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Minimal Polynomial Extrapolation (MPE) and Reduced Rank Extrapolation (RRE)are two polynomial methods used for accelerating the convergence of sequencesof vectors $\{{x}_m\}$. They are applied successfully in conjunction withfixed-point iterative schemes in the solution of large and sparse systems oflinear and nonlinear equations in different disciplines of science andengineering. Both methods produce approximations $s_k$ to the limit orantilimit 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 twomethods are derived suggests that they might, somehow, be related to eachother; this has not been explored so far, however. In this work, we tackle thisissue 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 independentlyof the way the $x_m$ are generated. One of our results states that RREstagnates, in the sense that ${s}_k^\text{RRE}={s}_{k-1}^\text{RRE}$, if andonly 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 aredefined in any weighted inner product and the norm induced by it. They alsocontain as special cases the known results pertaining to the connection betweenthe method of Arnoldi and the method of generalized minimal residuals, twoimportant Krylov subspace methods for solving nonsingular linear systems.

Minimal Polynomial and Reduced Rank Extrapolation Methods Are Related