This function, which is related to the logarithm or inverse hyperbolic functions for $y<x$ and to inverse circular functions if $x<y$, arises as a degenerate form of the elliptic integral of the first kind. If $y<0$, the result computed is the Cauchy principal value of the integral.
The basic algorithm, which is due to Carlson (1979) and Carlson (1988), is to reduce the arguments recursively towards their mean by the system:
The quantity $\left|{S}_{n}\right|$ for $n=0,1,2,3,\dots \text{}$ decreases with increasing $n$, eventually $\left|{S}_{n}\right|\sim 1/{4}^{n}$. For small enough ${S}_{n}$ the required function value can be approximated by the first few terms of the Taylor series about the mean. That is
The truncation error involved in using this approximation is bounded by $16{\left|{S}_{n}\right|}^{6}/(1-2\left|{S}_{n}\right|)$ and the recursive process is stopped when ${S}_{n}$ is small enough for this truncation error to be negligible compared to 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.
Carlson B C (1979) Computing elliptic integrals by duplication Numerische Mathematik33 1–16
Carlson B C (1988) A table of elliptic integrals of the third kind Math. Comput.51 267–280
5Arguments
1: $\mathbf{x}$ – Real (Kind=nag_wp)Input
2: $\mathbf{y}$ – Real (Kind=nag_wp)Input
On entry: the arguments $x$ and $y$ of the function, respectively.
Constraint:
${\mathbf{x}}\ge 0.0$ and ${\mathbf{y}}\ne 0.0$.
3: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ 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 $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: on soft failure the routine returns zero.
${\mathbf{ifail}}=1$
On entry, ${\mathbf{x}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{x}}\ge 0.0$. The function is undefined.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{y}}=0.0$. Constraint: ${\mathbf{y}}\ne 0.0$. The function is undefined and returns zero.
${\mathbf{ifail}}=-99$
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.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
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.
8Parallelism and Performance
s21baf is not threaded in any implementation.
9Further Comments
You should consult the S Chapter Introduction which shows the relationship of this function to the classical definitions of the elliptic integrals.
10Example
This example simply generates a small set of nonextreme arguments which are used with the routine to produce the table of low accuracy results.