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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_ellipint_legendre_1 (s21be)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_specfun_ellipint_legendre_1 (s21be) returns a value of the classical (Legendre) form of the incomplete elliptic integral of the first kind, via the function name.


[result, ifail] = s21be(phi, dm)
[result, ifail] = nag_specfun_ellipint_legendre_1(phi, dm)


nag_specfun_ellipint_legendre_1 (s21be) calculates an approximation to the integral
Fϕm = 0ϕ 1-m sin2θ -12 dθ ,  
where 0ϕ π2 , msin2ϕ1  and m  and sinϕ  may not both equal one.
The integral is computed using the symmetrised elliptic integrals of Carlson (Carlson (1979) and Carlson (1988)). The relevant identity is
Fϕm = RF q,r,1 sinϕ ,  
where q=cos2ϕ , r=1-m sin2ϕ  and RF  is the Carlson symmetrised incomplete elliptic integral of the first kind (see nag_specfun_ellipint_symm_1 (s21bb)).


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 ;
  • Only one of sinphi  and dm may be 1.0.
Note that dm × sin2phi = 1.0  is allowable, as long as dm1.0 .

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:

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

Constraint: 0phiπ2.
On soft failure, the function returns zero.
Constraint: dm×sin2phi1.0.
On soft failure, the function returns zero.
W  ifail=3
On entry, sinphi=1 and dm=1.0; the integral is infinite.
On soft failure, the function returns the largest machine number (see nag_machine_real_largest (x02al)).
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_1 (s21be) 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 algorithm used to compute RF , see the function document for nag_specfun_ellipint_symm_1 (s21bb).
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, F-ϕ|m=-Fϕ|m and Fsπ±ϕ|m=2sKm±Fϕ|m where s is an integer and Km is the complete elliptic integral given by nag_specfun_ellipint_complete_1 (s21bh).
A parameter m>1 can be replaced by one less than unity using Fϕ|m=1mFθ|1m, sinθ=msinϕ.


This example simply generates a small set of nonextreme arguments that are used with the function to produce the table of results.
function s21be_example

fprintf('s21be 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] = s21be(phi(j), dm(j));

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


function s21be_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] = s21be(phi(i), m(j));

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

s21be example results

    phi      m        F(phi|m)
    0.52    0.25       0.5294
    1.05    0.50       1.1424
    1.57    0.75       2.1565

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Chapter Contents
Chapter Introduction
NAG Toolbox

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