NAG Library Routine Document

Integer, Intent (In)	::	ia, ib, ic, jw
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	c(ic)
Real (Kind=nag_wp), Intent (Out)	::	a(ia), b(ib), w(jw)

C Header Interface

#include <nagmk26.h>

void	e02raf_ (const Integer ia, const Integer ib, const double c[], const Integer ic, double a[], double b[], double w[], const Integer jw, Integer *ifail)

3

Description

Given a power series

c_{0} + c_{1} x + c_{2} x^{2} + \dots + c_{l + m} x^{l + m} + \dots

e02raf uses the coefficients

c_{i}

, for

i = 0, 1, \dots, l + m

, to form the

[l / m]

Padé approximant of the form

\frac{a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{l} x^{l}}{b_{0} + b_{1} x + b_{2} x^{2} + \dots + b_{m} x^{m}}

with

b_{0}

defined to be unity. The two sets of coefficients

a_{j}

, for

j = 0, 1, \dots, l

, and

b_{k}

, for

k = 0, 1, \dots, m

, in the numerator and denominator are calculated by direct solution of the Padé equations (see Graves–Morris (1979)); these values are returned through the argument list unless the approximant is degenerate.

Padé approximation is a useful technique when values of a function are to be obtained from its Maclaurin expansion but convergence of the series is unacceptably slow or even nonexistent. It is based on the hypothesis of the existence of a sequence of convergent rational approximations, as described in Baker and Graves–Morris (1981) and Graves–Morris (1979).

Unless there are reasons to the contrary (as discussed in Chapter 4, Section 2, Chapters 5 and 6 of Baker and Graves–Morris (1981)), one normally uses the diagonal sequence of Padé approximants, namely

\{[m / m], m = 0, 1, 2, \dots\} .

Subsequent evaluation of the approximant at a given value of

x

may be carried out using e02rbf.

4

References

Baker G A Jr and Graves–Morris P R (1981) Padé approximants, Part 1: Basic theory encyclopaedia of Mathematics and its Applications Addison–Wesley

Graves–Morris P R (1979) The numerical calculation of Padé approximants Padé Approximation and its Applications. Lecture Notes in Mathematics (ed L Wuytack) 765 231–245 Adison–Wesley

5

Arguments

1: $ia$ – IntegerInput
2: $ib$ – IntegerInput: On entry: ia must specify $l + 1$ and ib must specify $m + 1$ , where $l$ and $m$ are the degrees of the numerator and denominator of the approximant, respectively.

Constraint: $ia \geq 1$ and $ib \geq 1$ .
3: $c (ic)$ – Real (Kind=nag_wp) arrayInput: On entry: $c (i)$ must specify, for $i = 1, 2, \dots, l + m + 1$ , the coefficient of $x^{i - 1}$ in the given power series.
4: $ic$ – IntegerInput: On entry: the dimension of the array c as declared in the (sub)program from which e02raf is called.

Constraint: $ic \geq ia + ib - 1$ .
5: $a (ia)$ – Real (Kind=nag_wp) arrayOutput: On exit: $a (j + 1)$ , for $j = 1, 2, \dots, l + 1$ , contains the coefficient $a_{j}$ in the numerator of the approximant.
6: $b (ib)$ – Real (Kind=nag_wp) arrayOutput: On exit: $b (k + 1)$ , for $k = 1, 2, \dots, m + 1$ , contains the coefficient $b_{k}$ in the denominator of the approximant.
7: $w (jw)$ – Real (Kind=nag_wp) arrayWorkspace
8: $jw$ – IntegerInput: On entry: the dimension of the array w as declared in the (sub)program from which e02raf is called.

Constraint: $jw \geq ib \times (2 \times ib + 3)$ .
9: $ifail$ – IntegerInput/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $ia = 〈value〉$ .
Constraint: $ia \geq 1$ .

On entry, $ib = 〈value〉$ .
Constraint: $ib \geq 1$ .

On entry, $ic = 〈value〉$ , $ia = 〈value〉$ and $ib = 〈value〉$ .
Constraint: $ic \geq ia + ib - 1$ .

On entry, $jw = 〈value〉$ and $ib = 〈value〉$ .
Constraint: $jw \geq ib \times (2 \times ib + 3)$ .

$ifail = 2$: The Pade approximant is degenerate.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

The solution should be the best possible to the extent to which the solution is determined by the input coefficients. It is recommended that you determine the locations of the zeros of the numerator and denominator polynomials, both to examine compatibility with the analytic structure of the given function and to detect defects. (Defects are nearby pole-zero pairs; defects close to

x = 0.0

characterise ill-conditioning in the construction of the approximant.) Defects occur in regions where the approximation is necessarily inaccurate. The example program calls c02agf to determine the above zeros.

It is easy to test the stability of the computed numerator and denominator coefficients by making small perturbations of the original Maclaurin series coefficients (e.g.,

c_{l}

c_{l + m}

). These questions of intrinsic error of the approximants and computational error in their calculation are discussed in Chapter 2 of Baker and Graves–Morris (1981).

8

Parallelism and Performance

e02raf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

e02raf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

Further Comments

The time taken is approximately proportional to

m^{3}

10

Example

This example calculates the

[4 / 4]

Padé approximant of

e^{x}

(whose power-series coefficients are first stored in the array c). The poles and zeros are then calculated to check the character of the

[4 / 4]

Padé approximant.

10.1

Program Text

Program Text (e02rafe.f90)

10.2

Program Data

None.

10.3

Program Results

Program Results (e02rafe.r)

NAG Library Manual, Mark 26

NAG AD Library Manual, Mark 26

NAG C Library Manual, Mark 26

E02 (fit) Chapter Contents

E02 (fit) Chapter Introduction

NAG Library Routine Document

e02raf (pade_app)

▸▿ 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

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