Abstract
The solution of linear systems represented by Ax = b is fundamental in many numerical simulations in science and engineering. Reducing the profile of A can reduce the storage requirements and time processing costs of solving such linear systems. In this work, we propose a generalized algorithm for finding pseudo–peripheral vertices for Snay’s heuristic. In experiment performed on 36 instances contained in the Harwell-Boeing and SuiteSparse matrix collections, it has been found that the number of pseudo– peripheral vertices selected in Snay’s heuristic may be suitable for small instances, but it is insufficient to obtain reasonable results in instances that are not small. This paper recommends to select up to 26% (0.3%) of pseudo–peripheral vertices in relation to the instance size when applied to instances smaller than 3,000 (larger than 20,000) vertices.
Highlights
Several real-world problems reduce into a linear system in the form Ax = b, where A is an n × n large-scale sparse matrix, x is the unknown n-vector solution which is sough, and b is a known n-vector
Snay’s heuristic with starting vertices given by the generalized algorithm for finding pseudo–peripheral vertices [Snay(ν)] achieved better profile results than Snay’s heuristic with starting vertices given by the original Snay’s algorithm for finding pseudo–peripheral vertices [19], at a higher execution time
A previous publication [14] reports that in certain cases several reordering algorithms do not reduce the computational times of the Jacobi-preconditioned conjugate gradient method
Summary
Several real-world problems reduce into a linear system in the form Ax = b, where A is an n × n large-scale sparse matrix, x is the unknown n-vector solution which is sough, and b is a known n-vector. The resolution of large sparse linear systems in this form is crucial in various engineering and science applications. It is normally the part of the simulation that requires the highest processing cost. If the coefficient matrix A is dense, users employ a direct method (e.g., Gaussian Elimination, LU factorization, Cholesky factorization, etc.) to solve the linear system. Reducing the profile of a sparse symmetric matrix A can benefit the storage requirements and processing times to solve the linear system
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