Volume 3, Issue 2
A Subspace-Projected Approximate Matrix Method for Systems of Linear Equations

East Asian J. Appl. Math., 3 (2013), pp. 120-137.

Published online: 2018-02

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• Abstract

Given two n×n matrices A and A0 and a sequence of subspaces {0} = V0 ⊂ ··· ⊂ Vn = R n with dim(Vk ) = k, the k-th subspace-projected approximated matrix Ak is defined as Ak = A + Πk (A0 − A)Πk , where Πk 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.

• Keywords

Linear system Galerkin method subspace projected approximate matrix minimal residual method

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@Article{EAJAM-3-120, author = {Jan Brandts and Ricardo R. da Silva}, title = {A Subspace-Projected Approximate Matrix Method for Systems of Linear Equations}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {3}, number = {2}, pages = {120--137}, abstract = {

Given two n×n matrices A and A0 and a sequence of subspaces {0} = V0 ⊂ ··· ⊂ Vn = R n with dim(Vk ) = k, the k-th subspace-projected approximated matrix Ak is defined as Ak = A + Πk (A0 − A)Πk , where Πk 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.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.070213.280513a}, url = {http://global-sci.org/intro/article_detail/eajam/10851.html} }
TY - JOUR T1 - A Subspace-Projected Approximate Matrix Method for Systems of Linear Equations AU - Jan Brandts & Ricardo R. da Silva JO - East Asian Journal on Applied Mathematics VL - 2 SP - 120 EP - 137 PY - 2018 DA - 2018/02 SN - 3 DO - http://doi.org/10.4208/eajam.070213.280513a UR - https://global-sci.org/intro/article_detail/eajam/10851.html KW - Linear system KW - Galerkin method KW - subspace projected approximate matrix KW - minimal residual method AB -

Given two n×n matrices A and A0 and a sequence of subspaces {0} = V0 ⊂ ··· ⊂ Vn = R n with dim(Vk ) = k, the k-th subspace-projected approximated matrix Ak is defined as Ak = A + Πk (A0 − A)Πk , where Πk 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.

Jan Brandts & Ricardo R. da Silva. (1970). A Subspace-Projected Approximate Matrix Method for Systems of Linear Equations. East Asian Journal on Applied Mathematics. 3 (2). 120-137. doi:10.4208/eajam.070213.280513a
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