An enhancement of the convergence of the IDR method

Abstract

In this paper, we consider a family of algorithms, called IDR, based on the induced dimension reduction theorem. IDR is a family of efficient short recurrence methods introduced by Sonneveld and Van Gijzen for solving large systems of nonsymmetric linear equations. These methods generate residual vectors that live in a sequence of nested subspaces. We present the IDR(s) method and give two improvements of its convergence. We also define and give a global version of the IDR(s) method and describe a partial and a complete improvement of its convergence. Moreover, we recall the block version and state its improvements. Numerical experiments are provided to illustrate the performances of the derived algorithms compared to the well-known classical GMRES method and the bi-conjugate gradient stabilized method for systems with a single right-hand side, as well as the global GMRES, the global bi-conjugate gradient stabilized, the block GMRES, and the block bi-conjugate gradient stabilized methods for systems with multiple right-hand sides.