Structured backward error and condition number for linear systems of the type \(A^* Ax = b\)
This paper presents a summary of a thorough study of the structured backward error analysis and the condition number of systems \[ A^*Ax=b,\tag{1} \] where \(A\) is an \(m\times n\) matrix of rank \(n\). It is important to point out that the system (1) is not the system of normal equations resulting from a least-squares problem. The paper derives a formulation of the structured condition number of (1.1) when the matrix \(A\) is the object of normwise perturbations. Then the corresponding backward error analysis is obtained, whose result can be viewed as the solution of a minimization problem over a sphere. The proofs of these results, whose mere enunciation is quite involved, can be found in technical reports by the same authors. A major feature in the paper is the section devoted to reporting results on numerical experiments performed with MATLAB on test matrices taken from Nick Higham's collection. The experiments show that the structured analysis proposed succeeds in predicting errors faithfully.
scientific article; zbMATH DE number 1471139