(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: $x$ – Real (Kind=nag_wp) Input
2: $y$ – Real (Kind=nag_wp) Input
3: $z$ – Real (Kind=nag_wp) 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: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $- 1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $- 1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $- 1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: 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.

$ifail = 2$: 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.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

In principle s21bbf 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.