On the Sensitivity of Solution Components in Linear Systems of Equations
- 1 January 1995
- journal article
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Matrix Analysis and Applications
- Vol. 16 (1) , 93-112
- https://doi.org/10.1137/s0895479892231255
Abstract
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 us...Keywords
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