Algorithm 845: EIGIFP: a MATLAB program for solving large symmetric generalized eigenvalue problems

ABSTRACT

eigifp is a MATLAB program for computing a few extreme eigenvalues and eigenvectors of the large symmetric generalized eigenvalue problem Ax = λ Bx. It is a black-box implementation of an inverse free preconditioned Krylov subspace projection method developed by Golub and Ye [2002]. It has important features that allow it to solve some difficult problems without any input from users. It is particularly suitable for problems where preconditioning by the standard shift-and-invert transformation is not feasible.

REFERENCES

• 1: J. Baglama , D. Calvetti , L. Reichel, Algorithm 827: irbleigs: A MATLAB program for computing a few eigenpairs of a large sparse Hermitian matrix, ACM Transactions on Mathematical Software (TOMS), v.29 n.3, p.337-348, September 2003 [doi>10.1145/838250.838257]
• 2: Zhaojun Bai , James Demmel , Jack Dongarra , Axel Ruhe , Henk van der Vorst, Templates for the solution of algebraic eigenvalue problems: a practical guide, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2000
• 3: I. S. Duff , Roger G. Grimes , John G. Lewis, Sparse matrix test problems, ACM Transactions on Mathematical Software (TOMS), v.15 n.1, p.1-14, March 1989 [doi>10.1145/62038.62043]
• 4: Diederik R. Fokkema , Gerard L. G. Sleijpen , Henk A. Van der, Jacobi--Davidson Style QR and QZ Algorithms for the Reduction of Matrix Pencils, SIAM Journal on Scientific Computing, v.20 n.1, p.94-125, Aug. 1998 [doi>10.1137/S1064827596300073]
• 5: Golub, G. and Van Loan, C. 1983. Matrix Computations. The Johns Hopkins University Press, Baltimore.
• 6: Gene H. Golub , Qiang Ye, An Inverse Free Preconditioned Krylov Subspace Method for Symmetric Generalized Eigenvalue Problems, SIAM Journal on Scientific Computing, v.24 n.1, p.312-334, 2002 [doi>10.1137/S1064827500382579]
• 7: Knyazev, A. 1987. Convergence rate estimates for iterative methods for a mesh symmetric eigenvalue problem Soviet J. Numer. Anal. Math. Modelling 2, 371--396.
• 8: Knyazev, A. 1998. Preconditioned eigensolvers---an oxymoron? Electronic Trans. Numer. Anal. 7, 104--123.
• 9: Andrew V. Knyazev, Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method, SIAM Journal on Scientific Computing, v.23 n.2, p.517-541, 2001 [doi>10.1137/S1064827500366124]
• 10: Lehoucq, R., Sorenson, D., and Yang, C. 1998. ARPACK Users' Guides, Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Method. SIAM, Philadelphia.
• 11: Notay Y. 2002. Combination of Jacobi-Davidson and conjugate gradient for the partial symmetric eigenproblem. Numer. Linear Alg. Appl. 9, 21--44.
• 12: Beresford N. Parlett, The symmetric eigenvalue problem, Prentice-Hall, Inc., Upper Saddle River, NJ, 1998
• 13: Saad, Y. 1992. Numerical Methods for Large Eigenvalue Problems. Manchester University Press, Manchester, UK.
• 14: Gerard L. G. Sleijpen , Henk A. Van der, A Jacobi--Davidson Iteration Method for Linear EigenvalueProblems, SIAM Journal on Matrix Analysis and Applications, v.17 n.2, p.401-425, April 1996 [doi>10.1137/S0895479894270427]

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.3 Numerical Linear Algebra

Additional Classification:
  G. Mathematics of Computing
  G.4 MATHEMATICAL SOFTWARE
      Subjects: Documentation

Keywords:
Krylov subspace methods, eigenvalue, generalized eigenvalue problem, preconditioning