On Structured Pencils arising in Sonneveld Methods

The pencils arising in Sonneveld methods, e.g., methods based on the induced dimension reduction (IDR) principle by Sonneveld and van Gijzen, are highly structured and some eigenvalues are known. The other eigenvalues are approximations to eigenvalues of the matrix used to compute the Sonneveld pencil. In [SIAM J. Matrix Anal. Appl. 34(2), 2013, pp. 283–311] we proved that it is possible to purify the characteristic polynomial from the known values by moving them to infinity and to deflate the problem to obtain a smaller pencil that has only the other eigenvalues. Depending on the strategy used to select the known eigenvalues, this may result in large condition numbers or even break down due to a singular pencil. In this paper we prove that there are one-dimensional families of purified and deflated pencils that all have the same eigenvalues. We give a selection scheme to chose a pencil suitable for the stable computation of the wanted eigenvalues.

Schlagworte

Krylovraumverfahren

Sonneveldmethoden

strukturierte Büschel

Krylov subspace method

Sonneveld methods

structured pencils

eigenvalues

DDC Class

510: Mathematik

Lizenz

http://doku.b.tu-harburg.de/doku/lic_mit_pod.php

Name

Bericht186.pdf

Size

690.32 KB

Format

Adobe PDF

Options

On Structured Pencils arising in Sonneveld Methods