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NAG Toolbox: nag_specfun_ellipint_legendre_3 (s21bg)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

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;ϕm = 0ϕ 1-n sin2θ -1 1-m sin2θ -12 dθ ,  
where 0ϕ π2 , msin2ϕ1 , m  and sinϕ  may not both equal one, and nsin2ϕ1 .
The integral is computed using the symmetrised elliptic integrals of Carlson (Carlson (1979) and Carlson (1988)). The relevant identity is
Π n;ϕm = sinϕ RF q,r,1 + 13 n sin3ϕ RJ q,r,1,s ,  
where q=cos2ϕ , r=1-m sin2ϕ , s=1-n sin2ϕ , RF  is the Carlson symmetrised incomplete elliptic integral of the first kind (see nag_specfun_ellipint_symm_1 (s21bb)) and RJ  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:
  • 0.0phi π2;
  • dm× sin2phi 1.0 ;
  • Only one of sinphi  and dm may be 1.0;
  • dn× sin2phi 1.0 .
Note that dm × sin2phi = 1.0  is allowable, as long as dm1.0 .

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: 0phiπ/2.
   ifail=2
Constraint: dm×sin2phi1.0.
W  ifail=3
On entry, sinphi=1 and dm=1.0; the integral is infinite.
W  ifail=4
Constraint: dn×sin2phi1.0.
   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 RF  and RJ , 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.
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

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Chapter Introduction
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