Abstract
Solving large scale system of Simultaneous Linear Equations (SLE) has been (and continue to be) a major challenging problem for many real-world engineering and science applications. Solving SLE with singular coefficient matrices arises from various engineering and sciences applications [1]-[6]. In this paper, efficient numerical procedures for finding the generalized (or pseudo) inverse of a general (square/rectangle, symmetrical/unsymmetrical, non-singular/singular) matrix and solving systems of Simultaneous Linear Equations (SLE) are formulated and explained. The developed procedures and its associated computer software (under MATLAB [7] computer environment) have been based on “special Cholesky factorization schemes” (for a singular matrix). Test matrices from different fields of applications have been chosen, tested and compared with other existing algorithms. The results of the numerical tests have indicated that the developed procedures are far more efficient than the existing algorithms.
Highlights
Most computational time is spent on solving system of Simultaneous Linear Equations (SLE) which can be represented in matrix notations as
The generalized inverse of a matrix is an extension of the ordinary/regular square matrix inverse, which can be applied to any matrix
In this paper we introduce an efficient generalized inverse formulation to solve SLE with full or deficient rank of the coefficient matrix
Summary
Most computational time is spent on solving system of Simultaneous Linear Equations (SLE) which can be represented in matrix notations as. The generalized (or pseudo) inverse of a matrix is an extension of the ordinary/regular square (non-singular) matrix inverse, which can be applied to any matrix (such as singular, rectangular etc.). In this paper we introduce an efficient (in terms of computational time and computer memory requirement) generalized inverse formulation to solve SLE with full or deficient rank of the coefficient matrix. Extensive set of coefficient matrices (including rectangular, square, symmetrical, non-symmetrical, singular, non-singular matrices) obtained from well-established/popular websites [8] [9] were tested and the numerical performance in terms of timings, error norm were compared with other algorithms. (a) The “special Cholesky factorization” (for symmetrical/singular coefficient matrix), and (b) The generalized inverse of a product of 2 matrices [6] and can be described in the following paragraphs. Equation (3.9)] to be formed explicitly, our main idea is to solve SLE where b is a known right-hand-side vector
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