NAG Library Routine Document
s21bdf (ellipint_symm_3)
1
Purpose
s21bdf returns a value of the symmetrised elliptic integral of the third kind, via the function name.
2
Specification
Fortran Interface
Real (Kind=nag_wp)  ::  s21bdf  Integer, Intent (Inout)  ::  ifail  Real (Kind=nag_wp), Intent (In)  ::  x, y, z, r 

C Header Interface
#include nagmk26.h
double 
s21bdf_ (const double *x, const double *y, const double *z, const double *r, Integer *ifail) 

3
Description
s21bdf calculates an approximation to the integral
where
$x$,
$y$,
$z\ge 0$,
$\rho \ne 0$ and at most one of
$x$,
$y$ and
$z$ is zero.
If $\rho <0$, 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 rule:
For
$n$ sufficiently large,
and the function may be approximated by a fifth order power series
where
${S}_{n}^{\left(m\right)}=\left({X}_{n}^{m}+{Y}_{n}^{m}+{Z}_{n}^{m}+2{P}_{n}^{m}\right)/2m$.
The truncation error in this expansion is bounded by $3{\epsilon}_{n}^{6}/\sqrt{{\left(1{\epsilon}_{n}\right)}^{3}}$ 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: ${R}_{J}\left(x,x,x,x\right)={x}^{\frac{3}{2}}$, so there exists a region of extreme arguments for which the function value is not representable.
4
References
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
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: $\mathbf{x}$ – Real (Kind=nag_wp)Input
 2: $\mathbf{y}$ – Real (Kind=nag_wp)Input
 3: $\mathbf{z}$ – Real (Kind=nag_wp)Input
 4: $\mathbf{r}$ – Real (Kind=nag_wp)Input

On entry: the arguments $x$, $y$, $z$ and $\rho $ of the function.
Constraint:
${\mathbf{x}}$,
y,
${\mathbf{z}}\ge 0.0$,
${\mathbf{r}}\ne 0.0$ and at most one of
x,
y and
z may be zero.
 5: $\mathbf{ifail}$ – IntegerInput/Output

On entry:
ifail must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ 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 $\mathbf{1}\text{ or}\mathbf{1}$ 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).
6
Error Indicators and Warnings
If on entry
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
 ${\mathbf{ifail}}=1$

On entry, at least one of
x,
y and
z is negative, or at least two of them are zero; the function is undefined.
 ${\mathbf{ifail}}=2$

${\mathbf{r}}=0.0$; the function is undefined.
 ${\mathbf{ifail}}=3$

On entry, either
r is too close to zero, or any two of
x,
y and
z are too close to zero; there is a danger of setting overflow. See also the
Users' Note for your implementation.
 ${\mathbf{ifail}}=4$

On entry, at least one of
x,
y,
z and
r is too large; there is a danger of setting underflow. See also the
Users' Note for your implementation.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.9 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.8 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
7
Accuracy
In principle the routine is capable of producing full machine precision. However roundoff 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 roundoff error. It is reasonable to assume that the result is accurate to within a small multiple of the machine precision.
8
Parallelism and Performance
s21bdf is not threaded in any implementation.
You should consult the
S Chapter Introduction which shows the relationship of this function to the classical definitions of the elliptic integrals.
If the argument
r is equal to any of the other arguments, the function reduces to the integral
${R}_{D}$, computed by
s21bcf.
10
Example
This example simply generates a small set of nonextreme arguments which are used with the routine to produce the table of low accuracy results.
10.1
Program Text
Program Text (s21bdfe.f90)
10.2
Program Data
None.
10.3
Program Results
Program Results (s21bdfe.r)