Block s‐step Krylov iterative methods

Block (including s-step) iterative methods for (non)symmetric linear systems have been studied and implemented in the past. In this article we present a (combined) block s-step Krylov iterative method for nonsymmetric linear systems. We then consider the problem of applying any block iterative method to solve a linear system with one right-hand side using many linearly independent initial residual vectors. We present a new algorithm which combines the many solutions obtained (by any block iterative method) into a single solution to the linear system. This approach of using block methods in order to increase the parallelism of Krylov methods is very useful in parallel systems. We implemented the new method on a parallel computer and we ran tests to validate the accuracy and the performance of the proposed methods. It is expected that the block s-step methods performance will scale well on other parallel systems because of their efficient use of memory hierarchies and their reduction of the number of global communication operations over the standard methods. Copyright q 2009 John Wiley & Sons, Ltd.

[1]  Yves Robert,et al.  Parallel conjugate gradient-like algorithms for solving sparse nonsymmetric linear systems on a vector multiprocessor , 1989, Parallel Comput..

[2]  G. Meurant.  Computer Solution of Large Linear Systems , 1999 .

[3]  Elizabeth R. Jessup,et al.  On Improving Linear Solver Performance: A Block Variant of GMRES , 2005, SIAM J. Sci. Comput..

[4]  Anthony T. Chronopoulos,et al.  Parallel Iterative S-Step Methods for Unsymmetric Linear Systems , 1996, Parallel Comput..

[5]  H. Sadok,et al.  A block version of BiCGSTAB for linear systems with multiple right-hand sides. , 2003 .

[6]  C. G. Broyden.  Block conjugate gradient methods , 1993 .

[7]  Efstratios Gallopoulos,et al.  An Iterative Method for Nonsymmetric Systems with Multiple Right-Hand Sides , 1995, SIAM J. Sci. Comput..

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

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

[10]  Rudnei Dias da Cunha,et al.  Dynamic block GMRES: an iterative method for block linear systems , 2007, Adv. Comput. Math..

[11]  Anthony T. Chronopoulos,et al.  On the Odir iterative method for non‐symmetric indefinite linear systems , 2001 .

[12]  Anthony T. Chronopoulos.  s-Step Iterative Methods for (Non) Symmetric (In) Definite Linear Systems , 1989, PPSC.

[13]  Jack Dongarra,et al.  Numerical Linear Algebra for High-Performance Computers , 1998 .