A Krylov Subspace Method to Solve a Sequence of Linear Systems with Dioeerent Right-Hand.pdfVIP

A Krylov Subspace Method to Solve a Sequenceof Linear Systems with Dierent Right-HandSidesDaniel Skoogh November 1998AbstractA new method to solve linear systems of equations with several right-hand sides is described. It uses the basis from a previous solution to reducethe number of matrix vector products needed to solve a linear system ofequations with a new right-hand side. It builds up a subspace of a union ofKrylov spaces. Some numerical examples are given where variants of themethod are compared to Krylov subspace methods, particularly a blockArnoldi (GMRES) algorithm.Keywords: Arnoldi, Krylov, block, right-hand sides, iterativeAMS subject classication 65F101 IntroductionConsider a set of k linear systems with the same matrix A, but with severalright hand sides bi; i = 1; : : : ; k:Axi = bi; i = 1; : : : ; k (1)A 2 Cnn; bi;xi 2 Cn: (2)We are going to describe a new method that reduces the total number of matrixvector products needed to solve the system (1) compared to when the system issolved for each right-hand side separately. We will discuss cases when all righthand sides are known before the iterations begin, as well as cases when they arenot.If all right-hand sides are known before the iterations begin, then the blockArnoldi method can be used to solve the set of systems (1). It generates or-thonormal basis vectors v1; : : : ;vm+k that span the subspaceSm+k = k[i=1K(mk +1)(A; bi); (3)Report No 1998-46 in Bl? serien preprint series, available at URL:http://www.math.chalmers.se/Math/Research/Preprints. Authors address: Depart-ment of Mathematics, Chalmers University of Technology and the University of G?teborg,S-41296 G?teborg, Sweden (skoogh@math.chalmers.se)1 which is a union of Krylov spaces. As usual, a Krylov subspace is dened byKk(A;v) = spanfv;Av;A2v; : : : ;Ak1vg: (4)In this report we will introduce new algorithms that generate orthonormal basisvectors which span a subspace of a union of Krylov spaces, whereas block Arnoldigenerates basis v