Implicit application of polynomial filters in a \(k\)-step Arnoldi method. (English) Zbl 0763.65025

Sorensen, Danny C.

SIAM J. Matrix Anal. Appl. 13, No. 1, 357-385 (1992).

The author describes and analyses a new implementation of the Arnoldi method for computing a few eigenvalues and the corresponding eigenvectors of a large general square matrix (which reduces to the Lanczos method in the symmetric case). Using a truncated variant of the implicitly shifted \(QR\)-iteration, the author applies a polynomial filter to the Arnoldi (Lanczos) vector on each iteration. This approach generalizes explicit restart methods. Advantages of the method are discussed and some preliminary computational results using parallel and vector computers are given.

Reviewer: A.L.Andrew (Bundoora)

MSC:

65F15

Numerical computation of eigenvalues and eigenvectors of matrices

Keywords:

Arnoldi method; eigenvalues; eigenvectors; Lanczos method; implicitly shifted \(QR\)-iteration; polynomial filter; explicit restart methods; parallel and vector computers

Software:

eigs; IRAM