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## stable, numerical, maclaurin series of analytic functions

**From**: |
Fotios Kasolis |

**Subject**: |
stable, numerical, maclaurin series of analytic functions |

**Date**: |
Tue, 14 Sep 2010 19:33:22 +0200 |

If i recall correctly that was asked time ago! Given an analytic function
f:R->R you can get an approximation of the coefficients of the maclaurin
polynomial by a function like
function [ p ] = maclaurin (f, n)
N = min (max (2 ^ n, 1024), 1048576);
x = [ 0:N - 1 ] / N;
z = exp (2 * i * pi * x);
p = real (fft (f (z) / N));
p(find (abs (p) <= eps)) = 0;
p = p(n:-1:1);
endfunction
This will not work for singular functions. For instance, it ll return Nans if
your function is f(x) = 1/(1-x) and garbage if there is a singularity. An
example is
>* p = maclaurin (@(x)exp (sin (x)), 10)*
>* x = 0:0.01:10;*
>* pv = polyval (p, x);*
>* plot (x, pv)*
>* semilogy (x, abs(pv-exp (sin (x))))*
The algorithm is stable since FFT is stable!
Enjoy!
/Fotios

**stable, numerical, maclaurin series of analytic functions**,
*Fotios Kasolis* **<=**