naginterfaces.library.interp.dim1_​ratnl_​eval

naginterfaces.library.interp.dim1_ratnl_eval(a, u, x)

dim1_ratnl_eval evaluates continued fractions of the form produced by dim1_ratnl().

For full information please refer to the NAG Library document for e01rb

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/e01/e01rbf.html

Parameters

afloat, array-like, shape $\left(m\right)$

$a[\text{j}-1]$ must be set to the value of the parameter $a_{\text{j}}$ in the continued fraction, for $\text{j}=1,2,\dots ,m$.

ufloat, array-like, shape $\left(m\right)$

$u[\text{j}-1]$ must be set to the value of the parameter $u_{\text{j}}$ in the continued fraction, for $\text{j}=1,2,\dots ,m-1$. (The element $u[m-1]$ is not used).

xfloat

The value of $x$ at which the continued fraction is to be evaluated.

Returns

ffloat

The value of the continued fraction corresponding to the value of $x$.

Raises

NagValueError

(errno $1$)

$x$ corresponds to a pole of $R\left(x\right)$, or is very close. $x=⟨value⟩$.

Notes

dim1_ratnl_eval evaluates 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

$$R_1\left(x\right)=0$$

for a prescribed value of $x$. dim1_ratnl_eval is intended to be used to evaluate the continued fraction representation (of an interpolatory rational function) produced by dim1_ratnl().

References

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