Volume 2, pp. 1-21, 1994.
An implicitly restarted Lanczos method for large symmetric eigenvalue problems
D. Calvetti, L. Reichel, and D. C. Sorensen
Abstract
The Lanczos process is a well known technique for computing a few, say ,
eigenvalues and associated eigenvectors of a large symmetric matrix.
However, loss of orthogonality of the computed Krylov subspace basis can reduce
the accuracy of the computed approximate eigenvalues. In the implicitly
restarted Lanczos method studied in the present paper, this problem is
addressed by fixing the number of steps in the Lanczos process at a
prescribed value, , where typically is not much larger, and may
be smaller, than . Orthogonality of the basis vectors of the
Krylov subspace is secured by reorthogonalizing these vectors when necessary.
The implicitly restarted Lanczos method exploits that the residual vector
obtained by the Lanczos process is a function of the initial Lanczos vector.
The method updates the initial Lanczos vector through an iterative scheme. The
purpose of the iterative scheme is to determine an initial vector such that
the associated residual vector is tiny. If the residual vector vanishes, then
an invariant subspace has been found. This paper studies several iterative
schemes, among them schemes based on Leja points.
The resulting algorithms are capable of computing a few of the largest or
smallest eigenvalues and associated eigenvectors. This is accomplished
using only storage locations in addition to the storage
required for the matrix, where the second term is independent of .
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Key words
Lanczos method, eigenvalue, polynomial acceleration.
AMS subject classifications
65F15.
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