The function may be called by the names: s21bac, nag_specfun_ellipint_symm_1_degen or nag_elliptic_integral_rc.
s21bac calculates an approximate value for the integral
where and .
This function, which is related to the logarithm or inverse hyperbolic functions for and to inverse circular functions if , arises as a degenerate form of the elliptic integral of the first kind. If , the result computed is the Cauchy principal value of the integral.
The basic algorithm, which is due to Carlson (1979) and Carlson (1988), is to reduce the arguments recursively towards their mean by the system:
The quantity for decreases with increasing , eventually . For small enough the required function value can be approximated by the first few terms of the Taylor series about the mean. That is
The truncation error involved in using this approximation is bounded by and the recursive process is stopped when is small enough for this truncation error to be negligible compared to the machine precision.
Within the domain of definition, the function value is itself representable for all representable values of its arguments. However, for values of the arguments near the extremes the above algorithm must be modified so as to avoid causing underflows or overflows in intermediate steps. In extreme regions arguments are prescaled away from the extremes and compensating scaling of the result is done before returning to the calling program.
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
On entry: the arguments and of the function, respectively.
3: – 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, . Constraint: . The function is undefined and returns zero.
On entry, . Constraint: . 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
s21bac 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.