In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Padé approximant is the "best" approximation of a function near a specific point by a
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
of given order. Under this technique, the approximant's
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
agrees with the power series of the function it is approximating. The technique was developed around 1890 by
Henri Padé
Henri Eugène Padé (; 17 December 1863 – 9 July 1953) was a French mathematician, who is now remembered mainly for his development of Padé approximation techniques for functions using rational functions.
Education and career
Pad ...
, 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
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, and it may still work where the Taylor series does not
converge. For these reasons Padé approximants are used extensively in computer
calculation
A calculation is a deliberate mathematical process that transforms one or more inputs into one or more outputs or ''results''. The term is used in a variety of senses, from the very definite arithmetical calculation of using an algorithm, to t ...
s. They have also been used as
auxiliary functions in
Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
The first problem was to know how well a real number can be approximated by r ...
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.
Definition
Given a function ''f'' and two
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s ''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
is expanded in a Maclaurin series (
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
at 0), its first
terms would cancel the first
terms of
, and as such
:
When it exists, the Padé approximant is unique as a formal power series for the given ''m'' and ''n''.
The Padé approximant defined above is also denoted as
:
Computation
For given ''x'', Padé approximants can be computed by
Wynn's epsilon algorithm and also other
sequence transformation
In mathematics, a sequence transformation is an operator acting on a given space of sequences (a sequence space). Sequence transformations include linear mappings such as convolution with another sequence, and resummation of a sequence and, mo ...
s from the partial sums
:
of the
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
of ''f'', i.e., we have
:
''f'' can also be a
formal power series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial s ...
, and, hence, Padé approximants can also be applied to the summation of
divergent series
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.
If a series converges, the individual terms of the series mus ...
.
One way to compute a Padé approximant is via the
extended Euclidean algorithm
In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers ''a'' and ''b'', also the coefficients of Bézout's id ...
for the
polynomial greatest common divisor
In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common ...
. The relation
:
is equivalent to the existence of some factor
such that
:
which can be interpreted as the
Bézout identity of one step in the computation of the extended greatest common divisor of the polynomials
and
.
Recall that, to compute the greatest common divisor 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 greatest common divisor 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 greatest common divisor computation, one would obtain an anti-diagonal of the
Pade table.
Riemann–Padé zeta function
To study the resummation of a
divergent series
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.
If a series converges, the individual terms of the series mus ...
, say
:
it can be useful to introduce the Padé or simply rational zeta function as
:
where
:
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 ''a
j'' and ''b
j'' are the coefficients in the Padé approximation. The subscript '0' means that the Padé is of order
/0and hence, we have the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
.
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
.
Generalizations
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 Padé 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
:
:
and 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 where
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 Padé approximant
A further extension of the 2-point Padé approximant is the multi-point Padé approximant.
This method treats singularity points
of a function
which is to be approximated. Consider the cases when singularities of a function are expressed with index
by
:
Besides the 2-point Padé approximant, which includes information at
, this method approximates to reduce 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.
Examples
;
:
;
:
;
Jacobi Jacobi may refer to:
* People with the surname Jacobi
Mathematics:
* Jacobi sum, a type of character sum
* Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations
* Jacobi eigenvalue algorithm, ...
:
;
Bessel
:
;
:
;
Fresnel
:
See also
*
Padé table
In complex analysis, a Padé table is an array, possibly of infinite extent, of the rational Padé approximants
:''R'm'', ''n''
to a given complex formal power series. Certain sequences of approximants lying within a Padé table can often b ...
*
Bhaskara I's sine approximation formula
In mathematics, Bhaskara I's sine approximation formula is a rational expression in one variable for the computation of the approximate values of the trigonometric sines discovered by Bhaskara I (c. 600 – c. 680), a seventh-century Indian m ...
References
Literature
* Baker, G. A., Jr.; and Graves-Morris, P. '' Padé Approximants''.
Cambridge U.P., 1996.
* Baker, G. A., Jr
Padé approximantScholarpedia 7(6):9756.
* Brezinski, C.;
Redivo Zaglia, M. ''Extrapolation Methods. Theory and Practice''.
North-Holland, 1991.
* .
* Frobenius, G.; ,
ournal für die reine und angewandte Mathematik (Crelle's Journal) Volume 1881, Issue 90, Pages 1–17.
* Gragg, W. B.; ''The Pade Table and Its Relation to Certain Algorithms of Numerical Analysis''
IAM Review Vol. 14, No. 1, 1972, pp. 1–62.
* Padé, H.; , Thesis, [Ann. École Nor. (3), 9, 1892, pp. 1–93 supplement.
available online->
* .
External links
*
Padé Approximants Oleksandr Pavlyk, The Wolfram Demonstrations Project.
Data Analysis BriefBook: Pade Approximation Rudolf K. Bock
European Laboratory for Particle Physics
The European Organization for Nuclear Research, known as CERN (; ; ), is an intergovernmental organization that operates the largest particle physics laboratory in the world. Established in 1954, it is based in a northwestern suburb of G ...
,
CERN
The European Organization for Nuclear Research, known as CERN (; ; ), is an intergovernmental organization that operates the largest particle physics laboratory in the world. Established in 1954, it is based in a northwestern suburb of Gen ...
.
Sinewave Scott Dattalo, last accessed 2010-11-11.
for Padé approximation of models with time delays.
{{DEFAULTSORT:Pade approximant
Continued fractions
Numerical analysis
Rational functions