NAG Library Manual, Mark 27

Interfaces: FL CL AD

NAG CL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s21ba: FL CL AD

NAG CL Interface
s21bac (ellipint_symm_1_degen)

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NAG Library Manual, Mark 27

Interfaces: FL CL AD

NAG CL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s21ba: FL CL AD

▸▿ Contents

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

© The Numerical Algorithms Group Ltd. 2019

1 Purpose

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>

double

s21bac (double x, double y, NagError *fail)

The function may be called by the names: s21bac, nag_specfun_ellipint_symm_1_degen or nag_elliptic_integral_rc.

3 Description

s21bac 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

NIST Digital Library of Mathematical Functions

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: $x$ – double Input
2: $y$ – double Input: 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 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL_ARG_EQ: On entry, $y = 0.0$ .
Constraint: $y \neq 0.0$ .
The function is undefined and returns zero.
NE_REAL_ARG_LT: On entry, $x = 〈value〉$ .
Constraint: $x \geq 0.0$ .
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 Parallelism and Performance

s21bac is not threaded in any implementation.

9 Further Comments

You should consult the S Chapter Introduction which shows the relationship of this function to the classical definitions of the elliptic integrals.

10 Example

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

10.1 Program Text

Program Text (s21bace.c)

10.2 Program Data

None.

10.3 Program Results

Program Results (s21bace.r)

NAG Library Manual, Mark 27

Interfaces: FL CL AD

NAG CL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s21ba: FL CL AD

© The Numerical Algorithms Group Ltd. 2019