advertisement

Homework 3 Problem 1 Write a Gaussian elimination program for solving Ax = b. Consider the following test problems. 1) The matrix A (n × n) is a tri-diagonal matrix with 2 on the main diagonal and −1 on sub- and super- diagonal (in matlab A = 2 ∗ diag(ones(n, 1)) + (−1) ∗ diag(ones(n − 1, 1), 1) + (−1) ∗ diag(ones(n − 1, 1), −1)). In this example, try to take an advantage of the banded structure of the matrix and store only nonzero elements. Take n = 50, 100 and b = (1, . . . , 1)T . Problem 2 1) Compute condition numbers using norms kAk2 and kAk∞ : a+1 a 1 2 α 1 a) A = , b) A = , c) A = , a a−1 1 2.01 1 1 2) In solving Ax = b for the matrix A from part b) predict how slight changes in b will 0 affect the solution x, and test your prediction in the concrete case b = (4, 4)T and b = (3, 5)T . Problem 3 Show that if a square matrix A satisfies kAxk ≥ θkxk for all x, with θ > 0, then A is nonsingular and kA−1 k ≤ θ −1 . This is valid for any vector norm and its subordinate matrix norm. Remark.PNote that if A is diagonally dominant, then kAxk∞ ≥ θkxk∞ , where θ = mink (|akk | − nj=1,j6=k |akj ). 1