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New approaches to large scale eigenanalysis. (English) Zbl 0911.65031

Bristeau, M.-O. (ed.) et al., Computational science for the 21st century. Dedicated to Prof. Roland Glowinski on the occasion of his 60th birthday. Symposium, Tours, France, May 5–7, 1997. Chichester: John Wiley & Sons. 62-71 (1997).
The past few years have seen significant advances in numerical techniques to compute partial eigendecompositions of large matrices. These new approaches have led to critical advances in several application areas including computational chemistry, semi-conductor laser design, linear stability analysis, and reduced basis techniques for large state space control systems.
This paper will survey some of these new techniques. In particular, we shall discuss the implicitly restarted Arnoldi method which is the foundation for the eigenvalue software package ARPACK. This package has been used extensively in many application areas that require the solution of large scale symmetric and nonsymmetric (generalized) eigenvalue problems. A brief introduction to Krylov subspace projection is given and the Lanczos/Arnoldi factorization is introduced. Implicit restarting is presented as a means for controlling computational cost, maintaining numerical accuracy, avoiding “ghost” eigenvalues, and computing selected eigenvalues of specific interest. The large scale eigenvalue software ARPACK that is based on this computational framework is discussed. Finally, new approaches currently under development are presented that promise to address some of the most difficult challenges that remain in this active research area.
For the entire collection see [Zbl 0889.00026].

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65F50 Computational methods for sparse matrices

Software:

ARPACK
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