An elegant IDR ( s ) variant that efficiently exploits bi - orthogonality properties

The IDR(s) method that is proposed in [7] is a very competitive 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 [7] is a direct translation of the theorem into an algorithm. This translation is not unique. This paper derives a new IDR(s) variant. This new variant imposes bi-orthogonalization conditions on the iteration vectors, which results in a very elegant method with 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 will show through numerical experiments that our new variant is more accurate than the original IDR(s) for large values of s. We will also present a numerical comparison with GMRES [3], Bi-CGSTAB [8], and BiCGstab(l) [4] to illustrate the efficiency of IDR(s).