A Jacobi-Davidson iteration method for linear eigenvalue problems. (English) Zbl 0860.65023

Sleijpen, Gerard L. G.; van der Vorst, Henk A.

SIAM J. Matrix Anal. Appl. 17, No. 2, 401-425 (1996).

The algorithm of E. R. Davidson [J. Comput. Phys. 17, 87-94 (1975; Zbl 0293.65022)] for computing the smallest eigenvalue of a large symmetric matrix, is shown to be closely related to an algorithm proposed by C. G. J. Jacobi [Über ein leichtes Verfahren, die in der Theorie der Säcularstörungen vorkommenden Gleichungen numerisch aufzulösen, J. Reine Angew. Math. 30, 51-94 (1846)]. This leads to a new algorithm, where a preconditioner is applied to a shifted matrix, which is projected away from the approximate eigendirection. Different preconditioners can then be used both in symmetric and nonsymmetric cases. Numerical tests are reported using ILU-GMRES to precondition both the Davidson, the new Jacobi-Davidson algorithm and shift invert iteration.

Reviewer: A.Ruhe (Göteborg)

MSC:

65F15

Numerical computation of eigenvalues and eigenvectors of matrices

Keywords:

Jacobi-Davidson iteration method; smallest eigenvalue; large symmetric matrix; algorithm; preconditioner; shift invert iteration

Software:

JDQZ