The subspace projected approximate matrix (SPAM) modification of the Davidson method

Abstract A modification of the iterative matrix diagonalization method of Davidson is presented that is applicable to the symmetric eigenvalue problem. This method is based on subspace projections of a sequence of one or more approximate matrices. The purpose of these approximate matrices is to improve the efficiency of the solution of the desired eigenpairs by reducing the number of matrix–vector products that must be computed with the exact matrix. Several applications are presented. These are chosen to show the range of applicability of the method, the convergence behavior for a wide range of matrix types, and also the wide range of approaches that may be employed to generate approximate matrices.

[1]  J. G. Lewis,et al.  A Shifted Block Lanczos Algorithm for Solving Sparse Symmetric Generalized Eigenproblems , 1994, SIAM J. Matrix Anal. Appl..

[2]  W. Miller.  Semiclassical limit of quantum mechanical transition state theory for nonseparable systems , 1975 .

[3]  J. Cullum,et al.  Computing eigenvalues of very large symmetric matrices—An implementation of a Lanczos algorithm with no reorthogonalization , 1981 .

[4]  C. Moler,et al.  Numerical algorithms in chemistry: algebraic methods. [Workshop, August 9-11, 1978] , 1978 .

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

[6]  Ron Shepard,et al.  Elimination of the diagonalization bottleneck in parallel Direct-SCF methods , 1993 .

[7]  H. V. D. Vorst,et al.  Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems , 1995 .

[8]  B. Parlett.  The Algebraic Eigenvalue Problem (J. H. Wilkinson) , 1966 .

[9]  F. L. Pilar,et al.  Elementary Quantum Chemistry , 1968 .

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

[11]  Thomas Müller,et al.  High-level multireference methods in the quantum-chemistry program system COLUMBUS: Analytic MR-CISD and MR-AQCC gradients and MR-AQCC-LRT for excited states, GUGA spin–orbit CI and parallel CI density , 2001 .

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

[13]  W. Miller,et al.  Quantum mechanical calculations of the rate constant for the H2+OH→H+H2O reaction: Full‐dimensional results and comparison to reduced dimensionality models , 1994 .

[14]  Iain S. Duff,et al.  Users' guide for the Harwell-Boeing sparse matrix collection (Release 1) , 1992 .

[15]  C. Bender,et al.  The iterative calculation of several of the lowest or highest eigenvalues and corresponding eigenvectors of very large symmetric matrices , 1973 .

[16]  William H. Miller,et al.  The cumulative reaction probability as eigenvalue problem , 1993 .

[17]  J. Olsen,et al.  Passing the one-billion limit in full configuration-interaction (FCI) calculations , 1990 .

[18]  E. Davidson,et al.  Improved Algorithms for the Lowest Few Eigenvalues and Associated Eigenvectors of Large Matrices , 1992 .

[19]  J. H. van Lenthe,et al.  A space‐saving modification of Davidson's eigenvector algorithm , 1990 .

[20]  Ernest R. Davidson,et al.  Super-matrix methods , 1989 .

[21]  Robert J. Harrison,et al.  Approaches to large‐scale parallel self‐consistent field calculations , 1995, J. Comput. Chem..

[22]  Russell M. Pitzer,et al.  A progress report on the status of the COLUMBUS MRCI program system , 1988 .

[23]  H.J.J. Van Dam,et al.  An improvement of Davidson's iteration method: Applications to MRCI and MRCEPA calculations , 1996 .

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

[25]  Bernard Philippe,et al.  The Davidson Method , 1994, SIAM J. Sci. Comput..

[26]  Robert J. Harrison,et al.  High-Performance Computational Chemistry: Hartree-Fock Electronic Structure Calculations on Massively Parallel Processors , 1999, Int. J. High Perform. Comput. Appl..

[27]  M. Almond,et al.  Computers in Physics , 1971 .

[28]  William J. Thompson,et al.  Monster Matrices: Their Eigenvalues and Eigenvectors , 1993 .