nag_elliptic_integral_complete_E (s21bjc) returns a value of the classical (Legendre) form of the complete elliptic integral of the second kind.
nag_elliptic_integral_complete_E (s21bjc) calculates an approximation to the integral
where
$m\le 1$.
The integral is computed using the symmetrised elliptic integrals of Carlson (
Carlson (1979) and
Carlson (1988)). The relevant identity is
where
${R}_{F}$ is the Carlson symmetrised incomplete elliptic integral of the first kind (see
nag_elliptic_integral_rf (s21bbc)) and
${R}_{D}$ is the Carlson symmetrised incomplete elliptic integral of the second kind (see
nag_elliptic_integral_rd (s21bcc)).
In principle nag_elliptic_integral_complete_E (s21bjc) 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.
nag_elliptic_integral_complete_E (s21bjc) is not threaded in any implementation.
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_elliptic_integral_rf (s21bbc) and
nag_elliptic_integral_rd (s21bcc), respectively.
This example simply generates a small set of nonextreme arguments that are used with the function to produce the table of results.
None.