e01 Chapter Contents
e01 Chapter Introduction
NAG C Library Manual

NAG Library Function Documentnag_1d_ratnl_interp (e01rac)

1  Purpose

nag_1d_ratnl_interp (e01rac) produces, from a set of function values and corresponding abscissae, the coefficients of an interpolating rational function expressed in continued fraction form.

2  Specification

 #include #include
 void nag_1d_ratnl_interp (Integer n, const double x[], const double f[], Integer *m, double a[], double u[], NagError *fail)

3  Description

nag_1d_ratnl_interp (e01rac) 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 $\mathit{i}=1,2,\dots ,n$. More specifically, nag_1d_ratnl_interp (e01rac) determines the parameters ${a}_{j}$, for $\mathit{j}=1,2,\dots ,m$ and ${u}_{j}$, for $\mathit{j}=1,2,\dots ,m-1$, in the continued fraction
 $Rx=a1+Rmx$ (1)
where
 $Rix=am-i+ 2x-um-i+ 1 1+Ri- 1x , for ​ i=m,m- 1,…,2,$
and
 $R1x=0,$
such that $R\left({x}_{i}\right)={f}_{i}$, for $\mathit{i}=1,2,\dots ,n$. 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 nag_1d_ratnl_eval (e01rbc).
The computational method employed in nag_1d_ratnl_interp (e01rac) is the modification of the Thacher–Tukey algorithm described in Graves–Morris and Hopkins (1981).

4  References

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

5  Arguments

1:     nIntegerInput
On entry: $n$, the number of data points.
Constraint: ${\mathbf{n}}>0$.
2:     x[n]const doubleInput
On entry: ${\mathbf{x}}\left[\mathit{i}-1\right]$ must be set to the value of the $\mathit{i}$th data abscissa, ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
Constraint: the ${\mathbf{x}}\left[i-1\right]$ must be distinct.
3:     f[n]const doubleInput
On entry: ${\mathbf{f}}\left[\mathit{i}-1\right]$ must be set to the value of the data ordinate, ${f}_{\mathit{i}}$, corresponding to ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
4:     mInteger *Output
On exit: $m$, the number of terms in the continued fraction representation of $R\left(x\right)$.
5:     a[n]doubleOutput
On exit: ${\mathbf{a}}\left[\mathit{j}-1\right]$ contains the value of the parameter ${a}_{\mathit{j}}$ in $R\left(x\right)$, for $\mathit{j}=1,2,\dots ,m$. The remaining elements of a, if any, are set to zero.
6:     u[n]doubleOutput
On exit: ${\mathbf{u}}\left[\mathit{j}-1\right]$ contains the value of the parameter ${u}_{\mathit{j}}$ in $R\left(x\right)$, for $\mathit{j}=1,2,\dots ,m-1$. 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.
7:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_CONT_FRAC
A continued fraction of the required form does not exist.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>0$.
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.
NE_REAL_ARRAY
On entry, ${\mathbf{x}}\left[i-1\right]$ is very close to ${\mathbf{x}}\left[j-1\right]$: $i=〈\mathit{\text{value}}〉$, ${\mathbf{x}}\left[i-1\right]=〈\mathit{\text{value}}〉$, $j=〈\mathit{\text{value}}〉$ and ${\mathbf{x}}\left[j-1\right]=〈\mathit{\text{value}}〉$.

7  Accuracy

Usually, it is not the accuracy of the coefficients produced by this function which is of prime interest, but rather the accuracy of the value of $R\left(x\right)$ that is produced by the associated function nag_1d_ratnl_eval (e01rbc) when subsequently it evaluates the continued fraction (1) for a given value of $x$. This final accuracy will depend mainly on the nature of the interpolation being performed. If interpolation of a ‘well-behaved smooth’ function is attempted (and provided the data adequately represents the function), high accuracy will normally ensue, but, if the function is not so ‘smooth’ or extrapolation is being attempted, high accuracy is much less likely. Indeed, in extreme cases, results can be highly inaccurate.
There is no built-in test of accuracy but several courses are open to you to prevent the production or the acceptance of inaccurate results.
1. If the origin of a variable is well outside the range of its data values, the origin should be shifted to correct this; and, if the new data values are still excessively large or small, scaling to make the largest value of the order of unity is recommended. Thus, normalization to the range $-1.0$ to $+1.0$ is ideal. This applies particularly to the independent variable; for the dependent variable, the removal of leading figures which are common to all the data values will usually suffice.
2. To check the effect of rounding errors engendered in the functions themselves, nag_1d_ratnl_interp (e01rac) should be re-entered with ${x}_{1}$ interchanged with ${x}_{i}$ and ${f}_{1}$ with ${f}_{i}$, $\left(i\ne 1\right)$. This will produce a completely different vector $a$ and a reordered vector $u$, but any change in the value of $R\left(x\right)$ subsequently produced by nag_1d_ratnl_eval (e01rbc) will be due solely to rounding error.
3. Even if the data consist of calculated values of a formal mathematical function, it is only in exceptional circumstances that bounds for the interpolation error (the difference between the true value of the function underlying the data and the value which would be produced by the two functions if exact arithmetic were used) can be derived that are sufficiently precise to be of practical use. Consequently, you are recommended to rely on comparison checks: if extra data points are available, the calculation may be repeated with one or more data pairs added or exchanged, or alternatively, one of the original data pairs may be omitted. If the algorithms are being used for extrapolation, the calculations should be performed repeatedly with the $2,3,\dots \text{}$ nearest points until, hopefully, successive values of $R\left(x\right)$ for the given $x$ agree to the required accuracy.

The time taken by nag_1d_ratnl_interp (e01rac) is approximately proportional to ${n}^{2}$.
The continued fraction (1) when expanded produces a rational function in $x$, the degree of whose numerator is either equal to or exceeds by unity that of the denominator. Only if this rather special form of interpolatory rational function is needed explicitly, would this function be used without subsequent entry (or entries) to nag_1d_ratnl_eval (e01rbc).

9  Example

This example reads in the abscissae and ordinates of $5$ data points and prints the arguments ${a}_{j}$ and ${u}_{j}$ of a rational function which interpolates them.

9.1  Program Text

Program Text (e01race.c)

9.2  Program Data

Program Data (e01race.d)

9.3  Program Results

Program Results (e01race.r)