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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_ellipint_complete_1 (s21bh)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_specfun_ellipint_complete_1 (s21bh) returns a value of the classical (Legendre) form of the complete elliptic integral of the first kind, via the function name.


[result, ifail] = s21bh(dm)
[result, ifail] = nag_specfun_ellipint_complete_1(dm)


nag_specfun_ellipint_complete_1 (s21bh) calculates an approximation to the integral
Km = 0 π2 1-m sin2θ -12 dθ ,  
where m<1 .
The integral is computed using the symmetrised elliptic integrals of Carlson (Carlson (1979) and Carlson (1988)). The relevant identity is
Km = RF 0,1-m,1 ,  
where 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:     dm – double scalar
The argument m of the function.
Constraint: dm<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:

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

Constraint: dm<1.0.
On soft failure, the function returns zero.
W  ifail=2
On entry, 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_complete_1 (s21bh) 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).


This example simply generates a small set of nonextreme arguments that are used with the function to produce the table of results.
function s21bh_example

fprintf('s21bh example results\n\n');

m = [0.25     0.5     0.75];
result = m;

for j=1:numel(m)
  [result(j), ifail] = s21bh(m(j));

disp('      m        K(m)');
fprintf('%8.2f%12.4f\n',[m; result]);

s21bh example results

      m        K(m)
    0.25      1.6858
    0.50      1.8541
    0.75      2.1565

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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