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Question

Math 6610 • Analysis of Numerical Methods I

Fall 1999 • Question 4

Question

Math 6610 • Analysis of Numerical Methods I Fall 1999 • Question 4

Sensitivity Analysis Consider the (square, non-singular) linear system \[ Ax=b. \tag {3} \] Suppose you compute (by any method) a numerical solution \(\hat {x}\) and you are concerned about the relative error \[ E=\frac {\lVert x-\hat {x}\rVert }{\lVert x\rVert } \tag {4} \] where \(\lVert \cdot \rVert \) denotes a suitable vector norm. Derive upper and lower bounds on \(E\) in terms of the residual \(r=b-A\hat {x}\) and the condition number of \(A\) with respect to \(\lVert \cdot \rVert \).

Raw LaTeX

{Sensitivity Analysis}
Consider the (square, non-singular) linear system
\[
  Ax=b. \tag{3}
\]
Suppose you compute (by any method) a numerical solution \(\hat{x}\) and you are
concerned about the relative error
\[
  E=\frac{\lVert x-\hat{x}\rVert}{\lVert x\rVert} \tag{4}
\]
where \(\lVert\cdot\rVert\) denotes a suitable vector norm. Derive upper and lower
bounds on \(E\) in terms of the residual \(r=b-A\hat{x}\) and the condition number of
\(A\) with respect to \(\lVert\cdot\rVert\).

Topic Tags

Conditioning, Sensitivity, and Perturbation Bounds Iterative Methods for Linear Systems Linear Programming and Optimization Matrix Norms and Induced Norms