The function may be called by the names: s21bcc, nag_specfun_ellipint_symm_2 or nag_elliptic_integral_rd.
s21bcc calculates an approximate value for the integral
where , , at most one of and is zero, and .
The basic algorithm, which is due to Carlson (1979) and Carlson (1988), is to reduce the arguments recursively towards their mean by the rule:
For sufficiently large,
and the function may be approximated adequately by a fifth order power series
where The truncation error in this expansion is bounded by and the recursive process is terminated when this quantity is negligible compared with the machine precision.
The function may fail either because it has been called with arguments outside the domain of definition, or with arguments so extreme that there is an unavoidable danger of setting underflow or overflow.
Note: , so there exists a region of extreme arguments for which the function value is not representable.
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
1: – doubleInput
2: – doubleInput
3: – doubleInput
On entry: the arguments , and of the function.
, , and only one of x and y may be zero.
4: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
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.
On entry, x and y are both . Constraint: at most one of x and y is . The function is undefined.
On entry, , , and . Constraint: and and . There is a danger of setting underflow and the function returns zero.
On entry, . Constraint: . The function is undefined.
On entry, , , and . Constraint: and ( or ). The function is undefined.
On entry, and . Constraint: and . The function is undefined.
In principle the function 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
s21bcc is not threaded in any implementation.
You should consult the S Chapter Introduction which shows the relationship of this function to the classical definitions of the elliptic integrals.
This example simply generates a small set of nonextreme arguments which are used with the function to produce the table of low accuracy results.