nag_elliptic_integral_pi (s21bgc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_elliptic_integral_pi (s21bgc)


    1  Purpose
    7  Accuracy

1  Purpose

nag_elliptic_integral_pi (s21bgc) returns a value of the classical (Legendre) form of the incomplete elliptic integral of the third kind.

2  Specification

#include <nag.h>
#include <nags.h>
double  nag_elliptic_integral_pi (double dn, double phi, double dm, NagError *fail)

3  Description

nag_elliptic_integral_pi (s21bgc) calculates an approximation to the integral
Π n;ϕm = 0ϕ 1-n sin2θ -1 1-m sin2θ -12 dθ ,  
where 0ϕ π2 , msin2ϕ1 , m  and sinϕ  may not both equal one, and nsin2ϕ1 .
The integral is computed using the symmetrised elliptic integrals of Carlson (Carlson (1979) and Carlson (1988)). The relevant identity is
Π n;ϕm = sinϕ RF q,r,1 + 13 n sin3ϕ RJ q,r,1,s ,  
where q=cos2ϕ , r=1-m sin2ϕ , s=1-n sin2ϕ , RF  is the Carlson symmetrised incomplete elliptic integral of the first kind (see nag_elliptic_integral_rf (s21bbc)) and RJ  is the Carlson symmetrised incomplete elliptic integral of the third kind (see nag_elliptic_integral_rj (s21bdc)).

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:     dn doubleInput
2:     phi doubleInput
3:     dm doubleInput
On entry: the arguments n, ϕ and m of the function.
  • 0.0phi π2;
  • dm× sin2phi 1.0 ;
  • Only one of sinphi  and dm may be 1.0;
  • dn× sin2phi 1.0 .
Note that dm × sin2phi = 1.0  is allowable, as long as dm1.0 .
4:     fail NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in the Essential Introduction 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 3.6.6 in the Essential Introduction for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
On entry, phi=value.
Constraint: 0phiπ/2.
On entry, phi=value and dm=value; the integral is undefined.
Constraint: dm×sin2phi1.0.
On entry, phi=value and dn=value; the integral is infinite.
Constraint: dn×sin2phi1.0.
On entry, sinphi=1 and dm=1.0; the integral is infinite.

7  Accuracy

In principle nag_elliptic_integral_pi (s21bgc) 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

Not applicable.

9  Further Comments

You should consult the s Chapter Introduction, which shows the relationship between this function and the Carlson definitions of the elliptic integrals. In particular, the relationship between the argument-constraints for both forms becomes clear.
For more information on the algorithms used to compute RF  and RJ , see the function documents for nag_elliptic_integral_rf (s21bbc) and nag_elliptic_integral_rj (s21bdc), respectively.
If you wish to input a value of phi outside the range allowed by this function you should refer to Section 17.4 of Abramowitz and Stegun (1972) for useful identities.

10  Example

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

10.1  Program Text

Program Text (s21bgce.c)

10.2  Program Data


10.3  Program Results

Program Results (s21bgce.r)

nag_elliptic_integral_pi (s21bgc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2015