Abstract
This paper presents an experimental performance study of implementations of three symbolic algorithms for solving band matrix systems of linear algebraic equations with heptadiagonal, pentadiagonal, and tridiagonal coefficient matrices. The only assumption on the coefficient matrix in order for the algorithms to be stable is nonsingularity. These algorithms are implemented using the GiNaC library of C++ and the SymPy library of Python, considering five different data storing classes. Performance analysis of the implementations is done using the high-performance computing (HPC) platforms “HybriLIT” and “Avitohol”. The experimental setup and the results from the conducted computations on the individual computer systems are presented and discussed. An analysis of the three algorithms is performed.
Highlights
Systems of linear algebraic equations (SLAEs) with heptadiagonal (HD), pentadiagonal (PD) and tridiagonal (TD) coefficient matrices may arise after many different scientific and engineering problems, as well as problems of the computational linear algebra where finding the solution of a SLAE is considered to be one of the most important problems
This paper presents an experimental performance study of implementations of three symbolic algorithms for solving band matrix systems of linear algebraic equations with heptadiagonal, pentadiagonal, and tridiagonal coefficient matrices
It must be mentioned that the matrix class in SymPy is a subclass of the ndarray. This means that every call on a matrix object requires a few extra Python calls
Summary
Systems of linear algebraic equations (SLAEs) with heptadiagonal (HD), pentadiagonal (PD) and tridiagonal (TD) coefficient matrices may arise after many different scientific and engineering problems, as well as problems of the computational linear algebra where finding the solution of a SLAE is considered to be one of the most important problems. The latter two points explain why there is a need of methods for solving of SLAEs which take into account the band structure of the matrices and do not have any other special requirements to them. One possible approach to this problem is the symbolic algorithms. An overview of some of the symbolic algorithms which exist in the literature is done by us in [1]. What is common for all of them, is that they are implemented using Computer Algebra Systems (CASs) such as Maple, Mathematica, and Matlab
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