A class of linear solvers built on the Biconjugate A-Orthonormalization Procedure for solving unsymmetric linear systems

We present economical iterative algorithms built on the Biconjugate $A$-Orthonormalization Procedure for real unsymmetric and complex non-Hermitian systems. The principal characteristics of the developed solvers is that they are fast convergent and cheap in memory. We report on a large combination of numerical experiments to demonstrate that the proposed family of methods is highly competitive and often superior to other popular algorithms built upon the Arnoldi method and the biconjugate Lanczos procedures for unsymmetric linear sytems.

[1]  Jiming Song,et al.  Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects , 1997 .

[2]  知広 曽我部.  Extensions of the conjugate residual method , 2006 .

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

[4]  Alan Edelman,et al.  Large Dense Numerical Linear Algebra in 1993: the Parallel Computing Influence , 1993, Int. J. High Perform. Comput. Appl..

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

[6]  Richard Barrett,et al.  Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods , 1994, Other Titles in Applied Mathematics.

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

[8]  Zhishun A. Liu,et al.  A Look Ahead Lanczos Algorithm for Unsymmetric Matrices , 1985 .

[9]  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..

[10]  Guillaume Sylvand.  La méthode multipôle rapide en électromagnétisme. Performances, parallélisation, applications , 2002 .

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

[12]  Weng Cho Chew,et al.  The Fast Illinois Solver Code: requirements and scaling properties , 1998 .

[13]  Yousef Saad,et al.  A Flexible Inner-Outer Preconditioned GMRES Algorithm , 1993, SIAM J. Sci. Comput..

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

[15]  Beresford N. Parlett,et al.  Reduction to Tridiagonal Form and Minimal Realizations , 1992, SIAM J. Matrix Anal. Appl..

[17]  Jiming Song,et al.  Fast Illinois solver code (FISC) , 1998 .

[18]  R. Freund,et al.  QMR: a quasi-minimal residual method for non-Hermitian linear systems , 1991 .

[19]  L. Gurel,et al.  Sequential and parallel preconditioners for large-scale integral-equation problems , 2007, 2007 Computational Electromagnetics Workshop.

[20]  Alan Edelman,et al.  The first annual large dense linear system survey , 1991, SGNM.

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

[22]  D. Day,et al.  An Efficient Implementation of the Nonsymmetric Lanczos Algorithm , 1997 .

[23]  Ozgur Ergul,et al.  Fast and accurate solutions of extremely large integral-equation problems discretised with tens of millions of unknowns , 2007 .

[24]  L. Gurel,et al.  Efficient Parallelization of the Multilevel Fast Multipole Algorithm for the Solution of Large-Scale Scattering Problems , 2008, IEEE Transactions on Antennas and Propagation.

[25]  Bruno Carpentieri,et al.  Combining Fast Multipole Techniques and an Approximate Inverse Preconditioner for Large Electromagnetism Calculations , 2005, SIAM J. Sci. Comput..

[26]  Levent Gürel,et al.  Incomplete LU Preconditioning with the Multilevel Fast Multipole Algorithm for Electromagnetic Scattering , 2007, SIAM J. Sci. Comput..

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

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

[29]  H. V. D. Vorst,et al.  An overview of approaches for the stable computation of hybrid BiCG methods , 1995 .