NAG Library Function Document

nag_1d_pade_eval (e02rbc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_1d_pade_eval (e02rbc) evaluates a rational function at a user-supplied point, given the numerator and denominator coefficients.

2 Specification

#include <nag.h>

#include <nage02.h>

void	nag_1d_pade_eval (const double a[], Integer ia, const double b[], Integer ib, double x, double ans, NagError fail)

3 Description

Given a real value

x

and the coefficients

a_{j}

, for

j = 0, 1, \dots, l

and

b_{k}

, for

k = 0, 1, \dots, m

, nag_1d_pade_eval (e02rbc) evaluates the rational function

\frac{\sum_{j = 0}^{l} a_{j} x^{j}}{\sum_{k = 0}^{m} b_{k} x^{k}} .

using nested multiplication (see Conte and de Boor (1965)).

A particular use of nag_1d_pade_eval (e02rbc) is to compute values of the Padé approximants determined by nag_1d_pade (e02rac).

4 References

Conte S D and de Boor C (1965) Elementary Numerical Analysis McGraw–Hill

Peters G and Wilkinson J H (1971) Practical problems arising in the solution of polynomial equations J. Inst. Maths. Applics. 8 16–35

5 Arguments

1: a[ia] – const doubleInput: On entry: $a [j]$ , 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 – IntegerInput: On entry: the value of $l + 1$ , where $l$ is the degree of the numerator.
Constraint: $ia \geq 1$ .
3: b[ib] – const doubleInput: On entry: $b [k]$ , 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 $ib = 1$ , $b [0] \neq 0.0$ .
4: ib – IntegerInput: On entry: the value of $m + 1$ , where $m$ is the degree of the denominator.
Constraint: $ib \geq 1$ .
5: x – doubleInput: On entry: the point $x$ at which the rational function is to be evaluated.
6: ans – double *Output: On exit: the result of evaluating the rational function at the given point $x$ .
7: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $ia = ⟨value⟩$ .
Constraint: $ia \geq 1$ .
On entry, $ib = ⟨value⟩$ .
Constraint: $ib \geq 1$ .
NE_INT_ARRAY: The first ib entries in b are zero: $ib = ⟨value⟩$ .
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_POLE_PRESENT: Evaluation at or near a pole.

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.