3 4]'.

Problem 2.4 (pp63)/ #1, #3*, #4 (The Cholesky decomposition)
Problem 3.5 (pp.67)/ #2, #3. (The Householder & QR decomposition)

HW#3 --- due   Sept. 29
Problem 4.4 (pp72)/ #1, #3; (Vector & Matrix norms);
Problem 5.4 (pp77-78)/ #1, #3*, #4; ( Condition number & error bounds)
Problem 6.6 (pp90)/ #1, #3,   #5* ( SVD & Pseudo-normal solution)

HW#4 ---due Oct. 13
Problem 1.4 (pp98)/ #2, #6*;   (The Householder method & Newton method);
Problem 2.3 (pp 104)/ #2, #3, #4, #5*; (The Jacobi method & Eigenvalue estimate)
Problem 3.4 (pp109)/ #1, #2, #3*; (The power method & The Rayleigh Quotient);
Problem 4.3 (pp116)/#1, #3. (The QR Algorithm & LU method). The matrix (example 8.4.3) for #3 is
           A=[-4 1 0 0 1 0; 1 -4 1 0 0 0; 0 1 -4 1 0 0; 0 0 1 -4 0 0; 1 0 0 0 -4 1; 0 0 0 0 1 -4].

HW#5 -- due Oct. 27
Problem #5-1 ( General iterative methods, classical stationary methods, relaxation methods);

HW#6 --due Nov. 15
Problem #6 (Krylov Subspace Methods, Conjugate Gradient, GMRES).

HW#7 -- due Dec. 01
Problem #7 (Newton's method, Broyden's method and global convergence).

MATH 502 PROBLEM ASSIGNMENTS