Solving Linear Systems of the Form \({\boldsymbol{(A + \gamma UU^T)\, {x} = {b}}}\) by Preconditioned Iterative Methods

We consider the iterative solution of large linear systems of equations in which the coefficient matrix is the sum of two terms, a sparse matrix A and a possibly dense, rank deficient matrix of the form \gammaUUT , where \gamma > 0 is a parameter which in some applications may be taken to be 1. The matrix A itself can be singular, but we assume that the symmetric part of A is positive semidefinite and that A + \gammaUUT is nonsingular. Linear systems of this form arise frequently in fields like optimization, fluid mechanics, computational statistics, and others. We investigate preconditioning strategies based on an alternating splitting approach combined with the use of the Sherman-Morrison-Woodbury matrix identity. The potential of the proposed approach is demonstrated by means of numerical experiments on linear systems from different application areas.