nag_specfun_ellipint_legendre_3 (s21bg) returns a value of the classical (Legendre) form of the incomplete elliptic integral of the third kind, via the function name.

Syntax

[result, ifail] = s21bg(dn, phi, dm)

[result, ifail] = nag_specfun_ellipint_legendre_3(dn, phi, dm)

Description

nag_specfun_ellipint_legendre_3 (s21bg) calculates an approximation to the integral

Π (n; ϕ ∣ m) = \int_{0}^{ϕ} {(1 - n \sin^{2} θ)}^{- 1} {(1 - m \sin^{2} θ)}^{- \frac{1}{2}} d θ,

where

0 \leq ϕ \leq \frac{π}{2}

m \sin^{2} ϕ \leq 1

m

and

\sin ϕ

may not both equal one, and

n \sin^{2} ϕ \neq 1

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

Π (n; ϕ ∣ m) = \sin ϕ R_{F} (q, r, 1) + \frac{1}{3} n \sin^{3} ϕ R_{J} (q, r, 1, s),

where

q = \cos^{2} ϕ

r = 1 - m \sin^{2} ϕ

s = 1 - n \sin^{2} ϕ

R_{F}

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

R_{J}

is the Carlson symmetrised incomplete elliptic integral of the third kind (see nag_specfun_ellipint_symm_3 (s21bd)).

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: $dn$ – double scalar

2: $phi$ – double scalar

3: $dm$ – double scalar

The arguments

n

ϕ

and

m

of the function.

Constraints:

$0.0 \leq phi \leq \frac{π}{2}$ ;
$dm \times \sin^{2} (phi) \leq 1.0$ ;
Only one of $\sin (phi)$ and dm may be $1.0$ ;
$dn \times \sin^{2} (phi) \neq 1.0$ .

Note that

dm \times \sin^{2} (phi) = 1.0

is allowable, as long as

dm \neq 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:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$ifail = 1$: Constraint: $0 \leq phi \leq (π / 2)$ .

$ifail = 2$: Constraint: $dm \times \sin^{2} (phi) \leq 1.0$ .

W $ifail = 3$: On entry, $\sin (phi) = 1$ and $dm = 1.0$ ; the integral is infinite.

W $ifail = 4$: Constraint: $dn \times \sin^{2} (phi) \neq 1.0$ .

$ifail = - 99$: 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_legendre_3 (s21bg) 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_{J}

, see the function documents for nag_specfun_ellipint_symm_1 (s21bb) and nag_specfun_ellipint_symm_3 (s21bd), respectively.

If you wish to input a value of phi outside the range allowed by this function you should refer to Section 17.4 of Abramowitz and Stegun (1972) for useful identities.

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: s21bg_example

function s21bg_example


fprintf('s21bg example results\n\n');

dn  = [0.1   -0.2   0.3];
phi = [pi/6  pi/3   pi/2];
dm  = [1/4   1/2    3/4];
result = phi;

for j = 1:numel(phi)
  [result(j), ifail] = s21bg(dn(j), phi(j), dm(j));
end

fprintf('    n       phi      m       Pi(n;phi|m)\n');
fprintf(' %7.2f %7.2f %7.2f %12.4f\n', [dn; phi; dm; result]);

s21bg example results

    n       phi      m       Pi(n;phi|m)
    0.10    0.52    0.25       0.5341
   -0.20    1.05    0.50       1.0778
    0.30    1.57    0.75       2.6568

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

Chapter Contents

Chapter Introduction

NAG Toolbox