NAG CL Interface
s21bfc (ellipint_legendre_2)
1
Purpose
s21bfc returns a value of the classical (Legendre) form of the incomplete elliptic integral of the second kind.
2
Specification
| double |
s21bfc (double phi,
double dm,
NagError *fail) |
|
The function may be called by the names: s21bfc, nag_specfun_ellipint_legendre_2 or nag_elliptic_integral_e.
3
Description
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
s21bbc) and
is the Carlson symmetrised incomplete elliptic integral of the second kind (see
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:
– double
Input
-
2:
– double
Input
-
On entry: the arguments and of the function.
Constraints:
- ;
- .
-
3:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
- NE_REAL
-
On entry, .
Constraint: .
- NE_REAL_2
-
On entry, and ; the integral is undefined.
Constraint: .
7
Accuracy
In principle 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
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
s21bbc and
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
10.2
Program Data
None.
10.3
Program Results