and the recursion process is terminated when this quantity is negligible compared with the machine precision. 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_{J} (x, x, x, x) = x^{- \frac{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
4: $r$ – Real (Kind=nag_wp) Input: On entry: the arguments $x$ , $y$ , $z$ and $ρ$ of the function.

Constraint: $x$ , y, $z \geq 0.0$ , $r \neq 0.0$ and at most one of x, y and z may be zero.
5: $ifail$ – 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: at most one of x, y and z is $0.0$ .
The function is undefined.

On entry, $x = ⟨ value ⟩$ , $y = ⟨ value ⟩$ and $z = ⟨ value ⟩$ .
Constraint: $x \geq 0.0$ and $y \geq 0.0$ and $z \geq 0.0$ .

$ifail = 2$: On entry, $r = 0.0$ .
Constraint: $r \neq 0.0$ .

$ifail = 3$: On entry, $L = ⟨ value ⟩$ , $r = ⟨ value ⟩$ , $x = ⟨ value ⟩$ , $y = ⟨ value ⟩$ and $z = ⟨ value ⟩$ .
Constraint: $| r | \geq L$ and at most one of x, y and z is less than $L$ .

$ifail = 4$: On entry, $U = ⟨ value ⟩$ , $r = ⟨ value ⟩$ , $x = ⟨ value ⟩$ , $y = ⟨ value ⟩$ and $z = ⟨ value ⟩$ .
Constraint: $| r | \leq U$ and $x \leq U$ and $y \leq U$ and $z \leq U$ .

$ifail = - 99$: 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.