nag_specfun_ellipint_complete_2 (s21bj) returns a value of the classical (Legendre) form of the complete elliptic integral of the second kind, via the function name.

Syntax

[result, ifail] = s21bj(dm)

[result, ifail] = nag_specfun_ellipint_complete_2(dm)

Description

nag_specfun_ellipint_complete_2 (s21bj) calculates an approximation to the integral

E (m) = \int_{0}^{\frac{π}{2}} {(1 - m \sin^{2} θ)}^{\frac{1}{2}} d θ,

where

m \leq 1

The integral is computed using the symmetrised elliptic integrals of Carlson (Carlson (1979) and Carlson (1988)). The relevant identity is

E (m) = R_{F} (0, 1 - m, 1) - \frac{1}{3} m R_{D} (0, 1 - m, 1),

where

R_{F}

is the Carlson symmetrised incomplete elliptic integral of the first kind (see nag_specfun_ellipint_symm_1 (s21bb)) and

R_{D}

is the Carlson symmetrised incomplete elliptic integral of the second kind (see nag_specfun_ellipint_symm_2 (s21bc)).

References

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

Carlson B C (1979) Computing elliptic integrals by duplication Numerische Mathematik 33 1–16

Carlson B C (1988) A table of elliptic integrals of the third kind Math. Comput. 51 267–280

Parameters

Compulsory Input Parameters

1: $dm$ – double scalar: The argument $m$ of the function.

Constraint: $dm \leq 1.0$ .

Optional Input Parameters

None.

Output Parameters

1: $result$ – double scalar: The result of the function.
2: $ifail$ – 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:

$ifail = 1$: Constraint: $dm \leq 1.0$ .

$ifail = - 99$: 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 nag_specfun_ellipint_complete_2 (s21bj) is capable of producing full machine precision. However round-off errors in internal arithmetic will result in slight loss of accuracy. This loss should never be excessive as the algorithm does not involve any significant amplification of round-off error. It is reasonable to assume that the result is accurate to within a small multiple of the machine precision.

Further Comments

You should consult the S Chapter Introduction, which shows the relationship between this function and the Carlson definitions of the elliptic integrals. In particular, the relationship between the argument-constraints for both forms becomes clear.

For more information on the algorithms used to compute

R_{F}

and

R_{D}

, see the function documents for nag_specfun_ellipint_symm_1 (s21bb) and nag_specfun_ellipint_symm_2 (s21bc), respectively.

Example

This example simply generates a small set of nonextreme arguments that are used with the function to produce the table of results.

Open in the MATLAB editor: s21bj_example

function s21bj_example


fprintf('s21bj example results\n\n');

m = [0.25     0.5     0.75];
result = m;

for j=1:numel(m)
  [result(j), ifail] = s21bj(m(j));
end

disp('      m        E(m)');
fprintf('%8.2f%12.4f\n',[m; result]);

s21bj example results

      m        E(m)
    0.25      1.4675
    0.50      1.3506
    0.75      1.2111

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

Chapter Contents

Chapter Introduction

NAG Toolbox