The routine 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:

R_{D} (x, x, x) = x^{- 3 / 2}

, so there exists a region of extreme arguments for which the function value is not representable.

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: $x$ , $y \geq 0.0$ , $z > 0.0$ and only one of x and y may be zero.
4: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$ . 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, $x = 〈value〉$ and $y = 〈value〉$ .
Constraint: $x \geq 0.0$ and $y \geq 0.0$ .
The function is undefined.

On entry, x and y are both $0.0$ .
Constraint: at most one of x and y is $0.0$ .
The function is undefined.

$ifail = 2$: On entry, $z = 〈value〉$ .
Constraint: $z > 0.0$ .
The function is undefined.

$ifail = 3$: On entry, $L = 〈value〉$ , $x = 〈value〉$ , $y = 〈value〉$ and $z = 〈value〉$ .
Constraint: $z \geq L$ and ( $x \geq L$ or $y \geq L$ ).
The function is undefined.

$ifail = 4$: On entry, $U = 〈value〉$ , $x = 〈value〉$ , $y = 〈value〉$ and $z = 〈value〉$ .
Constraint: $x < U$ and $y < U$ and $z < U$ .
There is a danger of setting underflow and the function 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 the routine 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.