NAG Library Routine Document

s21bgf  (ellipint_legendre_3)


    1  Purpose
    7  Accuracy


s21bgf returns a value of the classical (Legendre) form of the incomplete elliptic integral of the third kind, via the function name.


Fortran Interface
Function s21bgf ( dn, phi, dm, ifail)
Real (Kind=nag_wp):: s21bgf
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (In):: dn, phi, dm
C Header Interface
#include nagmk26.h
double  s21bgf_ ( const double *dn, const double *phi, const double *dm, Integer *ifail)


s21bgf 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 s21bbf) and RJ  is the Carlson symmetrised incomplete elliptic integral of the third kind (see s21bdf).


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


1:     dn – Real (Kind=nag_wp)Input
2:     phi – Real (Kind=nag_wp)Input
3:     dm – Real (Kind=nag_wp)Input
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:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1​ or ​1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, phi=value.
Constraint: 0phiπ/2.
On entry, phi=value and dm=value; the integral is undefined.
Constraint: dm×sin2phi1.0.
On entry, sinphi=1 and dm=1.0; the integral is infinite.
On entry, phi=value and dn=value; the integral is infinite.
Constraint: dn×sin2phi1.0.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 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 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


In principle s21bgf 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.

Parallelism and Performance

s21bgf is not threaded in any implementation.

Further Comments

You should consult the S Chapter Introduction, which shows the relationship between this routine 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 routine documents for s21bbf and s21bdf, respectively.
If you wish to input a value of phi outside the range allowed by this routine you should refer to Section 17.4 of Abramowitz and Stegun (1972) for useful identities.


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

Program Text

Program Text (s21bgfe.f90)

Program Data


Program Results

Program Results (s21bgfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017