The solution of linear systems by using the Sherman–Morrison formula

Abstract We consider the problem of solving the linear system A x  =  b , where A is the coefficient matrix, b is the known right hand side vector and x is the solution vector to be determined. Let us suppose that A is a nonsingular square matrix, so that the linear system A x  =  b is uniquely solvable. The well known Sherman–Morrison formula, that gives the inverse of a rank-one perturbation of a matrix from the knowledge of the unperturbed inverse matrix, is used to compute the numerical solution of arbitrary linear systems, in fact it can be repetitively applied to invert an arbitrary matrix. We describe some interesting properties of the method proposed. Finally we show some numerical results obtained with the method proposed.