Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 1997 • Question 1

Question

Math 6610 • Analysis of Numerical Methods I Fall 1997 • Question 1

Sensitivity of Solution of Linear System Suppose that \(A\) is a real non-singular \(n\times n\) matrix. Consider the family of linear systems \(Ax(\epsilon )=b+\epsilon c\), where \(b\) and \(c\) are given nonzero vectors in \(\mathbb {R}^n\), and \(\epsilon \) is a real parameter. Let \(x=x(0)\) be the solution for \(\epsilon =0\). Let \(\lVert \cdot \rVert _2\) denote the Euclidean vector norm. Show that \[ \frac {\lVert x(\epsilon )-x\rVert _2}{\lVert x\rVert _2} \le \lVert A\rVert _2\lVert A^{-1}\rVert _2 \frac {\lVert \epsilon c\rVert _2}{\lVert b\rVert _2}. \] The expression \(\lVert A\rVert _2\lVert A^{-1}\rVert _2\) is the condition number of \(A\). Interpret the above inequality and explain its significance?

{Sensitivity of Solution of Linear System}
Suppose that \(A\) is a real non-singular \(n\times n\) matrix. Consider the family of
linear systems \(Ax(\epsilon)=b+\epsilon c\), where \(b\) and \(c\) are given nonzero
vectors in \(\R^n\), and \(\epsilon\) is a real parameter. Let \(x=x(0)\) be the solution
for \(\epsilon=0\). Let \(\lVert\cdot\rVert_2\) denote the Euclidean vector norm. Show
that
\[
  \frac{\lVert x(\epsilon)-x\rVert_2}{\lVert x\rVert_2}
  \le
  \lVert A\rVert_2\lVert A^{-1}\rVert_2
  \frac{\lVert\epsilon c\rVert_2}{\lVert b\rVert_2}.
\]
The expression \(\lVert A\rVert_2\lVert A^{-1}\rVert_2\) is the condition number of
\(A\). Interpret the above inequality and explain its significance?

Topic Tags

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