Gutknecht, Martin H.
The Induced Dimension Reduction (IDR) method is a Krylov space method for solving linear systems that was developed by Peter Sonneveld around 1979. It was only noticed by few people, and mainly as the forerunner of Bi-CGSTAB, which was introduced a decade later. In 2007 Sonneveld and van Gijzen reconsidered IDR and generalized it to IDR(s), claiming that IDR(1) \approx IDR is equally fast but preferable to the closely related Bi-CGSTAB, and that IDR(s) with s > 1 may be much faster than Bi-CGSTAB. It also turned out that when s > 1, IDR(s) is related to ML(s)BiCGSTAB of Yeung and Chan, and that there is quite some flexibility in the IDR approach. This approach differs completely from traditional approaches to Krylov space methods, and therefore it requires an extra effort to get familiar with it and to understand the connections as well as the differences to better known Krylov space methods. This expository paper aims at providing some help in this and to make the method understandable even to non-experts. After presenting the history of IDR and related methods we summarize some of the basic facts on Krylov space methods. Then we present the original IDR(s) in detail and put it into perspective with other methods. Specifically, we analyze the differences between the IDR method published 1980, IDR(1) and Bi-CGSTAB. At the end, we discuss a recently proposed ingenious variant of IDR(s) whose residuals fulfill extra orthogonality conditions. There we dwell on details that have been left out in the publications of van Gijzen and Sonneveld.