NAG Library Function Document
nag_elliptic_integral_F (s21bec)
1 Purpose
nag_elliptic_integral_F (s21bec) returns a value of the classical (Legendre) form of the incomplete elliptic integral of the first kind.
2 Specification
| #include <nag.h> |
| #include <nags.h> |
| double |
nag_elliptic_integral_F (double phi,
double dm,
NagError *fail) |
|
3 Description
nag_elliptic_integral_F (s21bec) calculates an approximation to the integral
where
,
and
and
may not both equal one.
The integral is computed using the symmetrised elliptic integrals of Carlson (
Carlson (1979) and
Carlson (1988)). The relevant identity is
where
,
and
is the Carlson symmetrised incomplete elliptic integral of the first kind (see
nag_elliptic_integral_rf (s21bbc)).
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:
phi – doubleInput
- 2:
dm – doubleInput
On entry: the arguments and of the function.
Constraints:
- ;
- ;
- Only one of and dm may be .
Note that is allowable, as long as .
- 3:
fail – NagError *Input/Output
-
The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
- 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.
- NE_REAL
-
On entry, .
Constraint: .
On failure, the function returns zero.
- NE_REAL_2
-
On entry, and ; the integral is undefined.
Constraint: .
On failure, the function returns zero.
- NW_INTEGRAL_INFINITE
-
On entry,
and
; the integral is infinite.
On failure, the function returns the largest machine number (see
nag_real_largest_number (X02ALC)).
7 Accuracy
In principle nag_elliptic_integral_F (s21bec) 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
Not applicable.
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
, see the function document for
nag_elliptic_integral_rf (s21bbc).
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,
and
where
is an integer and
is the complete elliptic integral given by
nag_elliptic_integral_complete_K (s21bhc).
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 (s21bece.c)
10.2 Program Data
None.
10.3 Program Results
Program Results (s21bece.r)