Math gifts

- Art Gallery -

In mathematics, minimum polynomial extrapolation is a sequence transformation used for convergence acceleration of vector sequences, due to Sabay and Jackson.[1]

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:

\( x_{{k+1}}=f(x_{k}). \)

Given iterates \( x_{1},x_{2},...,x_{k} \) in \( \mathbb {R} ^{n} \), one constructs the \( n\times (k-1) \) matrix \( U=(x_{2}-x_{1},x_{3}-x_{2},...,x_{k}-x_{{k-1}}) \) whose columns are the k-1 differences. Then, one computes the vector \( c=-U^{+}(x_{{k+1}}-x_{k}) where U + {\displaystyle U^{+}} U^{+} \) denotes the Moore–Penrose pseudoinverse of U. The number 1 is then appended to the end of c, and the extrapolated limit is

\( s={Xc \over \sum _{{i=1}}^{k}c_{i}}, \)

where \( X=(x_{2},x_{3},...,x_{{k+1}}) \) is the matrix whose columns are the k 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);

References

Cabay, S.; Jackson, L.W. (1976), "A polynomial extrapolation method for finding limits and antilimits of vector sequences", SIAM Journal on Numerical Analysis, doi:10.1137/0713060

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

World

Index

Hellenica World - Scientific Library

Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License