Solving Band Systems of Linear Equations by Iterative Methods on Vector Machines

Abstract

The use of iterative methods in the solution of systems of linear algebraic equations whose coefficient matrices are band is an important problem, because such systems do appear in many fields of engineering and science. Band matrices, symmetric and positive definite as well as unsymmetric, can efficiently be treated on many vector machines. The possibility of exploiting zero diagonals within the band in order to reduce the storage and the computing time will be discussed (also such matrices appear very often when large-scale problems are to be handled). The treatment of some special matrices, as tri-diagonal and five-diagonal matrices, will be sketched. It will be shown that on vector machines iterative methods may perform better than the direct methods even in the case where the direct methods are better than the iterative methods on scalar machines. The reason for this phenomenon is the fact that many iterative methods can be designed so that one works with long and contiguous arrays during the whole computational process. This can not be achieved when systems of linear algebraic equations with band matrices are solved by direct methods.