# NAG FL Interfacee01rbf (dim1_​ratnl_​eval)

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## 1Purpose

e01rbf evaluates continued fractions of the form produced by e01raf.

## 2Specification

Fortran Interface
 Subroutine e01rbf ( m, a, u, x, f,
 Integer, Intent (In) :: m Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: a(m), u(m), x Real (Kind=nag_wp), Intent (Out) :: f
#include <nag.h>
 void e01rbf_ (const Integer *m, const double a[], const double u[], const double *x, double *f, Integer *ifail)
The routine may be called by the names e01rbf or nagf_interp_dim1_ratnl_eval.

## 3Description

e01rbf evaluates the continued fraction
 $R(x)=a1+Rm(x)$
where
 $Ri(x)=am-i+ 2(x-um-i+ 1) 1+Ri- 1(x) , for ​ i=m,m- 1,…,2.$
and
 $R1(x)=0$
for a prescribed value of $x$. e01rbf is intended to be used to evaluate the continued fraction representation (of an interpolatory rational function) produced by e01raf.

## 4References

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

## 5Arguments

1: $\mathbf{m}$Integer Input
On entry: $m$, the number of terms in the continued fraction.
Constraint: ${\mathbf{m}}\ge 1$.
2: $\mathbf{a}\left({\mathbf{m}}\right)$Real (Kind=nag_wp) array Input
On entry: ${\mathbf{a}}\left(\mathit{j}\right)$ must be set to the value of the parameter ${a}_{\mathit{j}}$ in the continued fraction, for $\mathit{j}=1,2,\dots ,m$.
3: $\mathbf{u}\left({\mathbf{m}}\right)$Real (Kind=nag_wp) array Input
On entry: ${\mathbf{u}}\left(\mathit{j}\right)$ 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 ${\mathbf{u}}\left(m\right)$ is not used).
4: $\mathbf{x}$Real (Kind=nag_wp) Input
On entry: the value of $x$ at which the continued fraction is to be evaluated.
5: $\mathbf{f}$Real (Kind=nag_wp) Output
On exit: the value of the continued fraction corresponding to the value of $x$.
6: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
x corresponds to a pole of $R\left(x\right)$, or is very close. ${\mathbf{x}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

See Section 7 in e01raf.

## 8Parallelism and Performance

e01rbf is not threaded in any implementation.

The time taken by e01rbf 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 e01raf) and evaluates the continued fraction at a point $x$.

### 10.1Program Text

Program Text (e01rbfe.f90)

### 10.2Program Data

Program Data (e01rbfe.d)

### 10.3Program Results

Program Results (e01rbfe.r)