nag_elliptic_integral_rd (s21bcc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_elliptic_integral_rd (s21bcc)


    1  Purpose
    7  Accuracy

1  Purpose

nag_elliptic_integral_rd (s21bcc) returns a value of the symmetrised elliptic integral of the second kind.

2  Specification

#include <nag.h>
#include <nags.h>
double  nag_elliptic_integral_rd (double x, double y, double z, NagError *fail)

3  Description

nag_elliptic_integral_rd (s21bcc) calculates an approximate value for the integral
RDx,y,z=320dt t+xt+y t+z 3  
where x, y0, at most one of x and y is zero, and z>0.
The basic algorithm, which is due to Carlson (1979) and Carlson (1988), is to reduce the arguments recursively towards their mean by the rule:
x0 = x,y0=y,z0=z μn = xn+yn+3zn/5 Xn = 1-xn/μn Yn = 1-yn/μn Zn = 1-zn/μn λn = xnyn+ynzn+znxn xn+1 = xn+λn/4 yn+1 = yn+λn/4 zn+1 = zn+λn/4  
For n sufficiently large,
εn=maxXn,Yn,Zn 14 n  
and the function may be approximated adequately by a fifth order power series
RDx,y,z= 3m= 0 n- 1 4-mzm+λnzm + 4-nμn3 1+ 37Sn 2 + 13Sn 3 + 322Sn 2 2+ 311Sn 4 + 313Sn 2 Sn 3 + 313Sn 5  
where Sn m =Xnm+Ynm+3Znm /2m. The truncation error in this expansion is bounded by 3εn6 1-εn 3  and the recursive process is terminated when this quantity is negligible compared with the machine precision.
The function 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:  RDx,x,x=x-3/2, so there exists a region of extreme arguments for which the function value is not representable.

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
3:     z doubleInput
On entry: the arguments x, y and z of the function.
Constraint: x, y0.0, z>0.0 and only one of x and y may be zero.
4:     fail NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
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.
An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
On entry, x and y are both 0.0.
Constraint: at most one of x and y is 0.0.
The function is undefined.
On entry, U=value, x=value, y=value and z=value.
Constraint: x<U and y<U and z<U.
There is a danger of setting underflow and the function returns zero.
On entry, z=value.
Constraint: z>0.0.
The function is undefined.
On entry, L=value, x=value, y=value and z=value.
Constraint: zL and (xL or yL).
The function is undefined.
On entry, x=value and y=value.
Constraint: x0.0 and y0.0.
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

nag_elliptic_integral_rd (s21bcc) 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 (s21bcce.c)

10.2  Program Data


10.3  Program Results

Program Results (s21bcce.r)

nag_elliptic_integral_rd (s21bcc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016