Gmres/Cr and Arnoldi/Lanczos as Matrix Approximation Problems (Classic Reprint) - Softcover

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Excerpt from Gmres/Cr and Arnoldi/Lanczos as Matrix Approximation Problems

A comparison of (1) and (2) suggests that from an approximation point of view, the difference between the gmres and Arnoldi algorithms is slight. This analogy is rarely brought out in accounts of these algorithms, partly for historical reasons and partly because the usual applications of the two algorithms are differ ent. Whereas gmres is applied to solve systems of equations A3: b, so that (1) comes quickly to mind as a description of it, the Arnoldi iteration is traditionally thought of as a method for estimating eigenvalues of A. The Arnoldi eigenvalue estimatesl at step n are the eigenvalues of H, that is, the roots of p. But what the Arnoldi iteration actually does is solve its connection with eigenvalues is indirect and approximate.

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