The Lanczos and conjugate gradient algorithms: from theory by Gérard Meurant

By Gérard Meurant

No current booklet comes close to this one within the variety and intensity of remedy of those vitally important methods—the Lanczos set of rules and the strategy of conjugate gradients.

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Extra info for The Lanczos and conjugate gradient algorithms: from theory to finite precision computations

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Bar-On proved the following result for unreduced (that is, with r)j ^ 0, 7 = 2 , . . ) tridiagonal matrices. 9. Let #^ denote the extended set of eigenvalues ofTj, j < k — 1, and let ft(j+2,k)foefne extenaea sej of eigenvalues ofTj+2,k- Let be the union o/# (y) and &(J+2

For earlier papers on that topic, see Householder [96] and Paige and Saunders [137]. If we already know the result, the equivalence between CG and Lanczos algorithms is easy to prove. In this section, for pedagogical reasons, we are going to pretend that we are not aware of the result and that we are just looking at simpler relations for computing the solution given by the Lanczos algorithm. Although the derivation is more involved it can be useful for the reader to see most of the details of this process.

The diamonds are the lower and upper bounds for a^ + i. The crosses on the jc-axis are the eigenvalues of A. 9. 10 shows for the same example the Iog10 of distances 0[ — X\ and Xn — 9^ as the middle solid curves as a function of the number of Lanczos iterations. Notice the scales are not the same for the two figures. 36. 23. 25, which are, in fact, not computable if we do not know the eigenvectors of A. We see that the bounds using the secular function are quite good. 11, where the dashed curve corresponds to the largest eigenvalue.

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