Programs in Physics & Physical Chemistry
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|Manuscript Title: A Davidson program for finding a few selected extreme eigenpairs of a large, sparse, real, symmetric matrix.|
|Authors: A. Stathopoulos, C.F. Fischer|
|Program title: DVDSON|
|Catalogue identifier: ACPZ_v1_0|
Distribution format: gz
|Journal reference: Comput. Phys. Commun. 79(1994)268|
|Programming language: Fortran.|
|Operating system: Sun OS Release 4.1.2, Unix System V/386 3.2.|
|Keywords: General purpose, Eigenvalue, Eigenvector, Sparse, Symmetric matrix, Matrix-vector Multiplication, Davidson method, Lanczos method, Electronic structure, Computational physics, Computational quantum Chemistry.|
Nature of problem:
Finding a few extreme eigenpairs of a real, symmetric matrix is of great importance in scientific computations. Examples abound in structural engineering, quantum chemistry and electronic structure physics [1,2]. The matrices involved are usually too large to be efficiently solved using standard methods. Moreover, their large size often prohibits full storage forcing various sparse representations. Even sparse representations cannot always be stored in main memory . Thus, an iterative method is needed that converges rapidly to the few eigenpairs required and takes advantage of the symmetry, sparsity and possibly the secondary storage of the matrix.
Library Routines used: LAPACK, BLAS
This program implements a version of the original Davidson method , focusing on high performance on vector processors and adopting many previously proposed extensions [3,5,6] which have not appeared together in a published program. The features of the current version are:
For performance improvements, DVDSON keeps the basis in memory which may cause memory overflow problems if exceedingly large matrices are used. This program bears two of the Davidson method's weaknesses: convergence is slow when the diagonal of the matrix is constant and not guaranteed for the required eigenpair when the matrix is permutationally equivalent to a block diagonal matrix.
For each eigenpair, the Davidson method takes 10-30 iterations if the block size is one. For larger block sizes the number of iterations decreases significantly.
|||B.N. Parlett, SIAM J. Sci. Stat. Comput. 5(1984) 590.|
|||C.F. Fischer, Comput. Phys. Commun. 64(1991)369.|
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|||E.R. Davidson, J. Comput. Phys. 17(1975)87.|
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|||J. Weber, R. Lacroix and G. Wanner, Comput. Chem. 4(1980)55.|
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