A hybridized iterative algorithm of the BiCORSTAB and GPBiCOR methods for solving non-Hermitian linear systems

In this study, we derive a new iterative algorithm (including its preconditioned version) which is a hybridized variant of the biconjugate A -orthogonal residual stabilized (BiCORSTAB) method and the generalized product-type solvers based on BiCOR (GPBiCOR) method. The proposed method, which is named GPBiCOR( m , ? ) similarly to the GPBiCG( m , ? ) method proposed by Fujino (2002), can be regarded as an extension of the BiCORSTAB2 method introduced by Zhao and Huang (2013). Inspired by Fujino's idea for improving the BiCGSTAB2 method, in the established GPBiCOR( m , ? ) method the parameters computed by the BiCORSTAB method are chosen at successive m iteration steps, and afterwards the parameters of the GPBiCOR method are utilized in the subsequent ? iteration steps. Therefore, the proposed method can inherit the low computational cost of BiCORSTAB and the attractive convergence of GPBiCOR. Extensive numerical convergence results on selected real and complex matrices are shown to assess the performances of the proposed GPBiCOR( m , ? ) method, also against other popular non-Hermitian Krylov subspace methods.

[1]  D. R. Fokkema,et al.  BiCGstab(ell) for Linear Equations involving Unsymmetric Matrices with Complex Spectrum , 1993 .

[2]  H. V. D. Vorst,et al.  Generalized conjugate gradient squared , 1996 .

[3]  Shao-Liang Zhang,et al.  Product-type Krylov-Subspace Methods for Solving Nonsymmetric Linear Systems(Domain Decomposition Methods and Related Topics) , 1997 .

[4]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[5]  Valeria Simoncini,et al.  Recent computational developments in Krylov subspace methods for linear systems , 2007, Numer. Linear Algebra Appl..

[6]  C. Lanczos.  Solution of Systems of Linear Equations by Minimized Iterations1 , 1952 .

[7]  Martin B. van Gijzen,et al.  IDR(s): A Family of Simple and Fast Algorithms for Solving Large Nonsymmetric Systems of Linear Equations , 2008, SIAM J. Sci. Comput..

[8]  Jun Tang,et al.  An efficient and flexible computational model for solving the mild slope equation , 2004 .

[9]  S. Eisenstat,et al.  Variational Iterative Methods for Nonsymmetric Systems of Linear Equations , 1983 .

[10]  Ting-Zhu Huang,et al.  The BiCOR and CORS Iterative Algorithms for Solving Nonsymmetric Linear Systems , 2011, SIAM J. Sci. Comput..

[11]  Jianhua Zhang,et al.  A new quasi-minimal residual method based on a biconjugate A-orthonormalization procedure and coupled two-term recurrences , 2015, Numerical Algorithms.

[12]  Timothy A. Davis,et al.  Direct methods for sparse linear systems , 2006, Fundamentals of algorithms.

[13]  Martin H. Gutknecht,et al.  Variants of BICGSTAB for Matrices with Complex Spectrum , 1993, SIAM J. Sci. Comput..

[14]  M. Sugihara,et al.  An extension of the conjugate residual method to nonsymmetric linear systems , 2009 .

[15]  P. Sonneveld.  CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems , 1989 .

[16]  Jens-Peter M. Zemke,et al.  IDR ( s ) and IDR ( s ) Eig in Parallel Computing , 2010 .

[17]  Shao-Liang Zhang,et al.  GPBi-CG: Generalized Product-type Methods Based on Bi-CG for Solving Nonsymmetric Linear Systems , 1997, SIAM J. Sci. Comput..

[18]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[19]  Michael K. Ng,et al.  Weighted Toeplitz Regularized Least Squares Computation for Image Restoration , 2014, SIAM J. Sci. Comput..

[20]  Zhong-Zhi Bai,et al.  Motivations and realizations of Krylov subspace methods for large sparse linear systems , 2015, J. Comput. Appl. Math..

[21]  Ting-Zhu Huang,et al.  A generalized product-type BiCOR method and its application in signal deconvolution , 2013, Comput. Math. Appl..

[22]  Gerard L. G. Sleijpen,et al.  DELFT UNIVERSITY OF TECHNOLOGY REPORT 09-02 Exploiting BiCGstab ( l ) strategies to induce dimension reduction , 2009 .

[23]  Ting-Zhu Huang,et al.  Experiments with Lanczos biconjugate Aorthonormalization methods for MoM discretizations of Maxwell’s equations , 2009 .

[24]  Pavel Jiránek,et al.  How to Make Simpler GMRES and GCR More Stable , 2008, SIAM J. Matrix Anal. Appl..

[25]  Tony F. Chan,et al.  ML(k)BiCGSTAB: A BiCGSTAB Variant Based on Multiple Lanczos Starting Vectors , 1999, SIAM J. Sci. Comput..

[26]  Roland W. Freund,et al.  A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems , 1993, SIAM J. Sci. Comput..

[27]  R. Fletcher.  Conjugate gradient methods for indefinite systems , 1976 .

[28]  Liang Li,et al.  Quasi-Minimal Residual Variants of the COCG and COCR Methods for Complex Symmetric Linear Systems in Electromagnetic Simulations , 2014, IEEE Transactions on Microwave Theory and Techniques.

[29]  M. Gutknecht.  IDR Explained , 2008 .

[30]  Tongxiang Gu,et al.  Minimizing synchronizations in sparse iterative solvers for distributed supercomputers , 2014, Comput. Math. Appl..

[31]  Roland W. Freund,et al.  An Implementation of the Look-Ahead Lanczos Algorithm for Non-Hermitian Matrices , 1993, SIAM J. Sci. Comput..

[32]  Gerard L. G. Sleijpen,et al.  BiCR variants of the hybrid BiCG methods for solving linear systems with nonsymmetric matrices , 2010, J. Comput. Appl. Math..

[33]  Jianhua Zhang,et al.  A transpose-free quasi-minimal residual variant of the CORS method for solving non-Hermitian linear systems , 2015, J. Comput. Phys..

[34]  Markus Clemens,et al.  The SCBiCG class of algorithms for complex symmetric linear systems with applications in several electromagnetic model problems , 2015, Comput. Phys. Commun..

[35]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[36]  Serge Gratton,et al.  A Flexible Generalized Conjugate Residual Method with Inner Orthogonalization and Deflated Restarting , 2011, SIAM J. Matrix Anal. Appl..

[37]  Seiji Fujino,et al.  GPBiCG(m, l): a hybrid of BiCGSTAB and GPBiCG methods with efficiency and robustness , 2002 .

[38]  Gerard L. G. Sleijpen,et al.  A BiCGStab2 variant of the IDR(s) method for solving linear equations , 2012 .

[39]  Y. Saad,et al.  Experimental study of ILU preconditioners for indefinite matrices , 1997 .

[40]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[41]  Yu-Mei Huang,et al.  On $${\varvec{m}}$$m-step Hermitian and skew-Hermitian splitting preconditioning methods , 2015 .

[42]  Yong Duan,et al.  A comparative study of iterative solutions to linear systems arising in quantum mechanics , 2010, J. Comput. Phys..

[43]  P. Sonneveld,et al.  IDR(s): A family of simple and fast algorithms for solving large nonsymmetric linear systems , 2007 .

[44]  Ting-Zhu Huang,et al.  A hybrid variant of the BiCOR method for a nonsymmetric linear system with a complex spectrum , 2013, Appl. Math. Lett..

[45]  Jörg Liesen,et al.  A FRAMEWORK FOR DEFLATED AND AUGMENTED KRYLOV , 2013 .

[46]  Martin H. Gutknecht,et al.  Lanczos-type solvers for nonsymmetric linear systems of equations , 1997, Acta Numerica.

[47]  E. Sturler,et al.  Truncation Strategies for Optimal Krylov Subspace Methods , 1999 .

[48]  Ronald B. Morgan,et al.  GMRES WITH DEFLATED , 2008 .

[49]  Timothy A. Davis,et al.  The university of Florida sparse matrix collection , 2011, TOMS.

[50]  Hua Dai,et al.  GENERALIZED CONJUGATE A-ORTHOGONAL RESIDUAL SQUARED METHOD FOR COMPLEX NON-HERMITIAN LINEAR SYSTEMS * , 2014 .

[51]  Yong Zhang,et al.  Lanczos-type variants of the COCR method for complex nonsymmetric linear systems , 2009, J. Comput. Phys..