nag_elliptic_integral_rd (s21bcc) returns a value of the symmetrised elliptic integral of the second kind.

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 (1978) 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 5th-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.

"NE_REAL_ARG_EQ"

On entry,  must not be equal to 0.0: . Both x and y are zero and the function is undefined.

"NE_REAL_ARG_GE"

On entry, x must not be greater than or equal to : .

"NE_REAL_ARG_LE"

On entry, z must not be less than or equal to 0.0: . The function is undefined.

"NE_REAL_ARG_LT"

On entry, x must not be less than 0.0: .

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.

Symmetrised elliptic integrals returned by functions nag_elliptic_integral_rd (s21bcc), s21bac (nag_elliptic_integral_rc), s21bbc (nag_elliptic_integral_rf) and s21bdc (nag_elliptic_integral_rj) can be related to the more traditional canonical forms (see Abramowitz and Stegun (1972)), as described in the Chapter Introduction.