E01RAF produces, from a set of function values and corresponding abscissae, the coefficients of an interpolating rational function expressed in continued fraction form.
E01RAF produces the parameters of a rational function
Rx which assumes prescribed values
fi at prescribed values
xi of the independent variable
x, for
i=1,2,…,n. More specifically, E01RAF determines the parameters
aj, for
j=1,2,…,m and
uj,
j=1,2,…,m-1, in the continued fraction
where
and
such that
Rxi=fi, for
i=1,2,…,n. The value of
m in
(1) is determined by the routine; normally
m=n. The values of
uj form a reordered subset of the values of
xi 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
E01RBF.
The computational method employed in E01RAF is the modification of the Thacher–Tukey algorithm described in
Graves–Morris and Hopkins (1981).
Graves–Morris P R and Hopkins T R (1981) Reliable rational interpolation
Numer. Math. 36 111–128
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Usually, it is not the accuracy of the coefficients produced by this routine which is of prime interest, but rather the accuracy of the value of
Rx that is produced by the associated routine
E01RBF 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.
- 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.
- To check the effect of rounding errors engendered in the routines themselves, E01RAF should be re-entered with x1 interchanged with xi and f1 with fi, i≠1. This will produce a completely different vector a and a reordered vector u, but any change in the value of Rx subsequently produced by E01RBF will be due solely to rounding error.
- 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 routines 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,… nearest points until, hopefully, successive values of Rx for the given x agree to the required accuracy.
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 routine be used without subsequent entry (or entries) to
E01RBF.