A block IDR() method for nonsymmetric linear systems with multiple right-hand sides

The IDR( s ) based on the induced dimension reduction (IDR) theorem, is a new class of efficient algorithms for large nonsymmetric linear systems. IDR(1) is mathematically equivalent to BiCGStab at the even IDR(1) residuals, and IDR( s ) with s > 1 is competitive with most Bi-CG based methods. For these reasons, we extend the IDR( s ) to solve large nonsymmetric linear systems with multiple right-hand sides. In this paper, a variant of the IDR theorem is given at first, then the block IDR( s ), an extension of IDR( s ) based on the variant IDR( s ) theorem, is proposed. By analysis, the upper bound on the number of matrix-vector products of block IDR( s ) is the same as that of the IDR( s ) for a single right-hand side in generic case, i.e., the total number of matrix-vector products of IDR( s ) may be m times that of of block IDR( s ), where m is the number of right-hand sides. Numerical experiments are presented to show the effectiveness of our proposed method.