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NAG Toolbox: nag_specfun_ellipint_legendre_1 (s21be)
Purpose
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.
Syntax
Description
nag_specfun_ellipint_legendre_1 (s21be) calculates an approximation to the integral
where
,
and
and
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
where
,
and
is the Carlson symmetrised incomplete elliptic integral of the first kind (see
nag_specfun_ellipint_symm_1 (s21bb)).
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:
– double scalar
- 2:
– double scalar
-
The arguments and of the function.
Constraints:
- ;
- ;
- Only one of and dm may be .
Note that is allowable, as long as .
Optional Input Parameters
None.
Output Parameters
- 1:
– double scalar
The result of the function.
- 2:
– int64int32nag_int scalar
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: .
On soft failure, the function returns zero.
-
-
Constraint: .
On soft failure, the function returns zero.
- W
-
On entry,
and
; 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.
Accuracy
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
, 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,
and
where
is an integer and
is the complete elliptic integral given by
nag_specfun_ellipint_complete_1 (s21bh).
A parameter can be replaced by one less than unity using , .
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:
s21be_example
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));
end
fprintf(' phi m F(phi|m)\n');
fprintf(' %7.2f %7.2f %12.4f\n', [phi; dm; result]);
s21be_plot;
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));
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 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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