Math 6610 • Analysis of Numerical Methods I

Sensitivity Analysis Consider the linear system \(Ax=b\) where \(A\) is a square, non-singular matrix. A numerical solution \(\hat {x}\) actually satisfies \[ (A+\delta A)\hat {x}=b+\delta b. \] If we denote \(\delta x=\hat {x}-x\), derive an upper bound for the relative error \[ E_1=\frac {\lVert \delta x\rVert }{\lVert \hat {x}\rVert } \] in terms of the condition number of \(A\) and the norms of \(A,\hat {x},\delta A,\delta b\), where \(\lVert \cdot \rVert \) is a suitable vector norm. Also, we can define another relative error \[ E_2=\frac {\lVert \delta x\rVert }{\lVert x\rVert }. \] Derive an upper bound for \(E_2\) in terms of the condition number of \(A\) and the norms of \(\delta A,A,\delta b\) and \(b\). When are these two errors close to each bound? What’s the significance of the condition number?