Padé approximant

A Padé approximant approximates a function in one variable.

His doctoral thesis disclosed the now-known Padé approximant.

The Padé approximant is unique for given m and n, that is, the coefficients can be uniquely determined.

To obtain we have to apply boundary conditions at , which may be done by writing the series as a Padé approximant:

Similarly, the 3rd-order Padé approximant gives a more accurate answer over an even larger range of r, but it has a slightly more complicated formula:

One could determine the Padé approximant starting from the Taylor polynomial of f using Euclid's algorithm.

More precisely, in any Padé approximant, the degrees of numerator and denominator polynomials have to add to the order of the approximant.

Both Matlab and GNU Octave use Padé approximant.

The Padé approximant often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge.

A Padé approximant with numerator of degree m and denominator of degree n is A-stable if and only if m n m + 2.

Padé approximant is the "best" approximation of a function by a rational function of given order - under this technique, the approximant's power series agrees with the power series of the function it is approximating.