NAG Library Function Document

nag_elliptic_integral_rc (s21bac)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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

nag_elliptic_integral_rc (s21bac) returns a value of an elementary integral, which occurs as a degenerate case of an elliptic integral of the first kind,

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_elliptic_integral_rc (double x, double y, NagError *fail)

3 Description

nag_elliptic_integral_rc (s21bac) calculates an approximate value for the integral

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

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:

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

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 pre-scaled 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 Arguments

1: x – doubleInput
2: y – doubleInput: On entry: the arguments $x$ and $y$ of the function, respectively.
Constraint: $x \geq 0.0$ and $y \neq 0.0$ .
3: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_REAL_ARG_EQ: On entry, y must not be equal to 0.0: $y = 〈value〉$ .
The function is undefined and returns zero.
NE_REAL_ARG_LT: On entry, x must not be less than 0.0: $x = 〈value〉$ .
The function is undefined.

7 Accuracy

In principle the function 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.

8 Further Comments

Symmetrised elliptic integrals returned by functions nag_elliptic_integral_rc (s21bac), nag_elliptic_integral_rf (s21bbc), nag_elliptic_integral_rd (s21bcc) and nag_elliptic_integral_rj (s21bdc) can be related to the more traditional canonical forms (see Abramowitz and Stegun (1972)), as described in the s Chapter Introduction.

9 Example

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