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A Non-Krylov Subspace Method for Solving Large and Sparse Linear System of Equations

Published online by Cambridge University Press:  24 May 2016

Wujian Peng  and
Qun Lin
Wujian Peng*
Affiliation:
Department of Math. and Stats. Sciences, Zhaoqing Univ., Zhaoqing, 526061, China
Qun Lin*
Affiliation:
Academy of Math. and System Sciences, Chinese Academy of Sciences, 100081, China
*
*Corresponding author. Email addresses: [email protected] (W.-J. Peng), [email protected] (Q. Lin)
*Corresponding author. Email addresses: [email protected] (W.-J. Peng), [email protected] (Q. Lin)
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Abstract

Most current prevalent iterative methods can be classified into the socalled extended Krylov subspace methods, a class of iterative methods which do not fall into this category are also proposed in this paper. Comparing with traditional Krylov subspace methods which always depend on the matrix-vector multiplication with a fixed matrix, the newly introduced methods (the so-called (progressively) accumulated projection methods, or AP (PAP) for short) use a projection matrix which varies in every iteration to form a subspace from which an approximate solution is sought. More importantly an accelerative approach (called APAP) is introduced to improve the convergence of PAP method. Numerical experiments demonstrate some surprisingly improved convergence behavior. Comparison between benchmark extended Krylov subspace methods (Block Jacobi and GMRES) are made and one can also see remarkable advantage of APAP in some examples. APAP is also used to solve systems with extremely ill-conditioned coefficient matrix (the Hilbert matrix) and numerical experiments shows that it can bring very satisfactory results even when the size of system is up to a few thousands.

Keywords

15A0665F10Iterative methodaccumulated projectionKrylov subspace
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 9 , Issue 2 , May 2016 , pp. 289 - 314
Copyright
Copyright © Global-Science Press 2016 

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