Abstract
The Extrapolation Algorithm is a technique devised in 1962 for accelerating the rate of convergence of slowly converging Picard iterations for fixed point problems. Versions to this technique are now called Anderson Acceleration in the applied mathematics community and Anderson Mixing in the physics and chemistry communities, and these are related to several other methods extant in the literature. We seek here to broaden and deepen the conceptual foundations for these methods, and to clarify their relationship to certain iterative methods for root-finding problems. For this purpose, the Extrapolation Algorithm will be reviewed in some detail, and selected papers from the existing literature will be discussed, both from conceptual and implementation perspectives.
Highlights
We simplify the notation to A, b and c, and set m = min(, M ) and n = N
In 1962, during the course of my doctoral dissertation research, I devised a technique for accelerating the convergence of the Picard iteration associated with a fixed point problem, which I called the Extrapolation Algorithm
We have (x − y)∗y = x∗y − y∗y, so we find that −v∗y
Summary
Comments on "Anderson Acceleration, Mixing and Extrapolation." Numerical Algorithms 80 (1): 135-234. In 1962, during the course of my doctoral dissertation research, I devised a technique for accelerating the convergence of the Picard iteration associated with a fixed point problem, which I called the Extrapolation Algorithm. The Extrapolation Algorithm For g : RN → RN , consider the problem of finding a fixed point x ∈ RN such that g(x) = x. In outline form, the basic Extrapolation Algorithm proceeds as follows: Choose the maximal m( ) such that there are well-determined θk( ) , 0 ≤ k ≤ m( ), minimizing v( ) − u( ) , with 0 ≤ m( ) ≤ min( , M ) N , and satisfying θ0( ) > 0. There will always be a unique v( ) − u( ) in the affine subspace with minimal norm — closest to 0
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