Algorithm 913: An elegant IDR(s) variant that efficiently exploits biorthogonality properties

The IDR(s) method that is proposed in Sonneveld and van Gijzen [2008] is a very efficient limited memory method for solving large nonsymmetric systems of linear equations. IDR(s) is based on the induced dimension reduction theorem, that provides a way to construct subsequent residuals that lie in a sequence of shrinking subspaces. The IDR(s) algorithm that is given in Sonneveld and van Gijzen [2008] is a direct translation of the theorem into an algorithm. This translation is not unique. This article derives a new IDR(s) variant, that imposes (one-sided) biorthogonalization conditions on the iteration vectors. The resulting method has lower overhead in vector operations than the original IDR(s) algorithms. In exact arithmetic, both algorithms give the same residual at every (s + 1)-st step, but the intermediate residuals and also the numerical properties differ. We show through numerical experiments that the new variant is more stable and more accurate than the original IDR(s) algorithm, and that it outperforms other state-of-the-art techniques for realistic test problems.