1: $a (ia)$ – Real (Kind=nag_wp) array Input: On entry: $a (j + 1)$ , for $j = 1, 2, \dots, l + 1$ , must contain the value of the coefficient $a_{j}$ in the numerator of the rational function.
2: $ia$ – Integer Input: On entry: the value of $l + 1$ , where $l$ is the degree of the numerator.

Constraint: $ia \geq 1$ .
3: $b (ib)$ – Real (Kind=nag_wp) array Input: On entry: $b (k + 1)$ , for $k = 1, 2, \dots, m + 1$ , must contain the value of the coefficient $b_{k}$ in the denominator of the rational function.

Constraint: if $ib = 1$ , $b (1) \neq 0.0$ .
4: $ib$ – Integer Input: On entry: the value of $m + 1$ , where $m$ is the degree of the denominator.

Constraint: $ib \geq 1$ .
5: $x$ – Real (Kind=nag_wp) Input: On entry: the point $x$ at which the rational function is to be evaluated.
6: $ans$ – Real (Kind=nag_wp) Output: On exit: the result of evaluating the rational function at the given point $x$ .
7: $ifail$ – 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 $- 1$ 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).

6 Error Indicators and Warnings

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:

$ifail = 1$: Evaluation at or near a pole.

$ifail = 2$: On entry, $ia = ⟨ value ⟩$ .
Constraint: $ia \geq 1$ .

On entry, $ib = ⟨ value ⟩$ .
Constraint: $ib \geq 1$ .

The first ib entries in b are zero: $ib = ⟨ value ⟩$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

A running error analysis for polynomial evaluation by nested multiplication using the recurrence suggested by Kahan (see Peters and Wilkinson (1971)) is used to detect whether you are attempting to evaluate the approximant at or near a pole.