- Padé approximant
Padé approximant is the "best" approximation of a function by a
rational function of given order. Developed byHenri Padé , a Padé approximant often gives better approximation of the function than truncating itsTaylor series , and it may still work where the Taylor series does not converge. For these reasons Padé approximants are used extensively in computercalculation s. They have also been applied toDiophantine approximation , though for sharp results "ad hoc" methods in some sense inspired by the Padé theory typically replace them.Definition
Given a function "f" and two
integer s "m" ≥ 0 and "n" ≥ 0, the "Padé approximant" of order ("m", "n") is the rational function:
which agrees with to the highest possible order, which amounts to:::::
Equivalently, if is expanded in a
Taylor series at 0, its first "m" + "n" + 1 terms would cancel the first "m" + "n" + 1 terms of , and as such:The Padé approximant is unique for given "m" and "n", that is, the coefficients can be uniquely determined. It is for reasons of uniqueness that the zero-th order term at the denominator of was chosen to be 1, otherwise the numerator and denominator of would have been unique only
up to multiplication by a constant.The Padé approximant defined above is also denoted as
:
For given , Padé approximants can be computed by the
epsilon algorithm and also othersequence transformation s from the partial sums:
of the
Taylor series of , i.e., we have:
It should be noted that can also be a
formal power series , and, hence, Padé approximants can also be applied to the summation ofdivergent series .Riemann–Padé zeta function
To study the resummation of a
divergent series , say: it can be useful to introduce the Padé or simply rational zeta function as
:
where
:
is just the Padé approximation of order ("m", "n") of the function "f"("x"). The
zeta regularization value at "s" = 0 is taken to be the sum of the divergent series.The functional equation for this Padé zeta function is
:
where and are the coefficients in the Padé approximation. The subscript '0' means that the Padé is of order [0/0] and hence, we got the Riemann zeta function.
Generalizations
A Padé approximant approximates a function in one variable. An approximant in two variables is called a Chisholm approximant, in multiple variables a
Canterbury approximant (after Graves-Morris at the University of Kent).ee also
*
Padé table References
* Baker, G. A., Jr.; and Graves-Morris, P. " Padé Approximants". Cambridge U.P., 1996.
* Brezinski, C.; and Redivo Zaglia, M. "Extrapolation Methods. Theory and Practice". North-Holland, 1991
* Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Numerical recipes in C." Section 5.12, [http://www.nrbook.com/a/bookcpdf/c5-12.pdf available online] . Cambridge University Press.External links
*
* [http://math.fullerton.edu/mathews/n2003/PadeApproximationMod.html Module for Padé Approximation by John H. Mathews]
* [http://demonstrations.wolfram.com/PadeApproximants/ Padé Approximants] by Oleksandr Pavlyk,The Wolfram Demonstrations Project .
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