Performance Analysis of Effective Methods for Solving Band Matrix SLAEs After Parabolic Nonlinear PDEs

Abstract

This paper presents an experimental performance study of implementations of three different types of algorithms for solving band matrix systems of linear algebraic equations (SLAEs) after parabolic nonlinear partial differential equations -direct, symbolic, and iterative, the former two of which were introduced in Veneva and Ayriyan (arXiv:1710.00428v2). An iterative algorithm is presented -the strongly implicit procedure (SIP), also known as the Stone method. This method uses the incomplete LU (ILU(0)) decomposition. An application of the Hotelling-Bodewig iterative algorithm is suggested as a replacement of the standard forward-backward substitutions. The upsides and the downsides of the SIP method are discussed. The complexity of all the investigated methods is presented. Performance analysis of the implementations is done using the high-performance computing (HPC) clusters "HybriLIT" and "Avitohol". To that purpose, the experimental setup and the results from the conducted computations on the individual computer systems are presented and discussed.

Performance Analysis of Effective Symbolic Methods for Solving Band Matrix SLAEs

“…Here, we are going to focus on two of the most popular programming languages for scientific computations, namely C++ and Python. The aim of this paper, which is a logical continuation of [5] and [6], is to investigate the performance characteristics of the considered serial methods with their two implementations being executed on modern computer clusters.…”

Symbolic Algorithm for Solving SLAEs with Heptadiagonal Coefficient Matrices

“…A performance analysis of effective methods (both numerical and symbolic) for solving band matrix SLAEs (with three and five diagonals) being implemented in C++ and run on modern (as of 2018) computer systems is made by us in [9]. Different strategies (symbolic included) for solving band matrix SLAEs (with three and five diagonals) are explored by us in [11].…”

“…A performance analysis of effective methods (both numerical and symbolic) for solving band matrix SLAEs (with three and five diagonals) being implemented in C++ and run on modern (as of 2018) computer systems is made by us in [9]. Different strategies (symbolic included) for solving band matrix SLAEs (with three and five diagonals) are explored by us in [11]. A performance analysis of effective symbolic algorithms for solving band matrix SLAEs with coefficient matrices with three, five and seven diagonals being implemented in both C++ and Python and run on modern (as of 2018) computer systems is made by us in [10].…”