How to implement the spectral transformation

Authors:
Bahram Nour-Omid, Beresford N. Parlett, Thomas Ericsson and Paul S. Jensen

Journal:
Math. Comp. **48** (1987), 663-673

MSC:
Primary 65F15

DOI:
https://doi.org/10.1090/S0025-5718-1987-0878698-5

MathSciNet review:
878698

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Abstract: The general, linear eigenvalue equations , where **H** and **M** are real symmetric matrices with **M** positive semidefinite, must be transformed if the Lanczos algorithm is to be used to compute eigenpairs . When the matrices are large and sparse (but not diagonal) some factorization must be performed as part of the transformation step. If we are interested in only a few eigenvalues near a specified shift, then the spectral transformation of Ericsson and Ruhe [1] proved itself much superior to traditional methods of reduction.

The purpose of this note is to show that a small variant of the spectral transformation is preferable in all respects. Perhaps the lack of symmetry in our formulation deterred previous investigators from choosing it. It arises in the use of inverse iteration.

A second goal is to introduce a systematic modification of the computed Ritz vectors, which improves the accuracy when **M** is ill-conditioned or singular.

We confine our attention to the simple Lanczos algorithm, although the first two sections apply directly to the block algorithms as well.

**[1]**Thomas Ericsson and Axel Ruhe,*The spectral transformation Lánczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems*, Math. Comp.**35**(1980), no. 152, 1251–1268. MR**583502**, https://doi.org/10.1090/S0025-5718-1980-0583502-2**[2]**Cornelius Lanczos,*An iteration method for the solution of the eigenvalue problem of linear differential and integral operators*, J. Research Nat. Bur. Standards**45**(1950), 255–282. MR**0042791****[3]**B. Nour-Omid, B. N. Parlett & R. L. Taylor, "Lanczos versus subspace iteration for solution of eigenvalue problems,"*Internat. J. Numer. Methods Engrg.*, v. 19, 1983, pp. 859-871.**[4]**B. Nour-Omid,*The Lanczos Algorithm for the Solution of Large Generalized Eigenproblems*, Tech. Rep. UCB/SESM-84/04, Dept. of Civil Engineering, University of California, Berkeley, 1984.**[5]**Beresford N. Parlett,*The symmetric eigenvalue problem*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1980. Prentice-Hall Series in Computational Mathematics. MR**570116****[6]**B. N. Parlett and B. Nour-Omid,*The use of a refined error bound when updating eigenvalues of tridiagonals*, Linear Algebra Appl.**68**(1985), 179–219. MR**794821**, https://doi.org/10.1016/0024-3795(85)90213-7**[7]**D. S. Scott,*The advantages of inverted operators in Rayleigh-Ritz approximations*, SIAM J. Sci. Statist. Comput.**3**(1982), no. 1, 68–75. MR**651868**, https://doi.org/10.1137/0903006

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DOI:
https://doi.org/10.1090/S0025-5718-1987-0878698-5

Article copyright:
© Copyright 1987
American Mathematical Society