Suppose that while solving Az = b for z when A= $ instead, due to roundoff for 2. Consider t as an approximate to z. Verify error, you solve the related system Aż=6= the formula (b) ( 14 2 2.0001 ) (4) -K(A) 1 1 1.0001 (4) by evaluating the following: and b= (a) ê
Suppose that while solving Az = b for z when A= $ instead, due to roundoff for 2. Consider t as an approximate to z. Verify error, you solve the related system Aż=6= the formula (b) ( 14 2 2.0001 ) (4) -K(A) 1 1 1.0001 (4) by evaluating the following: and b= (a) ê
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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![1. Suppose that while solving \( Ax = b \) for \( x \) when \( A = \begin{bmatrix} 1 & 1 \\ 1 & 1.0001 \end{bmatrix} \) and \( b = \begin{bmatrix} 2 \\ 2 \end{bmatrix} \), instead, due to roundoff error, you solve the related system \( A\hat{x} = \hat{b} = \begin{bmatrix} 2 \\ 2.0001 \end{bmatrix} \) for \( \hat{x} \). Consider \(\hat{x}\) as an approximate to \( x \). Verify the formula
\[
\frac{\|x - \hat{x}\|}{\|x\|} \leq \kappa(A) \frac{\|\hat{b} - b\|}{\|b\|}
\]
by evaluating the following:
(a) \(\frac{\|x - \hat{x}\|}{\|x\|}\) __________
(b) \(\kappa(A) \frac{\|\hat{b} - b\|}{\|b\|} = \kappa(A) \frac{\|b - A\hat{x}\|}{\|b\|} \) __________
---
The formula expresses the stability of the solution \( \hat{x} \) to numerical perturbations in \( b \) and the error bounds in terms of the condition number \(\kappa(A)\) of the matrix \( A \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F50070ce1-6090-4114-8016-03310776fe88%2F4dd1eb30-1e06-4602-9db8-972c8888312d%2Fzns7s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Suppose that while solving \( Ax = b \) for \( x \) when \( A = \begin{bmatrix} 1 & 1 \\ 1 & 1.0001 \end{bmatrix} \) and \( b = \begin{bmatrix} 2 \\ 2 \end{bmatrix} \), instead, due to roundoff error, you solve the related system \( A\hat{x} = \hat{b} = \begin{bmatrix} 2 \\ 2.0001 \end{bmatrix} \) for \( \hat{x} \). Consider \(\hat{x}\) as an approximate to \( x \). Verify the formula
\[
\frac{\|x - \hat{x}\|}{\|x\|} \leq \kappa(A) \frac{\|\hat{b} - b\|}{\|b\|}
\]
by evaluating the following:
(a) \(\frac{\|x - \hat{x}\|}{\|x\|}\) __________
(b) \(\kappa(A) \frac{\|\hat{b} - b\|}{\|b\|} = \kappa(A) \frac{\|b - A\hat{x}\|}{\|b\|} \) __________
---
The formula expresses the stability of the solution \( \hat{x} \) to numerical perturbations in \( b \) and the error bounds in terms of the condition number \(\kappa(A)\) of the matrix \( A \).
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