Average condition number for solving linear equations (Q1082774)

This is a statistical consideration of the condition number of a disturbed system of linear equations \(Ax=b\). For a Lebesgue-measure \(\mu\) it is shown, that the average condition number in \(L_ p\) is comparable with the worst case condition number. For the Frobenius norm one gains about \(\log_ 2n+0.9\) bits on the average as compared to the worst case, n being the dimension of the linear system. Beside the Frobenius norm componentwise perturbations and the spectral norm are considered. Finally it is indicated how the result can be extended for perturbations of both A and b.