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

作者：Brandts Jan^*; da Silva Ricardo R

来源：East Asian Journal on Applied Mathematics, 2013, 3(2): 120-137.

DOI：10.4208/eajam.070213.280513a

• Linear system
• Galerkin method
• subspace projected approximate matrix
• minimal residual method

Given two n x n matrices A and Ao and a sequence of subspaces {0} = gamma(0) subset of ... subset of gamma(n) = R-n with dim(gamma(k)) = k, the k-th subspace-projected approximated matrix A(k) is defined as A(k) = A + Pi(k)(A(0) - A)Pi(k), where Ilk is the orthogonal projection on Consequently, A(k)nu = A nu and nu*A(k) = nu*A for all nu epsilon gamma(k). Thus A(k)nu = A nu ,0 is a sequence of matrices that gradually changes from A(0) into A A. In principle, the definition of Yk+1 may depend on properties of A(k), which can be exploited to try to force A(k+1) to be closer to A in some specific sense. By choosing Ao 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 A(k)x(k) = 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 (x(k))(k %26gt;= 0) approximates x, by performing some illustrative numerical tests.

• 出版日期2013-5