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.

Syntax

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

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

Description

nag_specfun_ellipint_legendre_2 (s21bf) calculates an approximation to the integral

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

where

0 \leq ϕ \leq \frac{π}{2}

and

m \sin^{2} ϕ \leq 1

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

E (ϕ ∣ m) = \sin ϕ R_{F} (q, r, 1) - \frac{1}{3} m \sin^{3} ϕ R_{D} (q, r, 1),

where

q = \cos^{2} ϕ

r = 1 - m \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_{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: $phi$ – double scalar

2: $dm$ – double scalar

The arguments

ϕ

and

m

of the function.

Constraints:

$0.0 \leq phi \leq \frac{π}{2}$ ;
$dm \times \sin^{2} (phi) \leq 1.0$ .

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: $0 \leq phi \leq \frac{π}{2}$ .

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

$ifail = - 99$: 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_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

R_{F}

and

R_{D}

, 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) = - E (ϕ | m)

. A parameter

m > 1

can be replaced by one less than unity using

E (ϕ | m) = \sqrt{m} E (ϕ \sqrt{m} | \frac{1}{m}) - (m - 1) ϕ

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: s21bf_example

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));
end

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

s21bf_plot;



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));
    end
  end

  fig1 = figure;
  [Y,X] = meshgrid(m,phi);
  surf(X,Y,F);
  xlabel('\Phi');
  ylabel('m');
  zlabel('F(\Phi,n)');
  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

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

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