Padé approximant

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Padé approximant is the "best" approximation of a function by a rational function of given order. A 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. For these reasons Padé approximants are used extensively in computer calculations. They have also been applied to Diophantine approximation, though for sharp results ad hoc methods in some sense inspired by the Padé theory typically replace them. 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).

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[edit] Definition

Given a function f and two integers m ≥ 0 and n ≥ 0, the Padé approximant of order (m, n) is the rational function

R(x)=\frac{p_0+p_1x+p_2x^2+\cdots+p_mx^m}{1+q_1 x+q_2x^2+\cdots q_nx^n}

which agrees with f(x) to the highest possible order, which amounts to

f(0)=R(0)\,
f'(0)=R'(0)\,
f''(0)=R''(0)\,
\vdots\,
f^{(m+n)}(0)=R^{(m+n)}(0).\,

Equivalently, if R(x) is expanded in a Taylor series at 0, its first m + n + 1 terms would cancel the first m + n + 1 terms of f(x), and as such

f(x)-R(x) = c_{m+n+1}x^{m+n+1}+c_{m+n+2}x^{m+n+2}+\cdots

The Padé approximant is unique for given m and n, that is, the coefficients p_0, p_1, \dots, p_m, q_1, \dots, q_n can be uniquely determined. It is for reasons of uniqueness that the zero-th order term at the denominator of R(x) was chosen to be 1, otherwise the numerator and denominator of R(x) would have been unique only up to multiplication by a constant.

The Padé approximant defined above is also denoted as

[m/n]_f(x). \,

For given x, Padé approximants can be computed by the epsilon algorithm and also other sequence transformations from the partial sums

s_n(x)=c_0 + c_1 x + c_2 x^2 + \cdots + c_n x^n

of the Taylor series of f, i.e., we have

c_k = \frac{f^{(k)}(0)}{k!}.

It should be noted that f can also be a formal power series, and, hence, Padé approximants can also be applied to the summation of divergent series.

[edit] Riemann–Padé zeta function

To study the resummation of a divergent series, say

\sum_{z=1}^{\infty}f(z),

it can be useful to introduce the Padé or simply rational zeta function as

\zeta _{R}(s) = \sum_{z=1}^{\infty} \frac{R(z)}{z^{s}},

where

R(x) = [m/n]_{f}(x),\,

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

\sum_{j=0}^{n}p_{j}\zeta _{R}(s-j)= \sum_{j=0}^{m}q_{j}\zeta_{0}(s-j),

where pj and qj 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.

[edit] See also

[edit] 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, available online. Cambridge University Press.

[edit] External links