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 $ρ$ of the function.

Constraint: $x$ , y, $z \geq 0.0$ , $r \neq 0.0$ and at most one of x, y and z may be zero.

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

Constraint: $x$ , y, $z \geq 0.0$ , $r \neq 0.0$ and at most one of x, y and z may be zero.

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

Constraint: $x$ , y, $z \geq 0.0$ , $r \neq 0.0$ and at most one of x, y and z may be zero.

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

Constraint: $x$ , y, $z \geq 0.0$ , $r \neq 0.0$ and at most 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

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

Description

s21bd calculates an approximation to the integral

R_{J} (x, y, z, ρ) = \frac{3}{2} \int_{0}^{\infty} \frac{d t}{(t + ρ) \sqrt{(t + x) (t + y) (t + z)}}

where

x

y

z \geq 0

ρ \neq 0

and at most one of

x

y

and

z

is zero.

ρ < 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:

\begin{array}{lcl} x_{0} & = & x, y_{0} = y, z_{0} = z, ρ_{0} = ρ \\ μ_{n} & = & (x_{n} + y_{n} + z_{n} + 2 ρ_{n}) / 5 \\ X_{n} & = & 1 - x_{n} / μ_{n} \\ Y_{n} & = & 1 - y_{n} / μ_{n} \\ Z_{n} & = & 1 - z_{n} / μ_{n} \\ P_{n} & = & 1 - ρ_{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 \\ ρ_{n + 1} & = & (ρ_{n} + λ_{n}) / 4 \\ α_{n} & = & {[ρ_{n} (\sqrt{x_{n}}, + \sqrt{y_{n}}, + \sqrt{z_{n}}) + \sqrt{x_{n} y_{n} z_{n}}]}^{2} \\ β_{n} & = & ρ_{n} {(ρ_{n} + λ_{n})}^{2} \end{array}

For

n

sufficiently large,

ε_{n} = \max (|X_{n}|, |Y_{n}|, |Z_{n}|, |P_{n}|) \sim \frac{1}{4^{n}}

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

\begin{array}{l} R_{J} (x, y, z, ρ) = & 3 \sum_{m = 0}^{n - 1} 4^{- m} R_{C} (α_{m}, β_{m}) \\ + \frac{4^{- n}}{\sqrt{μ_{n}^{3}}} [1 + \frac{3}{7} S_{n}^{(2)} + \frac{1}{3} S_{n}^{(3)} + \frac{3}{22} {(S_{n}^{(2)})}^{2} + \frac{3}{11} S_{n}^{(4)} + \frac{3}{13} S_{n}^{(2)} S_{n}^{(3)} + \frac{3}{13} S_{n}^{(5)}] \end{array}

where

S_{n}^{(m)} = (X_{n}^{m} + Y_{n}^{m} + Z_{n}^{m} + 2 P_{n}^{m}) / 2 m

The truncation error in this expansion is bounded by

3 ε_{n}^{6} / \sqrt{{(1 - ε_{n})}^{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} (x, x, x, x) = 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:

$ifail = 1$: On entry, at least one of x, y and z is negative, or at least two of them are zero; the function is undefined.

$ifail = 2$: $r = 0.0$ ; the function is undefined.

$ifail = 3$: 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.

$ifail = 4$: 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.

$ifail = -9000$: 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.

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 the parameter r is equal to any of the other arguments, the function reduces to the integral

R_{D}

, computed by s21bc.

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#): s21bde.cs

Example program results: s21bde.r