Advertisement

Solving Eigenproblems: From Arnoldi via Jacobi-Davidson to the Riccati Method

  • Jan H. Brandts
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2542)

Abstract

The formulation of eigenproblems as generalized algebraic Riccati equations removes the non-uniqueness problem of eigenvectors. This basic idea gave birth to the Jacobi-Davidson (JD) method of Sleijpen and Van der Vorst (1996). JD converges quadratically when the current iterate is close enough to the solution that one targets for. Unfortunately, it may take quite some effort to get close enough to this solution. In this paper we present a remedy for this. Instead of linearizing the Riccati equation (which is done in JD) and replacing the linearization by a low-dimensional linear system, we propose to replace the Riccati equation by a low-dimensional Riccati equation and to solve it exactly. The performance of the resulting Riccati algorithm compares extremely favorable to JD while the extra costs per iteration compared to JD are in fact negligible.

Keywords

Arnoldi Riccati Jacobi-Davidson Krylov subspace 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnoldi, W.E.: The Principle of Minimized Iteration in the Solution of the Matrix Eigenvalue Problem. Quart. Appl. Math., 9 (1951) 17–29.MathSciNetzbMATHGoogle Scholar
  2. 2.
    Brandts, J.H.: The Riccati Method for Eigenvalues and Invariant Subspaces of Matrices with Inexpensive Action. Linear Algebra Appl., (2002) accepted.Google Scholar
  3. 3.
    Demmel, J.: Three Methods for Refining Estimates of Invariant Subspaces. Computing, 38 (1987) 43–57.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Sleijpen, G.L.G. and van der Vorst, H.A.: Jacobi-Davidson Iteration Method for Linear Eigenvalue Problems. SIAM J. Matrix Anal. Applic., 17 (1996) 401–425.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Jan H. Brandts
    • 1
  1. 1.Korteweg-de Vries Institute, Faculty of ScienceUniversity of AmsterdamTV AmsterdamNetherlands

Personalised recommendations