Conference publications

Abstracts

XIV conference

The direct projection method for solving systems of linear equation with sparse unstructured matrices

Gogoleva S.Y.

The Samara State Aerospace University named after academician S.P. Korolev, applied mathematics department, Russia, 443086,.Samara, Moscow Shosse, 34. т. (846) 332-56-07, E-mail: gogoleva_s@mail.ru

The mathematical models of many practical problems result in systems of linear algebraic equations (SLAE) with large and sparse matrices of coefficients. When the large part of matrix coefficients consists of zero, it is quite obvious, that we try to store only nonzero elements. The serious problem at a storage and processing of sparse matrices is represented by fill-in, i.e. occurrence of new nonzero elements. Reduction of fill-in accompany reduction of the requirements to memory volume and work acceleration of method.

In the given work the direct projection method (DPM) of solving SLAE is considered [1]. In this method if to carry out analogy to methods based on decomposition of matrices (LU-decomposition), the matrix of equations system A is factored into two matrices L_A and R. [1] For realization DPM probably to store in operative memory only matrix R, that requires accordingly n(n+1)/2 machine words. Thus, DPM gives a prize in required operative memory volume twice, in comparison with the computation scheme based on LU - decomposition, that it is favourable to use at the solving of large sparse SLAE. The total number of arithmetic operations required for DPM is estimated by size 2n³/3+O(n²). It almost as much, how many in a method based on LU - decomposition.

At the solving SLAE with sparse matrices by DPM the fill-in occurs only in a matrix R, whereas in LU - decomposition the fill-in can occur, both in a matrix L, and in a matrix U.

In a case for SLAE with the sparse unstructured matrices the numerical researches of DPM were carried out on examples considered in [2]. The results of numerical researches have shown, that at the solving of SLAE with the given matrices DPM surpasses LU - decomposition in arithmetic operations number and operative memory.