Padé approximation

In mathematics, the Padé approximation describes the best approximation of a function using rational functions .

The Padé approximation is named after the French mathematician Henri Padé , who made it famous in 1892, although the German mathematician Georg Frobenius published his research on the rational approximation of power series in 1881.

The Padé approximation often leads to better results than the approximation using Taylor series . Sometimes approximations are obtained even if the Taylor series does not converge. Hence, it is widely used in computer calculations. It is also useful in the area of Diophantine approximation .

definition

Be a function and , natural numbers, then that is Padé approximation of order the rational function ${\displaystyle f}$ ${\displaystyle m\geq 0}$ ${\displaystyle n\geq 1}$ ${\displaystyle [m/n]}$

${\displaystyle R(x)={\frac {\sum _{j=0}^{m}a_{j}x^{j}}{1+\sum _{k=1}^{n}b_{k}x^{k}}}={\frac {a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{m}x^{m}}{1+b_{1}x+b_{2}x^{2}+\cdots +b_{n}x^{n}}}}$,

which corresponds to in the highest possible order, from which it follows: ${\displaystyle f(x)}$

${\displaystyle {\begin{array}{rcl}f(0)&=&R(0)\\f'(0)&=&R'(0)\\f''(0)&=&R''(0)\\&\vdots &\\f^{(m+n)}(0)&=&R^{(m+n)}(0).\end{array}}}$

An equivalent definition is: If one develops into a Maclaurin series , i. H. into a Taylor series around the point 0, then the first terms of and agree. From this it follows for the approximation error ${\displaystyle R(x)}$ ${\displaystyle m+n}$ ${\displaystyle R(x)}$ ${\displaystyle f(x)}$

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

For each given and the Padé approximation is unique, i. H. the coefficients are clear. ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle a_{0},a_{1},\dots ,a_{m},b_{1},\dots ,b_{n}}$

In the denominator of , the initial term was chosen without loss of generality . Otherwise the shape mentioned is obtained by suitable shortening. ${\displaystyle R(x)}$ ${\displaystyle b_{0}=1}$

The Padé approximation is also represented as

${\displaystyle [m/n]_{f}(x).\,}$

calculation

For a given one, the Padé approximation can be calculated using the so-called "Epsilon method" of the Belgian mathematician Peter Wynn, or other sequential transformations . The subtotals are used for this ${\displaystyle x}$

${\displaystyle T_{N}(x)=c_{0}+c_{1}x+c_{2}x^{2}+\cdots +c_{N}x^{N}}$

the Taylor series of ; the are so in accordance ${\displaystyle f}$ ${\displaystyle c_{k}}$

${\displaystyle c_{k}={\frac {f^{(k)}(0)}{k!}}}$

by determined. ${\displaystyle f}$

The function can also be a formal power series , so that Padé approximations can also be applied to the summation of divergent series. ${\displaystyle f}$

The extended Euclidean algorithm for the greatest common polynomial divisor can be used to calculate the Padé approximation . The relationship

${\displaystyle R(x)=P(x)/Q(x)=T_{m+n}(x){\text{ mod }}x^{m+n+1}}$

is equivalent to the existence of a factor such that ${\displaystyle K(x)}$

${\displaystyle P(x)=Q(x)T_{m+n}(x)+K(x)x^{m+n+1}}$.

This can be interpreted as the Bézout equation of a step in the computation of the greatest common polynomial divisor:

${\displaystyle T_{m+n}(x)}$and . ${\displaystyle x^{m+n+1}}$

For the approximation one applies the extended Euclidean algorithm for ${\displaystyle [m/n]}$

${\displaystyle r_{0}=x^{m+n+1},\;r_{1}=T_{m+n}(x)}$

and stops when the grade is less than or equal to . Then the polynomials represent the -Padé approximation. ${\displaystyle v_{k}}$ ${\displaystyle n}$ ${\displaystyle P=r_{k},\;Q=v_{k}}$ ${\displaystyle [m/n]}$

Riemann – Padé zeta function

For examining divergent series, for example

${\displaystyle \sum _{z=1}^{\infty }f(z)}$

it can be helpful to introduce the padé or rational zeta function:

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

in which

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

is the Padé approximation of the order of the function . The value for is the sum of the divergent series. The functional equation for this zeta function is: ${\displaystyle [m/n]}$ ${\displaystyle f}$ ${\displaystyle s=0}$

${\displaystyle \sum _{j=0}^{n}a_{j}\zeta _{R}(s-j)=\sum _{j=0}^{m}b_{j}\zeta _{0}(s-j),}$

where and are the coefficients of the Padé approximation. The index 0 stands for the Padé approximation of the order [0/0] and thus results in the Riemann ζ function . ${\displaystyle a_{j}}$ ${\displaystyle b_{j}}$

DLog-Padé method

With Padé approximations, critical points and exponents of a function can be determined. In thermodynamics is a critical point and the associated critical exponent of when the function near a point as non-analytic behavior. Are sufficiently many terms of the series expansion of known, resulting approximate the critical points and the critical exponent of the poles and residues of the Padé approximations with . ${\displaystyle x=r}$ ${\displaystyle p}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle x=r}$ ${\displaystyle f(x)\sim \left|x-r\right|^{p}}$ ${\displaystyle f}$ ${\displaystyle \left[n/n+1\right]_{g}\left(x\right)}$ ${\displaystyle g={\frac {f'}{f}}}$

Generalizations

A Padé approximation approximates a function in a variable. An approximation in two variables is called the Chisholm approximation , in more than two variables the Canterbury approximation (named after Graves-Morris at the University of Kent ).

literature

• GA Baker, Jr., P. Graves-Morris: Padé Approximants. Cambridge UP, 1996, ISBN 0-521-45007-1 .
• C. Brezinski, M. Redivo Zaglia: Extrapolation Methods. Theory and Practice. North-Holland, 1991, ISBN 0-444-88814-4 .
• WH Press, SA Teukolsky, WT Vetterling, BP Flannery: Numerical Recipes. The Art of Scientific Computing. 3. Edition. Cambridge University Press, New York 2007, ISBN 978-0-521-88068-8 , Section 5.12 Padé Approximants. (apps.nrbook.com)
• WB Gragg: The Pade Table and Its Relation to Certain Algorithms of Numerical Analysis. In: SIAM Review. Vol. 14, No. 1, 1972, pp. 1-62.
• P. Wynn: Upon systems of recursions which obtain among the quotients of the Padé table. In: Numerical Mathematics. 8 (3), 1966, pp. 264-269.

Web links

• Eric W Weisstein: Padé Approximant from MathWorld (accessed June 3, 2014)
• Chapter 3.5 (accessed June 4, 2014)
• Examples of Padé approximations (accessed February 14, 2016)
• Padé approximation for sin (x) (Engl.) ( Memento of 1 March 2014 Internet Archive ) (accessed on 14 February 2016)

Individual evidence

1. ^ Henri Padé : Sur la représentation approchée d'une fonction par des fractions rationalles. In: Annales Scientifiques de l'Êcole Normale Supérieure. Volume 9 supplement, 1892, pp. 1-93.
2. ↑ Georg Frobenius : About relations between the approximate fractions of power series. In: Journal for pure and applied mathematics . Volume 90, 1881, pp. 1-17. (online , accessed June 3, 2014)
3. ↑ Elliot Ward Cheney: Introduction to Approximation Theory. McGraw-Hill Book Company, 1966, ISBN 0-07-010757-2 , p. 231.
4. ^ Peter Wynn: On the Convergence and Stability of the Epsilon Algorithm. In: SIAM Journal on Numerical Analysis. Volume 3 (1), March 1966, pp. 91-122 Theorem 1.
5. ↑ C. Brezenski: extrapolation algorithms and Padé approximations. In: Applied Numerical Mathematics. Volume 20 (3), 1996, pp. 299-318.
6. ↑ Dario Bini, Victor Pan: Polynomial and Matrix computations. Volume 1: Fundamental Algorithms. (= Progress in theoretical computer science. 12). Birkhäuser, 1994, ISBN 0-8176-3786-9 , p. 46, problem 5.2b and algorithm 5.2.