The function may be called by the names: s21bfc, nag_specfun_ellipint_legendre_2 or nag_elliptic_integral_e.
3Description
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).
4References
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 Mathematik33 1–16
Carlson B C (1988) A table of elliptic integrals of the third kind Math. Comput.51 267–280
5Arguments
1: – doubleInput
2: – doubleInput
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).
6Error 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: .
7Accuracy
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.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
s21bfc is not threaded in any implementation.
9Further Comments
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 s21bbcands21bcc, 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 .
10Example
This example simply generates a small set of nonextreme arguments that are used with the function to produce the table of results.