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Professor:
Prof. Martin H. Gutknecht
Date: 18.1.2006
Researcher: Damian Loher
Summary:
Since the early work of Cullum and Donath (1974) and Underwood and Golub
(1975,1977), there has been, from time to time, a renewed interest in
conjugate gradient methods for systems with multiple-right hand sides and
in Lanzos methods with several starting vectors for solving linear systems,
computing eigenvalues, or, more recently, system identification and model
reduction.
While the formal definition of block versions of CG and Lanczos is
quite easy, the true difficulties come when a dimension reduction of the
block Krylov space (so-called deflation) has to be treated properly,
or, in the nonsymmetric case, when the block Lanczos process breaks down.
Deflation for symmetric systems was treated by Cullum and Donath (1974),
Cullum and Willoughby (1985), Nikishin and Yeremin (1995), Bai and
Freund (2001), and Schmelzer (2004).
For the nonsymmetric case, the answer to both difficulties lies in an
adjustment of the look-ahead Lanczos algorithm for single systems.
The resulting nonsymmetric look-ahead (or cluster) band Lanczos
algorithm has been developped in 1994--1996 in partly independent and
partly cooperative efforts by Aliaga, Boley, Feldmann, Freund,
Hernandez, Malhorta, and even others.
The work culminates in the paper by Aliaga, Boley, Freund, and
Hernandez in {Math. Comp.} (1999/2000, submitted 1996).
One aspect of block solvers and look-ahead procedures alike is that there
is a trade-off between cost and stability.
One purpose of this project is to investigate the pros and cons of various
implementation options and to develop a particularly stable version of the
nonsymmetric block Lanczos process.