Modified Gaussian elimination technique using at each step an equation with original coefficients

Abstract

The first n-1 equations of a general n-by-n linear algebraic system are first made homogeneous. Then, the first unknown is eliminated from the equations 2, 3, ..., n-1, with the n-th equation kept in its original form. A first solution of the equations 2 to n-1 and of the n-th equation is obvious. If a second, independent solution of these equations is found, the solution of the original system will be determined from a linear combination of the two solutions by imposing the condition that its 1-st equation is also satisfied. This second solution is obtained in the same way, as the solution of an (n-1)-by-(n-1) system whose last equation only contains coefficients in the original n-th equation. After the elimination process is completed, the solution of the original system is derived by a backward substitution. The computational complexity in n3 is the same as in the classical Gaussian elimination, but the new method presents an improved stability and accuracy as shown in numerical experiments with infinite systems encountered in the analysis of electric fields in multiple-sphere configurations.