s21ba returns a value of an elementary integral, which occurs as a degenerate case of an elliptic integral of the first kind.

Syntax

C#
public static double s21ba( double x, double y, out int ifail )

Visual Basic
Public Shared Function s21ba ( _ x As Double, _ y As Double, _ <OutAttribute> ByRef ifail As Integer _ ) As Double

Visual C++
public: static double s21ba( double x, double y, [OutAttribute] int% ifail )

F#
static member s21ba : x : float * y : float * ifail : int byref -> float

Parameters

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

Constraint: $x \geq 0.0$ and $y \neq 0.0$ .

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

Constraint: $x \geq 0.0$ and $y \neq 0.0$ .

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

s21ba returns a value of an elementary integral, which occurs as a degenerate case of an elliptic integral of the first kind.

Description

s21ba calculates an approximate value for the integral

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

where

x \geq 0

and

y \neq 0

This function, which is related to the logarithm or inverse hyperbolic functions for

y < x

and to inverse circular functions if

x < y

, arises as a degenerate form of the elliptic integral of the first kind. If

y < 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 system:

\begin{matrix} x_{0} = x & y_{0} = y \\ μ_{n} = (x_{n} + 2 y_{n}) / 3, & S_{n} = (y_{n} - x_{n}) / 3 μ_{n} \\ λ_{n} = y_{n} + 2 \sqrt{x_{n} y_{n}} \\ x_{n + 1} = (x_{n} + λ_{n}) / 4, & y_{n + 1} = (y_{n} + λ_{n}) / 4 . \end{matrix}

The quantity

|S_{n}|

for

n = 0, 1, 2, 3, \dots

decreases with increasing

n

, eventually

|S_{n}| \sim 1 / 4^{n}

. For small enough

S_{n}

the required function value can be approximated by the first few terms of the Taylor series about the mean. That is

R_{C} (x, y) = (1 + \frac{3 S_{n}^{2}}{10} + \frac{S_{n}^{3}}{7} + \frac{3 S_{n}^{4}}{8} + \frac{9 S_{n}^{5}}{22}) / \sqrt{μ_{n}} .

The truncation error involved in using this approximation is bounded by

16 {|S_{n}|}^{6} / (1 - 2 |S_{n}|)

and the recursive process is stopped when

S_{n}

is small enough for this truncation error to be negligible compared to 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,

x < 0.0

; the function is undefined.

$ifail = 2$

On entry,

y = 0.0

; the function is undefined.

On failure the method returns zero.

$ifail = -9000$: 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.