Flexible and multi-shift induced dimension reduction algorithms for solving large sparse linear systems

We give two important generalizations of the Induced Dimension Reduction (IDR) approach for the solution of linear systems. We derive a flexible and a multi-shift Quasi-Minimal Residual IDR (QMRIDR) variant. Numerical examples are presented to show the effectiveness of these new IDR variants compared to existing ones and to other Krylov subspace...

Interpreting IDR(s) as a deflation method

In this paper the IDR(s) method is interpreted in the context of deflation methods. It is shown that IDR(s) can be seen as a Richardson iteration preconditioned by a variable deflation–type preconditioner. The main result of this paper is the IDR projection theorem, which relates the spectrum of the deflated system in each IDR(s) cycle to all...

Exploiting BiCGstab(?) strategies to induce dimension reduction

IDR(s) [P. Sonneveld and M. B. van Gijzen, SIAM J. Sci. Comput., 31 (2008), pp. 1035–1062] and BiCGstab(?) [G. L. G. Sleijpen and D. R. Fokkema, Electron. Trans. Numer. Anal., 1 (1993), pp. 11–32] are two of the most efficient short-recurrence iterative methods for solving large nonsymmetric linear systems of equations. Which of the two is best...

Exploiting BiCGstab(?) strategies to induce dimension reduction

Bi-CGSTAB as an induced dimension reduction method