The updating technique for the solution of a sequence of linear equations

In this paper we will present and analyze an algorithm for solving a sequence of linear equations of the form (left( {A + lambda _i B} right)x = b,i = 1,2,...,r,) wherex, b∈R ⁿ , andA, B aren x n real dense and large matrices. Two matrix factorizations are suggested that will allow efficient updating by changing λ _i . They are recommended ifB is singular or ill-conditioned. The implementation cost is evaluated and numerical results are included to present the performance of the two factorizations and in the solution of the above sequence of equations.