Solution of systems of linear algebraic equations with preliminary flattening (Q1580213)

A method for improving the computational accuracy of a system of linear equations having a coefficient matrix with nearly linearly depedent columns is given. Such an ill conditioned matrix has a small determinant. The absolute value of the determinant is equal to the volume of the simplex based on the column vectors of the matrix. In the proposed method, called the process of flattening, the volume of the simplex is increased. This leads to better conditioning. It is shown that the Gaussian elimination method with the proposed flattening gives significantly less relative error than the usual Gaussian elimination. This even holds for ill-conditioned Hilbert matrices.