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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_ellipint_legendre_2 (s21bf)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_specfun_ellipint_legendre_2 (s21bf) returns a value of the classical (Legendre) form of the incomplete elliptic integral of the second kind, via the function name.


[result, ifail] = s21bf(phi, dm)
[result, ifail] = nag_specfun_ellipint_legendre_2(phi, dm)


nag_specfun_ellipint_legendre_2 (s21bf) calculates an approximation to the integral
Eϕm = 0ϕ 1-m sin2θ 12 dθ ,  
where 0ϕ π2  and msin2ϕ1 .
The integral is computed using the symmetrised elliptic integrals of Carlson (Carlson (1979) and Carlson (1988)). The relevant identity is
Eϕm = sinϕ RF q,r,1 - 13 m sin3ϕ RD q,r,1 ,  
where q=cos2ϕ , r=1-m sin2ϕ , RF  is the Carlson symmetrised incomplete elliptic integral of the first kind (see nag_specfun_ellipint_symm_1 (s21bb)) and RD  is the Carlson symmetrised incomplete elliptic integral of the second kind (see nag_specfun_ellipint_symm_2 (s21bc)).


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


Compulsory Input Parameters

1:     phi – double scalar
2:     dm – double scalar
The arguments ϕ and m of the function.
  • 0.0phi π2;
  • dm× sin2phi 1.0 .

Optional Input Parameters


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:
Constraint: 0phiπ2.
Constraint: dm×sin2phi1.0.
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.


In principle nag_specfun_ellipint_legendre_2 (s21bf) 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 RD , see the function documents for nag_specfun_ellipint_symm_1 (s21bb) and nag_specfun_ellipint_symm_2 (s21bc), 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. For example, E-ϕ|m=-Eϕ|m. A parameter m>1 can be replaced by one less than unity using Eϕ|m=mEϕm|1m-m-1ϕ.


This example simply generates a small set of nonextreme arguments that are used with the function to produce the table of results.
function s21bf_example

fprintf('s21bf example results\n\n');

phi = [pi/6  pi/3   pi/2];
dm  = [1/4   1/2    3/4];
result = phi;

for j = 1:numel(phi)
  [result(j), ifail] = s21bf(phi(j), dm(j));

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


function s21bf_plot
  phi = [1:0.02:1.56];
  m = [0.5:0.02:0.98,0.982:0.002:1];
  F = zeros(numel(phi),numel(m));
  for i = 1:numel(phi)
    for j = 1:numel(m)
      [F(i,j), ifail] = s21bf(phi(i), m(j));

  fig1 = figure;
  [Y,X] = meshgrid(m,phi);
  title({'Incomplete Elliptic Integral of the second kind', ...
         'Classical (Legendre) form'});

s21bf example results

    phi      m        E(phi|m)
    0.52    0.25       0.5179
    1.05    0.50       0.9650
    1.57    0.75       1.2111

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Chapter Contents
Chapter Introduction
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