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Van Gijzen, M.B. (author), Sleijpen, G.L.G. (author), Zemke, J.P. (author)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...report 2011