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Name:
Znojil Miloslav
Reviewer number:
9689
Email:
znojil@ujf.cas.cz
Item's zbl-Number:
DE015199622
Author(s):
Nool, Margareet, van der Ploeg, Auke:
Shorttitle:
A parallel JD-type method
Source:
SIAM J. Sci. Comput. 22, No 1, 95-112 (2000).
Classification:
Primary Classification:
65F15Eigenvalues, eigenvectors
Secondary Classification:
65F50Sparse matrices
65N25Eigenvalue problems
65Y05Parallel computation
65Y20Complexity and performance of numerical algorithms
Keywords:
generalized eigenvalue problem, Jacobi-Davidson method, block-tridiagonal systems, parallelization
Review:

A key feature of the Jacobi-Davidson method (closely related to
the well known Arnoldi algorithm) lies in the step-by-step
constructive enlargement of the truncated basis, suitable for the
computation of a few selected eigenvalues. Authors assume being
given the generalized eigenvalue problem for some large and
complex block-tridiagonal matrices. Application to the plasma
stability in tokamaks is kept in mind. Their main purpose is to
pallalelize the corresponding algorithm.

In a preliminary step they employ the ``cheap" complete LU
decomposition [at a trial estimate of (a few) eigenvalues] and
reduce their generalized eigenvalue problem to the mere standard
eigenvalue equation. Then, the new basis elements are iteratively
sought from an augmented equation (tractable, say, by GMRES). The
LU decomposition itself is parallelized via a combination of the
domain decomposition and a cyclic reduction. It proves robust due
to partial pivoting.

In a detailed proposal, a compressed row storage strategy is
recommended and a re-computation of the matrix elements is
preferred to their input. Tests on Cray T3E are added to
illustrate the performance and speed-ups, with the wall clock
times determined, roughly speaking and in agreement with
predictions, by the matrix-vector multiplications.
Remarks to the editors: