Minimal Polynomial and Reduced Rank Extrapolation Methods Are Related

Abstract

Minimal Polynomial Extrapolation (MPE) and Reduced Rank Extrapolation (RRE) are two polynomial methods used for accelerating the convergence of sequences of vectors $\{{x}_m\}$. They are applied successfully in conjunction with fixed-point iterative schemes in the solution of large and sparse systems of linear and nonlinear equations in different disciplines of science and engineering. Both methods produce approximations $s_k$ to the limit or antilimit of $\{{x}_m\}$ that are of the form ${s}_k=\sum^k_{i=0}\gamma_i{x}_i$ with $\sum^k_{i=0}\gamma_i=1$, for some scalars $\gamma_i$. The way the two methods are derived suggests that they might, somehow, be related to each other; this has not been explored so far, however. In this work, we tackle this issue and show that the vectors $s_{k}^\text{MPE}$ and $s_k^\text{RRE}$ produced by the two methods are related in more than one way, and independently of the way the $x_m$ are generated. One of our results states that RRE stagnates, in the sense that ${s}_k^\text{RRE}={s}_{k-1}^\text{RRE}$, if and only if ${s}_{k}^\text{MPE}$ does not exist. Another result states that, when ${s}_{k}^\text{MPE}$ exists, there holds $$\mu_k{s}_k^\text{RRE} = \mu_{k-1}{s}_{k-1}^\text{RRE} + \nu_k{s}_{k}^\text{MPE} \quad \text{with} \quad \mu_k = \mu_{k-1} + \nu_k,$$ for some positive scalars $\mu_k$, $\mu_{k-1}$, and $\nu_k$ that depend only on ${s}_k^\text{RRE}$, ${s}_{k-1}^\text{RRE}$, and ${s}_{k}^\text{MPE}$, respectively. Our results are valid when MPE and RRE are defined in any weighted inner product and the norm induced by it. They also contain as special cases the known results pertaining to the connection between the method of Arnoldi and the method of generalized minimal residuals, two important Krylov subspace methods for solving nonsingular linear systems.