Modified quasi-Newton methods for solving systems of linear equations

Abstract

Quasi-Newton methods for unconstrained optimization problems are considered for solving a system of linear equations Ax = b where A ∈ R n×n , Rank(A )= n, b ∈ R n , and x ∈ R n is the vector of unknowns. This problem can be converted into an equivalent quadratic optimization problem. Based on the observation that if H ≈ (A T A) −1 = A −1 (A T ) −1 then ¯ x = HA T b can be taken as an approximate solution of the problem, we propose a modification to the Quasi-Newton method. The modified algorithm incorporates the above observation. Global convergence is ensured by adding the steepest descent direction into the combination. Numerical experiments are also reported.