s21bb returns a value of the symmetrised elliptic integral of the first kind.

Syntax

C#
public static double s21bb( double x, double y, double z, out int ifail )

Visual Basic
Public Shared Function s21bb ( _ x As Double, _ y As Double, _ z As Double, _ <OutAttribute> ByRef ifail As Integer _ ) As Double

Visual C++
public: static double s21bb( double x, double y, double z, [OutAttribute] int% ifail )

F#
static member s21bb : x : float * y : float * z : float * ifail : int byref -> float

Parameters

x: Type: System..::..Double
On entry: the arguments $x$ , $y$ and $z$ of the function.

Constraint: $x$ , y, $z \geq 0.0$ and only one of x, y and z may be zero.

y: Type: System..::..Double
On entry: the arguments $x$ , $y$ and $z$ of the function.

Constraint: $x$ , y, $z \geq 0.0$ and only one of x, y and z may be zero.

z: Type: System..::..Double
On entry: the arguments $x$ , $y$ and $z$ of the function.

Constraint: $x$ , y, $z \geq 0.0$ and only one of x, y and z may be zero.

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Return Value

s21bb returns a value of the symmetrised elliptic integral of the first kind.

Description

s21bb calculates an approximation to the integral

R_{F} (x, y, z) = \frac{1}{2} \int_{0}^{\infty} \frac{d t}{\sqrt{(t + x) (t + y) (t + z)}}

where

x

y

z \geq 0

and at most one is zero.

The basic algorithm, which is due to Carlson (1979) and Carlson (1988), is to reduce the arguments recursively towards their mean by the rule:

$x_{0} = \min (x, y, z)$ , $z_{0} = \max (x, y, z)$ ,
$y_{0} =$ remaining third intermediate value argument.

(This ordering, which is possible because of the symmetry of the function, is done for technical reasons related to the avoidance of overflow and underflow.)

\begin{array}{lcl} μ_{n} & = & (x_{n} + y_{n} + z_{n}) / 3 \\ X_{n} & = & (1 - x_{n}) / μ_{n} \\ Y_{n} & = & (1 - y_{n}) / μ_{n} \\ Z_{n} & = & (1 - z_{n}) / μ_{n} \\ λ_{n} & = & \sqrt{x_{n} y_{n}} + \sqrt{y_{n} z_{n}} + \sqrt{z_{n} x_{n}} \\ x_{n + 1} & = & (x_{n} + λ_{n}) / 4 \\ y_{n + 1} & = & (y_{n} + λ_{n}) / 4 \\ z_{n + 1} & = & (z_{n} + λ_{n}) / 4 \end{array}

ε_{n} = \max (|X_{n}|, |Y_{n}|, |Z_{n}|)

and the function may be approximated adequately by a fifth order power series:

R_{F} (x, y, z) = \frac{1}{\sqrt{μ_{n}}} (1 - \frac{E_{2}}{10} + \frac{E_{2}^{2}}{24} - \frac{3 E_{2} E_{3}}{44} + \frac{E_{3}}{14})

where

E_{2} = X_{n} Y_{n} + Y_{n} Z_{n} + Z_{n} X_{n}

E_{3} = X_{n} Y_{n} Z_{n}

The truncation error involved in using this approximation is bounded by

ε_{n}^{6} / 4 (1 - ε_{n})

and the recursive process is stopped when this truncation error is negligible compared with 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.

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:

$ifail = 1$: On entry, one or more of x, y and z is negative; the function is undefined.

$ifail = 2$: On entry, two or more of x, y and z are zero; the function is undefined. On failure, the method returns zero.

$ifail = -9000$: An error occured, see message report.

Accuracy

In principle s21bb 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.

Further Comments

You should consult (S not in this release) which shows the relationship of this function to the classical definitions of the elliptic integrals.

If two arguments are equal, the function reduces to the elementary integral

R_{C}

, computed by s21ba.

Example

This example simply generates a small set of nonextreme arguments which are used with the method to produce the table of low accuracy results.

Example program (C#): s21bbe.cs

Example program results: s21bbe.r