Eigenvector Computation with Spectral Transformation

in place of and rely on the fact that

to recover eigenvalues of the original matrix. The Arnoldi method is extremely effective when used in conjunction with SI. Convergence to eigenvectors corresponding to the eigenvalues of that are nearest to is generally very rapid. Hence, this approach should be used whenever it is possible to solve linear systems efficiently with as the coefficient matrix.

When a spectral transformation is used, additional considerations should be made with respect to stopping criteria to take advantage of the special nature of the transformed operator . Moreover, the quality of the approximate eigenvectors can be improved significantly with a minor amount of postprocessing. In order to compute eigenvalues of near to we will compute the eigenvalues of that are of largest magnitude. It is not really necessary for to be extremely close to a desired eigenvalue. However, for the following discussion it is worth keeping in mind that in practice, it is typical to take near to desired eigenvalues, and hence it is typical for to hold.

Let with
, where . Since

A simple rearrangement of (7.23) gives

Adding and subtracting
on the right-hand side of (7.24) and rearranging terms will
result in

A heuristic argument further supports the use of instead of .
From (7.23) it follows that

(132) |

This vector has not yet been scaled to have unit norm. However, , so the error bound will decrease after is normalized. Moreover, if , then the floating point computation of the norm will already result in without any rescaling.

From (7.23), the residual
is orthogonal to the Krylov space spanned by the
columns of . However, this Galerkin condition is lost upon transforming
the computed eigenpair to the original system,
regardless of whether we use or . This is because
is not the Rayleigh quotient
associated with or . However, from (7.25)

On the other hand, from
(7.24) we deduce that

Equations (7.25) and (7.27) imply that the vector is a better approximation than to the eigenvector associated with the approximate eigenvalue provided that is greater than 1. Moreover, when , only a moderately small Ritz estimate is needed to achieve an acceptably small direct residual and Rayleigh quotient error. If the is near the desired eigenvalues, then these eigenvalues are mapped by to large eigenvalues and typically .

The above analysis is based on that given by Ericsson and Ruhe [162] for the generalized symmetric definite eigenvalue problem.