Some Convalescent Methods for the Solution of Systems of Linear Equations

Abstract

In a variety of problems in the fields of physical sciences, engineering, economics, etc., we are led to systems of linear equations, Ax = b, comprising n linear equations in n unknowns x1, x2, …, xn, where A = [aij] is an nxn coefficient matrix, and x = [x1 x2 . . .xn]T, b = [b1 b2 . . .bn]T are the column vectors. There are many analytical as well as numerical methods[1}– [11] to solve such systems of equations, including Gauss elimination method, and its modifications namely Doolittle’s method, Crout’s method and Cholesky’s method, which employ LU-decomposition method, where L = [iij] and u = [uij] are the lower and upper triangular matrices respectively. The LU-decomposition method was first introduced by the mathematician Alan M. Turing[2]-[11] in 1948. Here, in this paper we have made an effort to modify the existing LU-decomposition methods to solve the above mentioned system Ax = b, with the least possible endeavour. It may be seen that the Gauss elimination method[1], [2], [3], [4] needs about 2n3/3 operations, while Doolittle’s and Crout’s methods require n2 operations. Accordingly, in these methods we are required to evaluate n2 number of unknown elements of the L and U matrices. Moreover, Cholesky’s method[1] requires 2n2/3 operations. Accordingly this method requires evaluation of 2n2/3 number of unknown elements of the L and U matrices But, in contrast, the improved Doolittle’s, Crout’s and Cholesky’s methods presented in this paper require evaluation of only (n–1)2 number of unknown elements of the L and U matrices. Moreover, an innovative method is also presented in this paper which requires evaluation of even less number of unknown elements of the L and U matrices. In this method we need to evaluate only (n–2)2 number of the said unknown elements. Thus, by employing these methods, the computational time and effort required for the purpose can substantially be reduced.