NAG Library Function Document
nag_elliptic_integral_E (s21bfc) returns a value of the classical (Legendre) form of the incomplete elliptic integral of the second kind.
||nag_elliptic_integral_E (double phi,
nag_elliptic_integral_E (s21bfc) calculates an approximation to the integral
The integral is computed using the symmetrised elliptic integrals of Carlson (Carlson (1979)
and Carlson (1988)
). The relevant identity is
is the Carlson symmetrised incomplete elliptic integral of the first kind (see nag_elliptic_integral_rf (s21bbc)
is the Carlson symmetrised incomplete elliptic integral of the second kind (see nag_elliptic_integral_rd (s21bcc)
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
phi – doubleInput
dm – doubleInput
On entry: the arguments and of the function.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG
On entry, .
On entry, and ;
the integral is undefined.
In principle nag_elliptic_integral_E (s21bfc) 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.
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
, see the function documents for nag_elliptic_integral_rf (s21bbc)
and nag_elliptic_integral_rd (s21bcc)
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,
. A parameter
can be replaced by one less than unity using
This example simply generates a small set of nonextreme arguments that are used with the function to produce the table of results.
9.1 Program Text
Program Text (s21bfce.c)
9.2 Program Data
9.3 Program Results
Program Results (s21bfce.r)