, 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 nag_fit_pade_eval (e02rb).

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

Parameters

Compulsory Input Parameters

1: $ia$ – int64int32nag_int scalar
2: $ib$ – int64int32nag_int scalar: 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: $c (ic)$ – double array: ic, the dimension of the array, must satisfy the constraint $ic \geq ia + ib - 1$ .
$c (i)$ must specify, for $i = 1, 2, \dots, l + m + 1$ , the coefficient of $x^{i - 1}$ in the given power series.

Optional Input Parameters

None.

Output Parameters

1: $a (ia)$ – double array: $a (j + 1)$ , for $j = 1, 2, \dots, l + 1$ , contains the coefficient $a_{j}$ in the numerator of the approximant.
2: $b (ib)$ – double array: $b (k + 1)$ , for $k = 1, 2, \dots, m + 1$ , contains the coefficient $b_{k}$ in the denominator of the approximant.
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,	$jw < ib \times (2 \times ib + 3)$ ,
or	$ia < 1$ ,
or	$ib < 1$ ,
or	$ic < ia + ib - 1$

(so there are insufficient coefficients in the given power series to calculate the desired approximant).

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

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

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 nag_zeros_poly_real (c02ag) 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).

Further Comments

The time taken is approximately proportional to

m^{3}

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.

Open in the MATLAB editor: e02ra_example

function e02ra_example


fprintf('e02ra example results\n\n');

ia = int64(5);
ib = int64(5);
ic = ia + ib - 1;
c  = ones(ic,1);
for j = 3:ic
  c(j) = c(j-1)/double(j-1);
end

[a, b, ifail] = e02ra(ia, ib, c);

fprintf('The given series coefficients are\n');
fmt = '%13.4e%13.4e%13.4e%13.4e%13.4e\n';
fprintf(fmt,c);
fprintf('\n\nNumerator coefficients\n');
fprintf(fmt,a);
fprintf('\nDenominator coefficients\n');
fprintf(fmt,b);

% Calculate zeros of the approximant using c02ag.
dd(ia:-1:1) = a(1:ia);
[z, ifail] = c02ag(dd,ia-1);

fprintf('\nZeros of approximant are at\n\n');
cz = z(1,:) + i*z(2,:);
disp(cz');

% Calculate poles of the approximant using c02ag
dd(ib:-1:1) = b(1:ib);
[z, ifail] = c02ag(dd,ib-1);

fprintf('\nPoles of approximant are at\n\n');
cz = z(1,:) + i*z(2,:);
disp(cz');

e02ra example results

The given series coefficients are
   1.0000e+00   1.0000e+00   5.0000e-01   1.6667e-01   4.1667e-02
   8.3333e-03   1.3889e-03   1.9841e-04   2.4802e-05

Numerator coefficients
   1.0000e+00   5.0000e-01   1.0714e-01   1.1905e-02   5.9524e-04

Denominator coefficients
   1.0000e+00  -5.0000e-01   1.0714e-01  -1.1905e-02   5.9524e-04

Zeros of approximant are at

  -5.7924 - 1.7345i
  -5.7924 + 1.7345i
  -4.2076 - 5.3148i
  -4.2076 + 5.3148i


Poles of approximant are at

   5.7924 - 1.7345i
   5.7924 + 1.7345i
   4.2076 - 5.3148i
   4.2076 + 5.3148i

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox