Minimum polynomial extrapolation
Minimum polynomial extrapolation (MPE) is a method to speed up the convergence of sequences of vectors. It was developed by Cabay and Jackson and works well for vector sequences, where simpler methods like Aitken’s often fail.
How it works (in simple terms)
- Start with a sequence of vectors x1, x2, ..., xk in R^n produced by some fixed-point iteration.
- Build a matrix U whose columns are the successive differences: U = [x2 − x1, x3 − x2, ..., xk − xk−1].
- Compute the Moore–Penrose pseudoinverse U+. Find the vector c0 = −U+ (xk+1 − xk).
- Append a 1 to c0 to form c = [c0; 1].
- Let X be the matrix of iterates X = [x2, x3, ..., xk+1].
- The MPE extrapolated limit is the product X c, which gives a single vector that estimates the limit of the iteration.
In short, MPE uses a small window of recent iterates to form a weighted combination that cancels out leading error terms and provides a faster convergence estimate.
This page was last edited on 3 February 2026, at 18:04 (CET).