# NAG C Library Function Document

## 1Purpose

nag_1d_ratnl_eval (e01rbc) evaluates continued fractions of the form produced by nag_1d_ratnl_interp (e01rac).

## 2Specification

 #include #include
 void nag_1d_ratnl_eval (Integer m, const double a[], const double u[], double x, double *f, NagError *fail)

## 3Description

nag_1d_ratnl_eval (e01rbc) evaluates the continued fraction
 $Rx=a1+Rmx$
where
 $Rix=am-i+ 2x-um-i+ 1 1+Ri- 1x , for ​ i=m,m- 1,…,2.$
and
 $R1x=0$
for a prescribed value of $x$. nag_1d_ratnl_eval (e01rbc) is intended to be used to evaluate the continued fraction representation (of an interpolatory rational function) produced by nag_1d_ratnl_interp (e01rac).

## 4References

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

## 5Arguments

1:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of terms in the continued fraction.
Constraint: ${\mathbf{m}}\ge 1$.
2:    $\mathbf{a}\left[{\mathbf{m}}\right]$const doubleInput
On entry: ${\mathbf{a}}\left[\mathit{j}-1\right]$ must be set to the value of the parameter ${a}_{\mathit{j}}$ in the continued fraction, for $\mathit{j}=1,2,\dots ,m$.
3:    $\mathbf{u}\left[{\mathbf{m}}\right]$const doubleInput
On entry: ${\mathbf{u}}\left[\mathit{j}-1\right]$ must be set to the value of the parameter ${u}_{\mathit{j}}$ in the continued fraction, for $\mathit{j}=1,2,\dots ,m-1$. (The element ${\mathbf{u}}\left[m-1\right]$ is not used).
4:    $\mathbf{x}$doubleInput
On entry: the value of $x$ at which the continued fraction is to be evaluated.
5:    $\mathbf{f}$double *Output
On exit: the value of the continued fraction corresponding to the value of $x$.
6:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_POLE_PRESENT
x corresponds to a pole of $R\left(x\right)$, or is very close. ${\mathbf{x}}=〈\mathit{\text{value}}〉$.

## 7Accuracy

See Section 7 in nag_1d_ratnl_interp (e01rac).

## 8Parallelism and Performance

nag_1d_ratnl_eval (e01rbc) is not threaded in any implementation.

The time taken by nag_1d_ratnl_eval (e01rbc) is approximately proportional to $m$.

## 10Example

This example reads in the arguments ${a}_{j}$ and ${u}_{j}$ of a continued fraction (as determined by the example for nag_1d_ratnl_interp (e01rac)) and evaluates the continued fraction at a point $x$.

### 10.1Program Text

Program Text (e01rbce.c)

### 10.2Program Data

Program Data (e01rbce.d)

### 10.3Program Results

Program Results (e01rbce.r)