naginterfaces.library.interp.dim1_​ratnl

naginterfaces.library.interp.dim1_ratnl(x, f)

dim1_ratnl produces, from a set of function values and corresponding abscissae, the coefficients of an interpolating rational function expressed in continued fraction form.

For full information please refer to the NAG Library document for e01ra

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/e01/e01raf.html

Parameters

xfloat, array-like, shape $\left(n\right)$

$x[\text{i}-1]$ must be set to the value of the $\text{i}$th data abscissa, $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

ffloat, array-like, shape $\left(n\right)$

$f[\text{i}-1]$ must be set to the value of the data ordinate, $f_{\text{i}}$, corresponding to $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

Returns

mint

$m$, the number of terms in the continued fraction representation of $R\left(x\right)$.

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

$a[\text{j}-1]$ contains the value of the parameter $a_{\text{j}}$ in $R\left(x\right)$, for $\text{j}=1,2,\dots ,m$. The remaining elements of $a$, if any, are set to zero.

ufloat, ndarray, shape $\left(n\right)$

$u[\text{j}-1]$ contains the value of the parameter $u_{\text{j}}$ in $R\left(x\right)$, for $\text{j}=1,2,\dots ,m-1$. The $u_j$ are a permuted subset of the elements of $x$. The remaining $n-m+1$ locations contain a permutation of the remaining $x_i$, which can be ignored.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n>0$.

(errno $2$)

On entry, $x[\text{I}-1]$ is very close to $x[\text{J}-1]$: $\text{I}=⟨value⟩$, $x[\text{I}-1]=⟨value⟩$, $\text{J}=⟨value⟩$ and $x[\text{J}-1]=⟨value⟩$.

(errno $3$)

A continued fraction of the required form does not exist.

Notes

dim1_ratnl produces the parameters of a rational function $R\left(x\right)$ which assumes prescribed values $f_i$ at prescribed values $x_i$ of the independent variable $x$, for $\text{i}=1,2,\dots ,n$. More specifically, dim1_ratnl determines the parameters $a_j$, for $\text{j}=1,2,\dots ,m$ and $u_j$, for $\text{j}=1,2,\dots ,m-1$, in the continued fraction

$$R\left(x\right)=a_1+R_m\left(x\right)$$

where

$$R_i\left(x\right)=\frac{a_{m-i+2}(x-u_{m-i+1})}{1+R_{i-1}\left(x\right)}\text{, for}i=m,m-1,\dots ,2\text{,}$$

and

$$R_1\left(x\right)=0\text{,}$$

such that $R\left(x_i\right)=f_i$, for $\text{i}=1,2,\dots ,n$. The value of $m$ in (1) is determined by the function; normally $m=n$. The values of $u_j$ form a reordered subset of the values of $x_i$ and their ordering is designed to ensure that a representation of the form (1) is determined whenever one exists.

The subsequent evaluation of (1) for given values of $x$ can be carried out using dim1_ratnl_eval().

The computational method employed in dim1_ratnl is the modification of the Thacher–Tukey algorithm described in Graves–Morris and Hopkins (1981).

References

Graves–Morris, P R and Hopkins, T R, 1981, Reliable rational interpolation, Numer. Math. (36), 111–128