A Rayleigh-Chebyshev procedure for finding the smallest eigenvalues and associated eigenvectors of large sparse Hermitian matrices

A procedure is presented for finding a number of the smallest eigenvalues and their associated eigenvectors of large sparse Hermitian matrices. The procedure, a modification of an inverse subspace iteration procedure, uses adaptively determined Chebyshev polynomials to approximate the required application of the inverse operator on the subspace. The method is robust, converges with acceptable rapidity, and can easily handle operators with eigenvalues of multiplicity greater than one. Numerical results are shown that demonstrate the utility of the procedure.

[1]  L. Eldén,et al.  Inexact Rayleigh Quotient-Type Methods for Eigenvalue Computations , 2002 .

[2]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .

[3]  C. Lanczos.  Applied Analysis , 1961 .

[4]  Alastair Spence,et al.  Inexact Inverse Subspace Iteration with Preconditioning Applied to Non-Hermitian Eigenvalue Problems , 2009, SIAM J. Matrix Anal. Appl..

[5]  Yousef Saad,et al.  A Chebyshev-Davidson Algorithm for Large Symmetric Eigenproblems , 2007, SIAM J. Matrix Anal. Appl..

[6]  Y. Saad,et al.  Numerical Methods for Large Eigenvalue Problems , 2011 .

[7]  E. Davidson.  The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices , 1975 .

[8]  Ed Anderson,et al.  LAPACK Users' Guide , 1995 .

[9]  J. Cullum,et al.  Lanczos algorithms for large symmetric eigenvalue computations , 1985 .

[10]  Christopher R. Anderson,et al.  Efficient solution of the Schroedinger-Poisson equations in layered semiconductor devices , 2009, J. Comput. Phys..

[11]  H. V. D. Vorst,et al.  EFFICIENT EXPANSION OF SUBSPACES IN THE JACOBI-DAVIDSON METHOD FOR STANDARD AND GENERALIZED EIGENPROBLEMS , 1998 .

[12]  Gene H. Golub,et al.  Matrix computations , 1983 .

[13]  D. Calvetti,et al.  Iterative methods for the computation of a few eigenvalues of a large symmetric matrix , 1996 .

[14]  R. Morgan,et al.  Generalizations of Davidson's method for computing eigenvalues of sparse symmetric matrices , 1986 .

[15]  Gerard L. G. Sleijpen,et al.  A Jacobi-Davidson Iteration Method for Linear Eigenvalue Problems , 1996, SIAM J. Matrix Anal. Appl..

[16]  Kesheng Wu,et al.  Thick-Restart Lanczos Method for Large Symmetric Eigenvalue Problems , 2000, SIAM J. Matrix Anal. Appl..

[17]  Ivan G. Graham,et al.  Inexact inverse iteration for symmetric matrices , 2006 .

[18]  N. Kosugi.  Modification of the Liu-Davidson method for obtaining one or simultaneously several eigensolutions of a large real-symmetric matrix , 1984 .

[19]  Axel Ruhe,et al.  The method of conjugate gradients used in inverse iteration , 1972 .

[20]  J. Cullum,et al.  Lanczos Algorithms for Large Symmetric Eigenvalue Computations Vol. I Theory , 1984 .

[21]  L. Borges,et al.  A Parallel Davidson-Type Algorithm for Several Eigenvalues , 1998 .

[22]  Zhishun A. Liu,et al.  A Look Ahead Lanczos Algorithm for Unsymmetric Matrices , 1985 .

[23]  B. Parlett,et al.  The Lanczos algorithm with selective orthogonalization , 1979 .