﻿ s21bd Method
s21bd returns a value of the symmetrised elliptic integral of the third kind.

# Syntax

C#
```public static double s21bd(
double x,
double y,
double z,
double r,
out int ifail
)```
Visual Basic
```Public Shared Function s21bd ( _
x As Double, _
y As Double, _
z As Double, _
r As Double, _
<OutAttribute> ByRef ifail As Integer _
) As Double```
Visual C++
```public:
static double s21bd(
double x,
double y,
double z,
double r,
[OutAttribute] int% ifail
)```
F#
```static member s21bd :
x : float *
y : float *
z : float *
r : float *
ifail : int byref -> float
```

#### Parameters

x
Type: System..::..Double
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.
y
Type: System..::..Double
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.
z
Type: System..::..Double
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.
r
Type: System..::..Double
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.
ifail
Type: System..::..Int32%
On exit: ${\mathbf{ifail}}={0}$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

#### Return Value

s21bd returns a value of the symmetrised elliptic integral of the third kind.

# Description

s21bd calculates an approximation to the integral
 $RJx,y,z,ρ=32∫0∞dtt+ρt+xt+yt+z$
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:
 $x0=x,y0=y,z0=z,ρ0=ρμn=xn+yn+zn+2ρn/5Xn=1-xn/μnYn=1-yn/μnZn=1-zn/μnPn=1-ρn/μnλn=xnyn+ynzn+znxnxn+1=xn+λn/4yn+1=yn+λn/4zn+1=zn+λn/4ρn+1=ρn+λn/4αn=ρnxn,+yn,+zn+xnynzn2βn=ρnρn+λn2$
For $n$ sufficiently large,
 $εn=maxXn,Yn,Zn,Pn∼14n$
and the function may be approximated by a fifth order power series
 $RJx,y,z,ρ=3∑m=0n-14-mRCαm,βm+4-nμn31+37Sn2+13Sn3+322Sn22+311Sn4+313Sn2Sn3+313Sn5$
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 method 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.

# 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

# Error Indicators and Warnings

Errors or warnings detected by the method:
${\mathbf{ifail}}=1$
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.
${\mathbf{ifail}}=-9000$
An error occured, see message report.

# Accuracy

In principle the method 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.

# Parallelism and Performance

None.

If the parameter r is equal to any of the other arguments, the function reduces to the integral ${R}_{D}$, computed by s21bc.