By R. Raman

ISBN-10: 0853343411

ISBN-13: 9780853343417

E-book through Raman, R.

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Dubois, A. H. Rodrigue, Approximating the inverse of a matrix for use on iterative algorithms on vectors processors, Computing 22 (1979) 257–268. S. A. Meurant, The e ect of ordering on preconditioned conjugate gradients, BIT 29 (1989) 635–657. [60] T. Dupont, R. Kendall, H. Rachford, An approximate factorization procedure for solving self-adjoint elliptic di erence equations, SIAM J. Numer. Anal. 5 (1968) 559–573. [61] T. Eirola, O. Nevanlinna, Accelerating with rank-one updates, Linear Algebra Appl.

These results show the superior quality of a Rayleigh quotient from a given subspace. It is exploited in modern iterative methods, such as the Lanczos algorithm, but it is also essential in the QR algorithm. Related to the perturbation analysis for Rayleigh quotients is work of Kaniel (1966) [69] for errors in the Ritz approximations computed in the Lanczos process. For a comprehensive discussion of this, see [100, Chapter 12]. 2 According to Parlett [100, p. ) results of Weinberger (1959). H. A.

185]). This leads to the observation that if a matrix A is perturbed by A, then the perturbation to i is in ÿrst order of terms of A given by i = 1 yi∗ xi yi∗ Axi ; where xi , and yi are the normalized right and left eigenvectors, respectively, corresponding to i , and yi∗ denotes the complex conjugate of yi . The factor 1=yi∗ xi is referred to as the condition number of the ith eigenvalue. The Bauer–Fike result (1960) [9], which is actually one of the more famous reÿnements of Gershgorin’s theorem, makes this more precise: the eigenvalues ˜j of A + A lie in discs Bi with centre i , and radius n( A 2 =|yi∗ xi |) (for normalized xi and yi ).

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