A Subspace-Projected Approximate Matrix Method for Systems of Linear Equations

Given two n×n matrices A and A0 and a sequence of subspaces {0}=V0 ⊂ · · · ⊂ Vn = Rn with dim(Vk) = k, the k-th subspace-projected approximated matrix Ak is defined as Ak = A + k(A0 − A)k , where k is the orthogonal projection on V ⊥ k . Consequently, Ak v = Av and v∗Ak = v∗A for all v ∈ Vk. Thus (Ak)n k≥0 is a sequence of matrices that gradually changes from A0 into An = A. In principle, the definition of Vk+1 may depend on properties of Ak, which can be exploited to try to force Ak+1 to be closer to A in some specific sense. By choosing A0 as a simple approximation of A, this turns the subspace-approximated matrices into interesting preconditioners for linear algebra problems involving A. In the context of eigenvalue problems, they appeared in this role in Shepard et al. (2001), resulting in their Subspace Projected Approximate Matrix method. In this article, we investigate their use in solving linear systems of equations Ax = b. In particular, we seek conditions under which the solutions xk of the approximate systems Ak xk = b are computable at low computational cost, so the
efficiency of the corresponding method is competitive with existing methods such as the Conjugate Gradient and the Minimal Residual methods. We also consider how well the sequence (xk)k≥0 approximates x, by performing some illustrative numerical tests. AMS subject classifications: 65F10, 65F08