A uniform object-oriented solution to the eigenvalue problem for real symmetric and Hermitian matrices

Abstract

We present a system of classes, SHMatrix, to deal in a unified way with the computation of eigenvalues and eigenvectors in real symmetric and Hermitian matrices. Thus, two descendant classes, one for the real symmetric and other for the Hermitian cases, override the abstract methods defined in a base class. The use of the inheritance relationship and polymorphism allows handling objects of any descendant class using a single reference of the base class. The system of classes is intended to be the core element of more sophisticated methods to deal with large eigenvalue problems, as those arising in the variational treatment of realistic quantum mechanical problems. The present system of classes allows computing a subset of all the possible eigenvalues and, optionally, the corresponding eigenvectors. Comparison with well established solutions for analogous eigenvalue problems, as those included in LAPACK, shows that the present solution is competitive against them. Program summary Program title: SHMatrix Catalogue identifier: AEHZ_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEHZ_v1_0.html Program obtainable from: CPC Program Library, Queenʼs University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 2616 No. of bytes in distributed program, including test data, etc.: 127 312 Distribution format: tar.gz Programming language: Standard ANSI C++. Computer: PCs and workstations. Operating system: Linux, Windows. Classification: 4.8. Nature of problem: The treatment of problems involving eigensystems is a central topic in the quantum mechanical field. Here, the use of the variational approach leads to the computation of eigenvalues and eigenvectors of real symmetric and Hermitian Hamiltonian matrices. Realistic models with several degrees of freedom leads to large (sometimes very large) matrices. Different techniques, such as divide and conquer, can be used to factorize the matrices in order to apply a parallel computing approach. However, it is still interesting to have a core procedure able to tackle the computation of eigenvalues and eigenvectors once the matrix has been factorized to pieces of enough small size. Several available software packages, such as LAPACK, tackled this problem under the traditional imperative programming paradigm. In order to ease the modelling of complex quantum mechanical models it could be interesting to apply an object-oriented approach to the treatment of the eigenproblem. This approach offers the advantage of a single, uniform treatment for the real symmetric and Hermitian cases. Solution method: To reach the above goals, we have developed a system of classes: SHMatrix. SHMatrix is composed by an abstract base class and two descendant classes, one for real symmetric matrices and the other for the Hermitian case. The object-oriented characteristics of inheritance and polymorphism allows handling both cases using a single reference of the base class. The basic computing strategy applied in SHMatrix allows computing subsets of eigenvalues and (optionally) eigenvectors. The tests performed show that SHMatrix is competitive, and more efficient for large matrices, than the equivalent routines of the LAPACK package. Running time: The examples included in the distribution take only a couple of seconds to run.