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In mathematics a 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. The technique was developed around 1890 by Henri Padé, but goes back to Georg Frobenius, who introduced the idea and investigated the features of rational approximations of power series.
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. For these reasons Padé approximants are used extensively in computer calculations. They have also been used as auxiliary functions, in Diophantine approximation and transcendental number theory, though for sharp results ad hoc methods, in some sense inspired by the Padé theory, typically replace them.
Since Padé approximant is a rational function, an artificial singular point may occur as an approximation, but this can be avoided by Borel-Padé analysis.
The reason why the Padé approximant tends to be a better approximation than a truncating Taylor series is clear from the viewpoint of the multi-point summation method. Since there are many cases in which the asymptotic expansion at infinity becomes 0 or a constant, it can be interpreted as the "incomplete two-point Padé approximation", in which the ordinary Padé approximation improves the method truncating a Taylor series.
Given a function f and two integers m ≥ 0 and n ≥ 1, the Padé approximant of order [m/n] is the rational function
which agrees with f(x) to the highest possible order, which amounts to
Equivalently, if R(x) is expanded in a Maclaurin series (Taylor series at 0), its first m + n terms would cancel the first m + n terms of f(x), 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-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
of the Taylor series of f, i.e., we have
is equivalent to the existence of some factor K(x) such that
which can be interpreted as the Bézout identity of one step in the computation of the extended gcd of the polynomials and .
To recapitulate: to compute the gcd of two polynomials p and q, one computes via long division the remainder sequence
k = 1, 2, 3, ... with , until . For the Bézout identities of the extended gcd one computes simultaneously the two polynomial sequences
to obtain in each step the Bézout identity
For the [m/n] approximant, one thus carries out the extended euclidean algorithm for
and stops it at the last instant that has degree n or smaller.
Then the polynomials give the [m/n] Padé approximant. If one were to compute all steps of the extended gcd computation, one would obtain an anti-diagonal of the Pade table.
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
is 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 aj and bj are the coefficients in the Padé approximation. The subscript '0' means that the Padé is of order [0/0] and hence, we have the Riemann zeta function.
DLog Padé method
Padé approximants can be used to extract critical points and exponents of functions. In thermodynamics, if a function f(x) behaves in a non-analytic way near a point x = r like , one calls x = r a critical point and p the associated critical exponent of f. If sufficient terms of the series expansion of f are known, one can approximately extract the critical points and the critical exponents from respectively the poles and residues of the Padé approximants where .
A Padé approximant approximates a function in one variable. An approximant in two variables is called a Chisholm approximant (after J. S. R. Chisholm), in multiple variables a Canterbury approximant (after Graves-Morris at the University of Kent).
Two points Pade approximant
The conventional Padé approximation is determined to reproduce the Maclaurin expansion up to a given order. Therefore, the approximation at the value apart from the expansion point may be poor. This is avoided by the 2-point Padé approximation, which is a type of multipoint summation method. At , consider a case that a function which is expressed by asymptotic behavior ,
Besides it, at ,additional asymptotic behavior
By selecting the major behavior of ,Approximate functions such that simultaneously reproduce asymptotic behavior by developing the Padé approximation can be found in various cases. As a result, at the point where the accuracy of the approximation may be the worst in the ordinary Pade approximation, Good accuracy of the 2-point Pade approximant is guaranteed. Therefore, the 2-point Pade approximant can be a method that gives a good approximation globally for .
In cases that are expressed by Polynomials or series of negative powers,exponential function,logarithmic function or , we can apply 2-point Padé approximant to . There is a method of using this to give an approximate solution of a differential equation with high accuracy. Also, for the nontrivial zeros of the Riemann zeta function, the first nontrivial zero can be estimated with some accuracy from the asymptotic behavior on the real axis.
Multi-point Pade approximant
A further extension of the 2-point Pade approximatnt is the multi-point Pade approximant. This method treats singularity points of a function which is wanted to be approximated. Consider cases singularities of a function is expressed with index by
Besides 2-point Pade approximant which includes information at, This is a method that approximates to reproduce the property of diverging at . As a result, since the information of the peculiarity of the function is captured, the approximation of a function can be performed with higher accuracy .
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