E01RAF : NAG Library, Mark 24

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

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

R (x)

which assumes prescribed values

f_{i}

at prescribed values

x_{i}

of the independent variable

x

, for

i = 1, 2, \dots, n

. More specifically, E01RAF determines the parameters

a_{j}

, for

j = 1, 2, \dots, m

and

u_{j}

, for

j = 1, 2, \dots, m - 1

, in the continued fraction

R (x) = a_{1} + R_{m} (x)

(1)

where

R_{i} (x) = \frac{a_{m - i + 2} (x - u_{m - i + 1})}{1 + R_{i - 1} (x)},   for ​ i = m, m - 1, \dots, 2,

and

R_{1} (x) = 0,

such that

R (x_{i}) = f_{i}

, for

i = 1, 2, \dots, n

. The value of

m

in (1) is determined by the routine; 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 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

On entry:

n

, the number of data points.

Constraint:

N > 0

On entry:

X (i)

must be set to the value of the

i

th data abscissa,

x_{i}

, for

i = 1, 2, \dots, n

Constraint: the

X (i)

must be distinct.

On entry:

F (i)

must be set to the value of the data ordinate,

f_{i}

, corresponding to

x_{i}

, for

i = 1, 2, \dots, n

On exit:

m

, the number of terms in the continued fraction representation of

R (x)

On exit:

A (j)

contains the value of the parameter

a_{j}

R (x)

, for

j = 1, 2, \dots, m

. The remaining elements of A, if any, are set to zero.

On exit:

U (j)

contains the value of the parameter

u_{j}

R (x)

, for

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.

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

At least one pair of the values

X (i)

are equal (or so nearly so that a subsequent division will inevitably cause overflow).

A continued fraction of the required form does not exist.

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

R (x)

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 $x_{1}$ interchanged with $x_{i}$ and $f_{1}$ with $f_{i}$ , $(i \neq 1)$ . This will produce a completely different vector $a$ and a reordered vector $u$ , but any change in the value of $R (x)$ 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 $2, 3, \dots$ nearest points until, hopefully, successive values of $R (x)$ for the given $x$ agree to the required accuracy.

The time taken by E01RAF 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 routine be used without subsequent entry (or entries) to E01RBF.

This example reads in the abscissae and ordinates of

5

data points and prints the parameters

a_{j}

and

u_{j}

of a rational function which interpolates them.

NAG Library Routine Document

E01RAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine DocumentE01RAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

E01RAF