On the Sensitivity of Solution Components in Linear Systems of Equations

Expressions are presented for the errors in individual components of the solution to systems of linear equations and linear least squares problems. No assumptions about the structure or distribution of the perturbations are made. The resulting "componentwise condition numbers" measure the sensitivity of each solution component to perturbations. It is shown that any linear system has at least one solution component whose sensitivity to perturbations is proportional to the condition number of the matrix; but there may exist many components that are much better conditioned. Unless the perturbations are restricted, no norm-based relative error bound can predict the presence of well-conditioned components, so these componentwise condition numbers are essential. For the class of componentwise perturbations, necessary and sufficient conditions are given under which Skeel's condition numbers are informative, and it is shown that these conditions are similar to conditions where componentwise condition numbers are useful. Numerical experiments not only confirm that these circumstances do occur frequently, they also illustrate that for many classes of matrices the ill conditioning of the matrix is due to a few rows of the inverse only. This means that many of the solution components are computed more accurately than current analyses predict.