nag_specfun_jacellip_real (s21ca) is based on a procedure given by Bulirsch (1960), and uses the process of the arithmetic-geometric mean (16.9 in Abramowitz and Stegun (1972)). Constraints are placed on the values of

u

and

m

in order to avoid the possibility of machine overflow.

References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

Bulirsch R (1960) Numerical calculation of elliptic integrals and elliptic functions Numer. Math. 7 76–90

Parameters

Compulsory Input Parameters

1: $u$ – double scalar

2: $m$ – double scalar

The argument

u

and the argument

m

of the functions, respectively.

Constraints:

$abs (u) \leq \sqrt{λ}$ , where $λ = 1 / x02am$ ;
if $abs (u) < 1 / \sqrt{λ}$ , $abs (m) \leq \sqrt{λ}$ .

Optional Input Parameters

None.

Output Parameters

1: $sn$ – double scalar
2: $cn$ – double scalar
3: $dn$ – double scalar: The values of the functions $sn u$ , $cn u$ and $dn u$ , respectively.
4: $ifail$ – 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,

abs (u) > \sqrt{λ}

, where

λ = 1 / x02am ()

$ifail = 2$

On entry,

abs (m) > \sqrt{λ}

and

abs (u) < 1 / \sqrt{λ}

$ifail = - 99$: 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

In principle the function is capable of achieving full relative precision in the computed values. However, the accuracy obtainable in practice depends on the accuracy of the standard elementary functions such as SIN and COS.

Further Comments

None.

Example

This example reads values of the argument

u

and argument

m

from a file, evaluates the function and prints the results.

Open in the MATLAB editor: s21ca_example

function s21ca_example


fprintf('s21ca example results\n\n');

u = [0.2   5  -0.5   10];
m = [0.3  -1  -0.1   11];
sn = u; cn = u; dn = u;

for j=1:numel(u)
  [sn(j), cn(j), dn(j), ifail] = s21ca(u(j),m(j));
end

disp('       u         m         sn        cn        dn');
fprintf('%10.2f%10.2f%10.4f%10.4f%10.4f\n',[u; m; sn; cn; dn]);

s21ca example results

       u         m         sn        cn        dn
      0.20      0.30    0.1983    0.9801    0.9941
      5.00     -1.00   -0.2440    0.9698    1.0293
     -0.50     -0.10   -0.4812    0.8766    1.0115
     10.00     11.00    0.2512    0.9679    0.5528

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

Chapter Contents

Chapter Introduction

NAG Toolbox