General scheme for solving linear algebraic problems by direct methods

Abstract Assume that A is an m × n real matrix with m⩾n and rank(A)=n. Let b be a real vector with m components and consider the problem of finding x=A † b , where A † =(A T A) −1 A T . A general scheme for solving this problem is described. Many well-known and commonly used in practice direct methods can be found as special cases within the general scheme. This is illustrated by four examples. The general scheme can be used to study the common properties of the direct methods. The usefulness of this approach in the efforts to improve the efficiency of the direct methods for sparse matrices is demonstrated by formulating an algorithm that can be applied to any particular method belonging to the general scheme. This algorithm is implemented in several subroutines solving linear algebraic problems by different direct methods. Numerical results, obtained in a wide range of runs with these subroutines, are given.

[1]  Z. Zlatev.  Use of Iterative Refinement in the Solution of Sparse Linear Systems , 1982 .

[2]  P. G. Thomsen,et al.  Application of backward differentiation methods to the finite element solution of time‐dependent problems , 1979 .

[3]  Åke Bjürck.  METHODS FOR SPARSE LINEAR LEAST SQUARES PROBLEMS , 1976 .

[4]  Michael A. Malcolm,et al.  Computer methods for mathematical computations , 1977 .

[5]  Z. Zlatev.  On Some Pivotal Strategies in Gaussian Elimination by Sparse Technique , 1980 .

[6]  Zahari Zlatev,et al.  The use of sparse matrix technique in the numerical integration of stiff systems of linear ordinary differential equations , 1980, Comput. Chem..

[7]  D. Faddeev,et al.  Computational methods of linear algebra , 1959 .

[8]  Z. Zlatev.  Comparison of two pivotal strategies in sparse plane rotations , 1980 .

[9]  G. Stewart.  Introduction to matrix computations , 1973 .

[10]  Åke Björck,et al.  Comment on the Iterative Refinement of Least-Squares Solutions , 1978 .

[11]  James Hardy Wilkinson,et al.  The Least Squares Problem and Pseudo-Inverses , 1970, Comput. J..

[12]  Zahari Zlatev,et al.  Comparison of Two Algorithms for Solving Large Linear Systems , 1982 .

[13]  Zahari Zlatev,et al.  A testing scheme for subroutines solving large linear problems , 1981, Comput. Chem..

[14]  Gary K. Leaf,et al.  A Preconditioned Conjugate Gradient Method for Solving a Class of Non-Symmetric Linear Systems , 1981 .

[15]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[16]  James Hardy Wilkinson,et al.  Error Analysis of Direct Methods of Matrix Inversion , 1961, JACM.