Abstract
We present variants of the block-GMRES( $m$ ) algorithms due to Vital and the block-LGMRES( $m$ , $k$ ) by Baker, Dennis and Jessup, obtained with replacing the standard QR factorization by a rank-revealing QR factorization in the Arnoldi process. The resulting algorithm allows for dynamic block deflation whenever there is a linear dependency between the Krylov vectors or the convergence of a right-hand-side occurs. $\textsc{Fortran 90}$ implementations of the algorithms were tested on a number of test matrices and the results show that in some cases a substantial reduction of the execution time is obtained. Also a parallel implementation of our variant of the block-GMRES( $m$ ) algorithm, using $\textsc{Fortran 90}$ and $\textsc{MPI}$ was tested on $\textsc{SunFire 15K}$ parallel computer, showing good parallel efficiency.
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