naginterfaces.library.fit.pade_​app

naginterfaces.library.fit.pade_app(ia, ib, c)

pade_app calculates the coefficients in a Padé approximant to a function from its user-supplied Maclaurin expansion.

For full information please refer to the NAG Library document for e02ra

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/e02/e02raf.html

Parameters

iaint

$ia$ must specify $l+1$ and $ib$ must specify $m+1$, where $l$ and $m$ are the degrees of the numerator and denominator of the approximant, respectively.

ibint

$ia$ must specify $l+1$ and $ib$ must specify $m+1$, where $l$ and $m$ are the degrees of the numerator and denominator of the approximant, respectively.

cfloat, array-like, shape $(ia+ib-1)$

$c[\text{i}-1]$ must specify, for $\text{i}=1,2,\dots ,l+m+1$, the coefficient of $x^{\text{i}-1}$ in the given power series.

Returns

afloat, ndarray, shape $\left(ia\right)$

$a\left[\text{j}\right]$, for $\text{j}=1,2,\dots ,l+1$, contains the coefficient $a_{\text{j}}$ in the numerator of the approximant.

bfloat, ndarray, shape $\left(ib\right)$

$b\left[\text{k}\right]$, for $\text{k}=1,2,\dots ,m+1$, contains the coefficient $b_{\text{k}}$ in the denominator of the approximant.

Raises

NagValueError

(errno $1$)

On entry, $\text{ic}=⟨value⟩$, $ia=⟨value⟩$ and $ib=⟨value⟩$.

Constraint: $\text{ic}\ge ia+ib-1$.

(errno $1$)

On entry, $ib=⟨value⟩$.

Constraint: $ib\ge 1$.

(errno $1$)

On entry, $ia=⟨value⟩$.

Constraint: $ia\ge 1$.

(errno $2$)

The Pade approximant is degenerate.

Notes

Given a power series

$$c_0+c_{1}x+c_2{x}^2+\cdots +c_{l+m}{x}^{l+m}+\cdots$$

pade_app uses the coefficients $c_i$, for $\text{i}=0,1,\dots ,l+m$, to form the $\left[l/m\right]$ Padé approximant of the form

$$\frac{a_0+a_{1}x+a_2{x}^2+\cdots +a_l{x}^l}{b_0+b_{1}x+b_2{x}^2+\cdots +b_m{x}^m}$$

with $b_0$ defined to be unity. The two sets of coefficients $a_j$, for $\text{j}=0,1,\dots ,l$, and $b_k$, for $\text{k}=0,1,\dots ,m$, in the numerator and denominator are calculated by direct solution of the Padé equations (see Graves–Morris (1979)); these values are returned through the argument list unless the approximant is degenerate.

Padé approximation is a useful technique when values of a function are to be obtained from its Maclaurin expansion but convergence of the series is unacceptably slow or even nonexistent. It is based on the hypothesis of the existence of a sequence of convergent rational approximations, as described in Baker and Graves–Morris (1981) and Graves–Morris (1979).

Unless there are reasons to the contrary (as discussed in Module 4, Section 2, Modules 5 and 6 of Baker and Graves–Morris (1981)), one normally uses the diagonal sequence of Padé approximants, namely

$$\{\left[m/m\right],m=0,1,2,\dots \}\text{.}$$

Subsequent evaluation of the approximant at a given value of $x$ may be carried out using pade_eval().

References

Baker, G A Jr and Graves–Morris, P R, 1981, Padé approximants, Part 1: Basic theory, encyclopaedia of Mathematics and its Applications, Addison–Wesley

Graves–Morris, P R, 1979, The numerical calculation of Padé approximants, Padé Approximation and its Applications. Lecture Notes in Mathematics, (ed L Wuytack) (765), 231–245, Adison–Wesley