# NAG Library Routine Document

## 1Purpose

s21bcf returns a value of the symmetrised elliptic integral of the second kind, via the function name.

## 2Specification

Fortran Interface
 Function s21bcf ( x, y, z,
 Real (Kind=nag_wp) :: s21bcf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x, y, z
#include <nagmk26.h>
 double s21bcf_ (const double *x, const double *y, const double *z, Integer *ifail)

## 3Description

s21bcf calculates an approximate value for the integral
 $RDx,y,z=32∫0∞dt t+xt+y t+z 3$
where $x$, $y\ge 0$, at most one of $x$ and $y$ is zero, and $z>0$.
The basic algorithm, which is due to Carlson (1979) and Carlson (1988), is to reduce the arguments recursively towards their mean by the rule:
 $x0 = x,y0=y,z0=z μn = xn+yn+3zn/5 Xn = 1-xn/μn Yn = 1-yn/μn Zn = 1-zn/μn λn = xnyn+ynzn+znxn xn+1 = xn+λn/4 yn+1 = yn+λn/4 zn+1 = zn+λn/4$
For $n$ sufficiently large,
 $εn=maxXn,Yn,Zn∼ 14 n$
and the function may be approximated adequately by a fifth order power series
 $RDx,y,z= 3∑m= 0 n- 1 4-mzm+λnzm + 4-nμn3 1+ 37Sn 2 + 13Sn 3 + 322Sn 2 2+ 311Sn 4 + 313Sn 2 Sn 3 + 313Sn 5$
where ${S}_{n}^{\left(m\right)}=\left({X}_{n}^{m}+{Y}_{n}^{m}+3{Z}_{n}^{m}\right)/2m\text{.}$ The truncation error in this expansion is bounded by $\frac{3{\epsilon }_{n}^{6}}{\sqrt{{\left(1-{\epsilon }_{n}\right)}^{3}}}$ and the recursive 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:  ${R}_{D}\left(x,x,x\right)={x}^{-3/2}$, so there exists a region of extreme arguments for which the function value is not representable.

## 4References

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

## 5Arguments

1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
2:     $\mathbf{y}$ – Real (Kind=nag_wp)Input
3:     $\mathbf{z}$ – Real (Kind=nag_wp)Input
On entry: the arguments $x$, $y$ and $z$ of the function.
Constraint: ${\mathbf{x}}$, ${\mathbf{y}}\ge 0.0$, ${\mathbf{z}}>0.0$ and only one of x and y may be zero.
4:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . 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  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  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 $-1$, 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, ${\mathbf{x}}=〈\mathit{\text{value}}〉$ and ${\mathbf{y}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\ge 0.0$ and ${\mathbf{y}}\ge 0.0$.
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.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{z}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{z}}>0.0$.
The function is undefined.
${\mathbf{ifail}}=3$
On entry, $L=〈\mathit{\text{value}}〉$, ${\mathbf{x}}=〈\mathit{\text{value}}〉$, ${\mathbf{y}}=〈\mathit{\text{value}}〉$ and ${\mathbf{z}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{z}}\ge L$ and (${\mathbf{x}}\ge L$ or ${\mathbf{y}}\ge L$).
The function is undefined.
${\mathbf{ifail}}=4$
On entry, $U=〈\mathit{\text{value}}〉$, ${\mathbf{x}}=〈\mathit{\text{value}}〉$, ${\mathbf{y}}=〈\mathit{\text{value}}〉$ and ${\mathbf{z}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}} and ${\mathbf{y}} and ${\mathbf{z}}.
There is a danger of setting underflow and the function returns zero.
${\mathbf{ifail}}=-99$
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.

## 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

s21bcf 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.

## 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.

### 10.1Program Text

Program Text (s21bcfe.f90)

None.

### 10.3Program Results

Program Results (s21bcfe.r)