hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_fit_pade_app (e02ra)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_fit_pade_app (e02ra) calculates the coefficients in a Padé approximant to a function from its user-supplied Maclaurin expansion.


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


Given a power series
nag_fit_pade_app (e02ra) uses the coefficients ci, for i=0,1,,l+m, to form the l/m Padé approximant of the form
a0+a1x+a2x2++alxl b0+b1x+b2x2++bmxm  
with b0 defined to be unity. The two sets of coefficients aj, for j=0,1,,l, and bk, for k=0,1,,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_fit_pade_eval (e02rb).


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


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: ia1 and ib1.
3:     cic – double array
ic, the dimension of the array, must satisfy the constraint icia+ib-1.
ci must specify, for i=1,2,,l+m+1, the coefficient of xi-1 in the given power series.

Optional Input Parameters


Output Parameters

1:     aia – double array
aj+1, for j=1,2,,l+1, contains the coefficient aj in the numerator of the approximant.
2:     bib – double array
bk+1, for k=1,2,,m+1, contains the coefficient bk 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:
On entry,jw<ib×2×ib+3,
(so there are insufficient coefficients in the given power series to calculate the desired approximant).
The Padé approximant is degenerate.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


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., cl or cl+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 m3.


This example calculates the 4/4 Padé approximant of ex (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.
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);

[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('\n\nNumerator coefficients\n');
fprintf('\nDenominator coefficients\n');

% 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,:);

% 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,:);

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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015