On the Convergence Behavior of IDR(s) and Related Methods

An explanation is given of the convergence behavior of IDR(s) methods. The convergence mechanism of these algorithms has two components. The first consists of damping properties of certain factors in the residual polynomials, which becomes less important for large values of s. The second component depends on the behavior of Lanczos polynomials that occur in the theoretical description. This part of the residual polynomials is related to Lanczos methods with s left starting vectors, as described in a paper by Yeung and Chan on their ML(k)BiCGSTAB method, in [SIAM J. Sci. Comput., 21 (1999), pp. 1263–1290]. In this paper, the behavior of the second component is compared with the full GMRES method [SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856–869] and an expected rate of convergence is given, based on a random choice of the s shadow vectors.