Algorithm 827: irbleigs: A MATLAB program for computing a few eigenpairs of a large sparse Hermitian matrix

ABSTRACT

irbleigs is a MATLAB program for computing a few eigenvalues and associated eigenvectors of a sparse Hermitian matrix of large order n. The matrix is accessed only through the evaluation of matrix-vector products. Working space of only a few n-vectors is required. The program implements a restarted block-Lanczos method. Judicious choices of acceleration polynomials make it possible to compute approximations of a few of the largest eigenvalues, a few of the smallest eigenvalues, or a few eigenvalues in the vicinity of a user-specified point on the real axis. irbleigs also can be applied to certain large generalized eigenproblems as well as to the computation of a few nearby singular values and associated right and left singular vectors of a large general matrix.

REFERENCES

• 1: Baglama, J. 2000. Dealing with linear dependence during the iterations of the restarted block Lanczos methods. Numer. Algorithms 25, 23--36.
• 2: Baglama, J., Calvetti, D., and Reichel, L. 1996. Iterative methods for the computation of a few eigenvalues of a large symmetric matrix. BIT 36, 400--421.
• 3: J. Baglama , D. Calvetti , L. Reichel, IRBL: An Implicitly Restarted Block-Lanczos Method for Large-Scale Hermitian Eigenproblems, SIAM Journal on Scientific Computing, v.24 n.5, p.1650-1677, 2002 [doi>10.1137/S1064827501397949]
• 4: J. Baglama , D. Calvetti , L. Reichel , A. Ruttan, Computation of a few small eigenvalues of a large matrix with application to liquid crystal modeling, Journal of Computational Physics, v.146 n.1, p.203-226, Oct. 10, 1998 [doi>10.1006/jcph.1998.6064 ]
• 5: Calvetti, D., Reichel, L., and Sorensen, D. C. 1994. An implicitly restarted Lanczos method for large symmetric eigenvalue problems. Elec. Trans. Numer. Anal. 2, 1--21.
• 6: Lehoucq, R. B., Sorensen, D. C., and Yang, C. 1998. ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. SIAM, Philadelphia, Pa. Code available at web site http://www.caam.rice.edu/software/ARPACK.
• 7: MathWorks. 1998. MATLAB Application Program Interface Guide, Version 5.
• 8: D. C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM Journal on Matrix Analysis and Applications, v.13 n.1, p.357-385, Jan. 1992 [doi>10.1137/0613025]

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

General Terms:
Algorithms, Documentation

Keywords:
Block Lanczos method, eigenvalue computation, generalized eigenproblem, polynomial acceleration, singular values