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20-08-2019 | Original Paper | Issue 3/2020

Numerical Algorithms 3/2020

A convergence study for reduced rank extrapolation on nonlinear systems

Journal:
Numerical Algorithms > Issue 3/2020
Author:
Avram Sidi
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Abstract

Reduced Rank Extrapolation (RRE) is a polynomial type method used to accelerate the convergence of sequences of vectors {xm}. 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 {xm} is generated via the fixed-point iterative scheme xm+ 1 = f(xm), m = 0,1,…, and next, RRE is applied to this sequence to accelerate its convergence. RRE produces approximations sn, k to s that are of the form \(\boldsymbol {s}_{n,k}={\sum }_{i=0}^{k} \gamma _{i} \boldsymbol {x}_{n+i}\) for some scalars γi depending (nonlinearly) on xn, xn+ 1,…,xn+k+ 1 and satisfying \({\sum }_{i=0}^{k} \gamma _{i}=1\). 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 sn, k obtained from RRE with nonlinear f(x) (i) when \(n\to \infty \) with fixed k, and (ii) in two so-called cycling modes.

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