In mathematics, minimum polynomial extrapolation is a sequence transformation used for convergence acceleration of vector sequences, due to Cabay and Jackson.
While Aitken's method is the most famous, it often fails for vector sequences. An effective method for vector sequences is the minimum polynomial extrapolation. It is usually phrased in terms of the fixed point iteration:
 formula_1
Given iterates formula_2 in formula_3, one constructs the formula_4 matrix formula_5 whose columns are the formula_6 differences. Then, one computes the vector formula_7 where formula_8 denotes the Moore–Penrose pseudoinverse of formula_9. The number 1 is then appended to the end of formula_10, and the extrapolated limit is
formula_11
where formula_12 is the matrix whose columns are the formula_13 iterates starting at 2.
The following 4 line MATLAB code segment implements the MPE algorithm:
U = x(:, 2:end - 1) - x(:, 1:end - 2);
c = - pinv(U) * (x(:, end) - x(:, end - 1));
c(end + 1, 1) = 1;
s = (x(:, 2:end) * c) / sum(c);