NAG Library Function Document
nag_1d_pade (e02rac)
1
Purpose
nag_1d_pade (e02rac) calculates the coefficients in a Padé approximant to a function from its usersupplied Maclaurin expansion.
2
Specification
#include <nag.h> 
#include <nage02.h> 
void 
nag_1d_pade (Integer ia,
Integer ib,
const double c[],
double a[],
double b[],
NagError *fail) 

3
Description
Given a power series
nag_1d_pade (e02rac) uses the coefficients
${c}_{i}$, for
$\mathit{i}=0,1,\dots ,l+m$, to form the
$\left[l/m\right]$ Padé approximant of the form
with
${b}_{0}$ defined to be unity. The two sets of coefficients
${a}_{j}$, for
$\mathit{j}=0,1,\dots ,l$, and
${b}_{k}$, for
$\mathit{k}=0,1,\dots ,m$, 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
Subsequent evaluation of the approximant at a given value of
$x$ may be carried out using
nag_1d_pade_eval (e02rbc).
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:
$\mathbf{ia}$ – IntegerInput
 2:
$\mathbf{ib}$ – 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:
${\mathbf{ia}}\ge 1$ and ${\mathbf{ib}}\ge 1$.
 3:
$\mathbf{c}\left[\left({\mathbf{ia}}+{\mathbf{ib}}1\right)\right]$ – const doubleInput

On entry: ${\mathbf{c}}\left[\mathit{i}1\right]$ must specify, for $\mathit{i}=1,2,\dots ,l+m+1$, the coefficient of ${x}^{\mathit{i}1}$ in the given power series.
 4:
$\mathbf{a}\left[{\mathbf{ia}}\right]$ – doubleOutput

On exit: ${\mathbf{a}}\left[\mathit{j}\right]$, for $\mathit{j}=1,2,\dots ,l+1$, contains the coefficient ${a}_{\mathit{j}}$ in the numerator of the approximant.
 5:
$\mathbf{b}\left[{\mathbf{ib}}\right]$ – doubleOutput

On exit: ${\mathbf{b}}\left[\mathit{k}\right]$, for $\mathit{k}=1,2,\dots ,m+1$, contains the coefficient ${b}_{\mathit{k}}$ in the denominator of the approximant.
 6:
$\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_DEGENERATE

The Pade approximant is degenerate.
 NE_INT_2

On entry, ${\mathbf{ib}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{ia}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ia}}\ge 1$ and ${\mathbf{ib}}\ge 1$.
 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.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 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 polezero pairs; defects close to
$x=0.0$ characterise illconditioning in the construction of the approximant.) Defects occur in regions where the approximation is necessarily inaccurate. The example program calls
nag_zeros_real_poly (c02agc) 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.,
${c}_{l}$ or
${c}_{l+m}$). 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
nag_1d_pade (e02rac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_1d_pade (e02rac) 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 function. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The time taken is approximately proportional to ${m}^{3}$.
10
Example
This example calculates the
$\left[4/4\right]$ Padé approximant of
${e}^{x}$ (whose powerseries coefficients are first stored in the array
c). The poles and zeros are then calculated to check the character of the
$\left[4/4\right]$ Padé approximant.
10.1
Program Text
Program Text (e02race.c)
10.2
Program Data
None.
10.3
Program Results
Program Results (e02race.r)