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Reduced Rank Extrapolation (RRE) is a polynomial type method used to accelerate the convergence of sequences of vectors { x m }. It is applied successfully in different disciplines of science and engineering in the solution of large and sparse systems of linear and nonlinear equations of very large dimension. If s is the solution to the system of equations x = f ( x ), first, a vector sequence { x m } is generated via the fixed-point iterative scheme x m + 1 = f ( x m ), m = 0,1,…, and next, RRE is applied to this sequence to accelerate its convergence. RRE produces approximations s n , k to s that are of the form s n , k = ∑ i = 0 k γ i x n + i $\boldsymbol {s}_{n,k}={\sum }_{i=0}^{k} \gamma _{i} \boldsymbol {x}_{n+i}$ for some scalars γ i depending (nonlinearly) on x n , x n + 1 ,…, x n + k + 1 and satisfying ∑ i = 0 k γ i = 1 ${\sum }_{i=0}^{k} \gamma _{i}=1$ . The convergence properties of RRE when applied in conjunction with linear f ( x ) have been analyzed in different publications. In this work, we discuss the convergence of the s n , k obtained from RRE with nonlinear f ( x ) (i) when n → ∞ $n\to \infty $ with fixed k , and (ii) in two so-called cycling modes.

A convergence study for reduced rank extrapolation on nonlinear systems