Flexible GPBi-CG Method for Nonsymmetric Linear Systems

Jia-Min Wang, Tong-Xiang Gu

Graduate School of Chinese Academy of Engineering Physics, Beijing, China.
Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, China.

DOI: 10.4236/am.2012.34050                

Abstract

We present a flexible version of GPBi-CG algorithm which allows for the use of a different preconditioner at each step of the algorithm. In particular, a result of the flexibility of the variable preconditioner is to use any iterative method. For example, the standard GPBi-CG algorithm itself can be used as a preconditioner, as can other Krylov subspace methods or splitting methods. Numerical experiments are conducted for flexible GPBi-CG for a few matrices including some nonsymmetric matrices. These experiments illustrate the convergence and robustness of the flexible iterative method.

Keywords

Krylov Subspace Method; Flexible Preconditioning; Inner-Outer Iteration; GPBi-CG

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Y. Notay, “Flexible Conjugate Gradients,” SIAM Journal on Scientific Computing, Vol. 22, No. 4, 2000, pp. 14441460. doi:10.1137/S1064827599362314
[2]	Y. Saad, “A Flexible Inner-Outer Preconditioned GMRES Algorithm,” SIAM Journal on Scientific Computing, Vol. 14, No. 2, 1993, pp. 461-469. doi:10.1137/0914028
[3]	D. B. Szyld and J. A. Vogel, “FQMR: A Flexible QuasiMinimal Residual Method with Inexact Preconditioning,” SIAM Journal on Scientific Computing, Vol. 23, No. 2, 2001, pp. 363-380. doi:10.1137/S106482750037336X
[4]	K. Abe and S.-L. Zhang, “A Variable Preconditioning Using The SOR Method For GCR-like Methods,” International Journal of Numerical Analysis and Modeling, Vol. 2, No.2, 2005, pp. 147-161.
[5]	J. A. Vogel, “Flexible BiCG and Flexible Bi-CGSTAB for Nonsymmetric Linear Systems,” Applied Mathematics and Computation, Vol. 188, No. 1, 2007, pp. 226-233. doi:10.1016/j.amc.2006.09.116
[6]	S.-L. Zhang, “GPBi-CG: Generalized Product-Type Methods Baseed on Bi-CG for Solving Nonsymentric Linear Systems,” SIAM Journal on Scientific Computing, Vol. 18, No. 2, 1997, pp. 537-551. doi:10.1137/S1064827592236313
[7]	Y. Saad, “Iterative Method for Solving Linear Systems,” 2nd Edition, SIAM, Philadelphia, 2003. doi:10.1137/1.9780898718003