NAG Library Routine Document

S21BAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

S21BAF returns a value of an elementary integral, which occurs as a degenerate case of an elliptic integral of the first kind, via the function name.

2 Specification

FUNCTION S21BAF (

X, Y, IFAIL)

REAL (KIND=nag_wp) S21BAF

INTEGER	IFAIL
REAL (KIND=nag_wp)	X, Y

3 Description

S21BAF 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.

4 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

5 Parameters

1: X – REAL (KIND=nag_wp)Input
2: Y – REAL (KIND=nag_wp)Input: On entry: the arguments $x$ and $y$ of the function, respectively.
Constraint: $X \geq 0.0$ and $Y \neq 0.0$ .
3: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit: $IFAIL = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,

X < 0.0

; the function is undefined.

$IFAIL = 2$

On entry,

Y = 0.0

; the function is undefined.

On soft failure the routine returns zero.

7 Accuracy

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.