Abstract
A direct as well as iterative method (called the orthogonally accumulated projection method, or the OAP for short) for solving linear system of equations with symmetric coefficient matrix is introduced in this paper. With the Lanczos process the OAP creates a sequence of mutually orthogonal vectors, on the basis of which the projections of the unknown vectors are easily obtained, and thus the approximations to the unknown vectors can be simply constructed by a combination of these projections. This method is an application of the accumulated projection technique proposed recently by the authors of this paper, and can be regarded as a match of conjugate gradient method (CG) in its nature since both the CG and the OAP can be regarded as iterative methods, too. Unlike the CG method which can be only used to solve linear systems with symmetric positive definite coefficient matrices, the OAP can be used to handle systems with indefinite symmetric matrices. Unlike classical Krylov subspace methods which usually ignore the issue of loss of orthogonality, OAP uses an effective approach to detect the loss of orthogonality and a restart strategy is used to handle the loss of orthogonality. Numerical experiments are presented to demonstrate the efficiency of the OAP.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bai Z Z. Rotated block triangular preconditioning based on PMHSS. Sci China Math, 2013, 56: 2523–2538
Benzi M. Preconditioning techniques for large linear systems: A survey. J Comput Phys, 2002, 182: 418–477
Bramley R, Sameh A. Row projection methods for large nonsymmetric linear systems. SIAM J Sci Comput, 1992, 13: 168–193
Ehrig R, Deuflhard P. GMERR—an error minimizing variant of GMRES. Technical Report SC-97-63, ZIB, 1997
Freund R W, Nachtigal N M. QMR: A quasi-minimal residual method for non-Hermitian linear systems. Numer Math, 1991, 60: 315–339
Galäntai A. Projectors and Projection Methods. Berlin: Springer, 2004
Golub G H, Van Loan C F. Matrix Computations. Baltimore-London: The Johns Hopkins University Press, 1996
Hackbusch W. Multi-Grid Methods and Applications. Berlin: Springer-Verlag, 1985
Jia Z X. Refined iterative algorithms based on Arnoldi’s process for large unsymmetric eigenproblems. Linear Algebra Appl, 1997, 259: 1–23
Jia Z X. On convergence of the inexact Rayleigh quotient iteration with the Lanczos method used for solving linear systems. Sci China Math, 2013, 56: 2145–2160
Paige C C, Saunders M A. Solution of sparse indefinite systems of linear equations. SIAM J Numer Anal, 1975, 12: 617–629
Peng W. A line-projection method for solving linear system of equations. Pacific J Appl Math, 2013, 5: 17–28
Peng W, Lin Q. A non-Krylov subspace method for solving large and sparse linear system of equations. Numer Math Theor Meth Appl, 2016, 9: 289–314
Peng W, Zhang S. A stationary accumulated projection method for linear system of equations. ArXiv:1603.05356, 2016
Saad Y. Iterative Methods for Sparse Linear Systems, 2nd ed. Philadelphia: SIAM, 2003
van der Vorst H A. Iterative Krylov Methods for Large Linear Systems. Cambridge: Cambridge University Press, 2003
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Peng, W., Lin, Q. & Zhang, S. An orthogonally accumulated projection method for symmetric linear system of equations. Sci. China Math. 59, 1235–1248 (2016). https://doi.org/10.1007/s11425-016-5142-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-016-5142-5