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GMRES on (nearly) singular systems. (English) Zbl 0876.65019
The authors’ purpose is to examine the behavior of the GMRES method when the matrix \(A\) is singular or nearly so, i.e., ill-conditioned, and to formulate practically effective ways of recognizing the singularity or the ill-conditioning when it might significantly affect the performance of the method. For singular \(A\) circumstances in which the GMRES iterates converge without breakdown to a least squares solution or the pseudoinverse solution of the linear system are determined. The obtained results apply not only to GMRES but also to any mathematically equivalent method. An efficient and reliable procedure for detecting the singularity or ill-conditioning when it threatens to cause breakdown of the method is outlined. Several numerical experiments are presented and discussed.
Reviewer: K.Zlateva (Russe)

65F10 Iterative numerical methods for linear systems
65F20 Numerical solutions to overdetermined systems, pseudoinverses
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