Let A be a 3 x 3 matrix. Suppose you run the steepest descent algorithm to solve Ax b with b (1, 1, 3)^t. Suppose your stopping condition is || Ax") -b|| < 0.01, |b|| || Ax6)-b|| and suppose your program stops at the 5th iteration where it computes x(5) ,and 0.0056 Suppose cond(A) = 1000. We learned that with a high condition number, ||x5) – x||/||x|| might still be large where x is the true solution to Ax = b. a) What is the largest that ||x(5) - x||/||x|| could theoretically be? (Hint: use the general inequality involving condition number) ||Ax)-b|| b) What should our tolerance (tol) be to guarantee that ||xk) – x||/||x|| < 0.01 ? Our program will still use the stopping condition < tol b| (Hint: use the general inequality involving condition number))
Let A be a 3 x 3 matrix. Suppose you run the steepest descent algorithm to solve Ax b with b (1, 1, 3)^t. Suppose your stopping condition is || Ax") -b|| < 0.01, |b|| || Ax6)-b|| and suppose your program stops at the 5th iteration where it computes x(5) ,and 0.0056 Suppose cond(A) = 1000. We learned that with a high condition number, ||x5) – x||/||x|| might still be large where x is the true solution to Ax = b. a) What is the largest that ||x(5) - x||/||x|| could theoretically be? (Hint: use the general inequality involving condition number) ||Ax)-b|| b) What should our tolerance (tol) be to guarantee that ||xk) – x||/||x|| < 0.01 ? Our program will still use the stopping condition < tol b| (Hint: use the general inequality involving condition number))
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Let A be a 3 x 3 matrix. Suppose you run the steepest descent algorithm to solve Ax = b with b =(1, 1, 3)^t. Suppose your stopping condition is
||Ax) -b||
|b
< 0.01,
||Ax5) -b||
||b||
and suppose your program stops at the 5th iteration where it computes x5), and
= 0.0056
Suppose cond(A) = 1000. We learned that with a high condition number, ||x5) – x||/||x|| might still be large where x is the true solution to Ax = b.
a) What is the largest that ||x(5) – x||/||x|| could theoretically be? (Hint: use the general inequality involving condition number))
|Ax)-b||
b) What should our tolerance (tol) be to guarantee that ||x*) – x||/l|x|| < 0.01 ? Our program will still use the stopping condition
< tol
|||
(Hint: use the general inequality involving condition number))
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