Comparison of Ten Methods for the Solution of Large and Sparse Linear Algebraic Systems

Abstract

The treatment of systems of linear algebraic equations is very often the most time-consuming part when large-scale applications arising in different fields of science and engineering are to be handled on computers. These systems can be very large, but in the most of the cases they are sparse (i.e. many of the elements in their coefficient matrices are zeros). Therefore, it is very important to select fast, robust and sufficiently accurate methods for the solution of large and sparse systems of linear algebraic equations. Tests with ten well-known methods have been carried out. Most of the methods are preconditioned conjugate gradient-type methods. Two important issues are mainly discussed: (i) the problem of finding automatically a good preconditioner and (ii) the development of robust and reliable stopping criteria. Numerical examples, which illustrate the efficiency of the developed algorithms for finding the preconditioner and for stopping the iterations when the required accuracy is achieved, are presented. The performance of the different methods for solving systems of linear algebraic equations is compared. Several conclusions are drawn, the main of them being the fact that it is necessary to include several different methods for the solution of large and sparse systems of linear algebraic equations in software designed to be used in the treatment of large-scale scientific and engineering problems.