E02RBF (PDF version)
E02 Chapter Contents
E02 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

E02RBF

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.

 Contents

    1  Purpose
    7  Accuracy

1  Purpose

E02RBF evaluates a rational function at a user-supplied point, given the numerator and denominator coefficients.

2  Specification

SUBROUTINE E02RBF ( A, IA, B, IB, X, ANS, IFAIL)
INTEGER  IA, IB, IFAIL
REAL (KIND=nag_wp)  A(IA), B(IB), X, ANS

3  Description

Given a real value x and the coefficients aj, for j=0,1,,l and bk, for k=0,1,,m, E02RBF evaluates the rational function
j=0lajxj k=0mbkxk .  
using nested multiplication (see Conte and de Boor (1965)).
A particular use of E02RBF is to compute values of the Padé approximants determined by E02RAF.

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  Parameters

1:     AIA – REAL (KIND=nag_wp) arrayInput
On entry: Aj+1, for j=1,2,,l+1, must contain the value of the coefficient aj 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: IA1.
3:     BIB – REAL (KIND=nag_wp) arrayInput
On entry: Bk+1, for k=1,2,,m+1, must contain the value of the coefficient bk in the denominator of the rational function.
Constraint: if IB=1, B10.0.
4:     IB – INTEGERInput
On entry: the value of m+1, where m is the degree of the denominator.
Constraint: IB1.
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 – INTEGERInput/Output
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).

6  Error Indicators and Warnings

If on entry IFAIL=0 or -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
The rational function is being evaluated at or near a pole.
IFAIL=2
On entry,IA<1,
orIB<1,
orB1=0.0 when IB=1 (so the denominator is identically zero).
IFAIL=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.
IFAIL=-399
Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.
IFAIL=-999
Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction 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.

8  Parallelism and Performance

Not applicable.

9  Further Comments

The time taken is approximately proportional to l+m.

10  Example

This example first calls E02RAF to calculate the 4/4 Padé approximant to ex, and then uses E02RBF to evaluate the approximant at x=0.1,0.2,,1.0.

10.1  Program Text

Program Text (e02rbfe.f90)

10.2  Program Data

None.

10.3  Program Results

Program Results (e02rbfe.r)

GnuplotProduced by GNUPLOT 4.6 patchlevel 3 0 2 4 6 8 10 12 14 16 18 20 22 0 0.5 1 1.5 2 2.5 3 1e−14 1e−12 1e−10 1e−08 1e−06 0.0001 0.01 y(x) Error x Example Program The [4|4] Pade Approximant of exp(x) gnuplot_plot_1 Pade gnuplot_plot_2 exp(x) gnuplot_plot_3 Error

E02RBF (PDF version)
E02 Chapter Contents
E02 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2015