hw6 - Homework 6 Math 471 Fall 2007 Assigned Friday Due...

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Homework 6 Math 471, Fall 2007 Assigned: Friday, October 19, 2006 Due: Friday, October 26, 2006 Include a cover page Clearly label all plots using title , xlabel , ylabel , legend Use the subplot command to compare multiple plots Include printouts of all Matlab code, labeled with your name, date, section, etc. (1) (Condition Numbers, P. 187 #1, 2) 1. Let A and B be n × n matrices, and α be a nonzero real number. (a) Show that κ ( AB ) κ ( A ) κ ( B ); (b) Show that κ ( αA ) = κ ( A ). 2. Let A be an n × n matrix, and suppose that A x = y for some vectors x and y . Show that κ ( A ) k A kk x k k y k . (2) (Error Estimates, P. 188 #7c, 8c) 7c. Compute κ for A = 1 - 1 - 1 0 1 - 1 0 0 1 8c. Compute the error e = ˜ x - x and the residual r = A ˜ x - b and then compare the relative error to the condition number times the relative residual with l norm. 1 - 1 - 1 0 1 - 1 0 0 1 x = 0 2 0 x = [2 2 0] T ˜ x = [1 . 9 2 . 1 - 0 . 1] T (3) (Positive Definite and Strictly Diagonally Dominant Matrices, P. 220 #3) Con- sider the matrix a - 1 0 - 1 4 1 0 1 5 (a)
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