NAG C Library Function Document
nag_elliptic_integral_E (s21bfc)
1
Purpose
nag_elliptic_integral_E (s21bfc) returns a value of the classical (Legendre) form of the incomplete elliptic integral of the second kind.
2
Specification
| #include <nag.h> |
| #include <nags.h> |
| double |
nag_elliptic_integral_E (double phi,
double dm,
NagError *fail) |
|
3
Description
nag_elliptic_integral_E (s21bfc) calculates an approximation to the integral
where
and
.
The integral is computed using the symmetrised elliptic integrals of Carlson (
Carlson (1979) and
Carlson (1988)). The relevant identity is
where
,
,
is the Carlson symmetrised incomplete elliptic integral of the first kind (see
nag_elliptic_integral_rf (s21bbc)) and
is the Carlson symmetrised incomplete elliptic integral of the second kind (see
nag_elliptic_integral_rd (s21bcc)).
4
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
5
Arguments
- 1:
– doubleInput
- 2:
– doubleInput
-
On entry: the arguments and of the function.
Constraints:
- ;
- .
- 3:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_INTERNAL_ERROR
-
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 for assistance.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- NE_REAL
-
On entry, .
Constraint: .
- NE_REAL_2
-
On entry, and ; the integral is undefined.
Constraint: .
7
Accuracy
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.
8
Parallelism and Performance
nag_elliptic_integral_E (s21bfc) 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
and
, see the function documents for
nag_elliptic_integral_rf (s21bbc) and
nag_elliptic_integral_rd (s21bcc), 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,
. A parameter
can be replaced by one less than unity using
.
10
Example
This example simply generates a small set of nonextreme arguments that are used with the function to produce the table of results.
10.1
Program Text
Program Text (s21bfce.c)
10.2
Program Data
None.
10.3
Program Results
Program Results (s21bfce.r)