(This ordering, which is possible because of the symmetry of the function, is done for technical reasons related to the avoidance of overflow and underflow.)

\begin{array}{lcl} μ_{n} & = & (x_{n} + y_{n} + z_{n}) / 3 \\ X_{n} & = & (1 - x_{n}) / μ_{n} \\ Y_{n} & = & (1 - y_{n}) / μ_{n} \\ Z_{n} & = & (1 - z_{n}) / μ_{n} \\ λ_{n} & = & \sqrt{x_{n} y_{n}} + \sqrt{y_{n} z_{n}} + \sqrt{z_{n} x_{n}} \\ x_{n + 1} & = & (x_{n} + λ_{n}) / 4 \\ y_{n + 1} & = & (y_{n} + λ_{n}) / 4 \\ z_{n + 1} & = & (z_{n} + λ_{n}) / 4 \end{array}

ε_{n} = \max (| X_{n} |, | Y_{n} |, | Z_{n} |)

and the function may be approximated adequately by a fifth order power series:

R_{F} (x, y, z) = \frac{1}{\sqrt{μ_{n}}} (1 - \frac{E_{2}}{10} + \frac{E_{2}^{2}}{24} - \frac{3 E_{2} E_{3}}{44} + \frac{E_{3}}{14})

where

E_{2} = X_{n} Y_{n} + Y_{n} Z_{n} + Z_{n} X_{n}

E_{3} = X_{n} Y_{n} Z_{n}

The truncation error involved in using this approximation is bounded by

ε_{n}^{6} / 4 (1 - ε_{n})

and the recursive process is stopped when this truncation error is negligible compared with 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.

4 References

NIST Digital Library of Mathematical Functions

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: $x$ – double Input
2: $y$ – double Input
3: $z$ – double Input: On entry: the arguments $x$ , $y$ and $z$ of the function.

Constraint: $x$ , y, $z \geq 0.0$ and only one of x, y and z may be zero.
4: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error 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_ARG_EQ: On entry, $x = ⟨ value ⟩$ , $y = ⟨ value ⟩$ and $z = ⟨ value ⟩$ .
Constraint: at most one of x, y and z is $0.0$ .
The function is undefined and returns zero.
NE_REAL_ARG_LT: On entry, $x = ⟨ value ⟩$ , $y = ⟨ value ⟩$ and $z = ⟨ value ⟩$ .
Constraint: $x \geq 0.0$ and $y \geq 0.0$ and $z \geq 0.0$ .
The function is undefined.

7 Accuracy

In principle s21bbc 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.