NAG CL Interface
s21bbc (ellipint_​symm_​1)

Settings help

CL Name Style:

1 Purpose

s21bbc returns a value of the symmetrised elliptic integral of the first kind.

2 Specification

#include <nag.h>
double  s21bbc (double x, double y, double z, NagError *fail)
The function may be called by the names: s21bbc, nag_specfun_ellipint_symm_1 or nag_elliptic_integral_rf.

3 Description

s21bbc calculates an approximation to the integral
RF(x,y,z)=120dt (t+x)(t+y)(t+z)  
where x, y, z0 and at most one is zero.
The basic algorithm, which is due to Carlson (1979) and Carlson (1988), is to reduce the arguments recursively towards their mean by the rule: (This ordering, which is possible because of the symmetry of the function, is done for technical reasons related to the avoidance of overflow and underflow.)
μn = (xn+yn+zn)/3 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  
εn=max(|Xn|,|Yn|,|Zn|) and the function may be approximated adequately by a fifth order power series:
RF(x,y,z)=1μn (1-E210+E2224-3E2E344+E314)  
where E2=XnYn+YnZn+ZnXn, E3=XnYnZn.
The truncation error involved in using this approximation is bounded by εn6/4(1-εn) and the recursive process is stopped when this truncation error is negligible compared with 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
3: z double Input
On entry: the arguments x, y and z of the function.
Constraint: x, y, z0.0 and only one of x, y and z may be zero.
4: 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

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
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.
On entry, x=value, y=value and z=value.
Constraint: at most one of x, y and z is 0.0.
The function is undefined and returns zero.
On entry, x=value, y=value and z=value.
Constraint: x0.0 and y0.0 and z0.0.
The function is undefined.

7 Accuracy

In principle s21bbc 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

Background information to multithreading can be found in the Multithreading documentation.
s21bbc 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.
If two arguments are equal, the function reduces to the elementary integral RC, computed by s21bac.

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 (s21bbce.c)

10.2 Program Data


10.3 Program Results

Program Results (s21bbce.r)