Abstract
A method for solving systems of linear equations is presented based on direct decomposition of the coefficient matrix using the form LAX = LB = B’ . Elements of the reducing lower triangular matrix L can be determined using either row wise or column wise operations and are demonstrated to be sums of permutation products of the Gauss pivot row multipliers. These sums of permutation products can be constructed using a tree structure that can be easily memorized or alternatively computed using matrix products. The method requires only storage of the L matrix which is half in size compared to storage of the elements in the LU decomposition. Equivalence of the proposed method with both the Gauss elimination and LU decomposition is also shown in this paper.
Highlights
Systems of linear equations or equations linearized for iterative solutions arise in many science and engineering problems [1]
A direct decomposition of the coefficient matrix forming part of a system of linear equations using a single lower triangular reducing matrix L has been demonstrated as shown in this paper
Elements of the reducing matrix L are shown to be sums of permutation products of the pivot row multipliers of the Gauss elimination technique. These sums of permutation products, for any element of the reducing matrix L, can be constructed using a tree diagram that is relatively easy to memorize besides using the formula developed for the purpose
Summary
Systems of linear equations or equations linearized for iterative solutions arise in many science and engineering problems [1]. Practical applications of systems of linear equations are many, examples of such application include applications in digital signal processing, linear programming problems, numerical analysis of non-linear problems and least square curve fitting [2]. Systems of equations are historically reported to have provided a motivation for the development of digital computer as less cumbersome way of solving the equations [3]. Gaussian elimination is a systematic way of reducing systems of linear equations into a triangularised matrix through addition of the independent equations.
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