NAG Library Routine Document

s21bff (ellipint_legendre_2)


    1  Purpose
    7  Accuracy


s21bff returns a value of the classical (Legendre) form of the incomplete elliptic integral of the second kind, via the function name.


Fortran Interface
Function s21bff ( phi, dm, ifail)
Real (Kind=nag_wp):: s21bff
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (In):: phi, dm
C Header Interface
#include nagmk26.h
double  s21bff_ (const double *phi, const double *dm, Integer *ifail)


s21bff calculates an approximation to the integral
Eϕm = 0ϕ 1-m sin2θ 12 dθ ,  
where 0ϕ π2  and msin2ϕ1 .
The integral is computed using the symmetrised elliptic integrals of Carlson (Carlson (1979) and Carlson (1988)). The relevant identity is
Eϕm = sinϕ RF q,r,1 - 13 m sin3ϕ RD q,r,1 ,  
where q=cos2ϕ , r=1-m sin2ϕ , RF  is the Carlson symmetrised incomplete elliptic integral of the first kind (see s21bbf) and RD  is the Carlson symmetrised incomplete elliptic integral of the second kind (see s21bcf).


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:     phi – Real (Kind=nag_wp)Input
2:     dm – Real (Kind=nag_wp)Input
On entry: the arguments ϕ and m of the function.
  • 0.0phi π2;
  • dm× sin2phi 1.0 .
3:     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: 0phiπ2.
On entry, phi=value and dm=value; the integral is undefined.
Constraint: dm×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 s21bff 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

s21bff 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 RD , see the routine documents for s21bbf and s21bcf, 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. For example, E-ϕ|m=-Eϕ|m. A parameter m>1 can be replaced by one less than unity using Eϕ|m=mEϕm|1m-m-1ϕ.


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 (s21bffe.f90)

Program Data


Program Results

Program Results (s21bffe.r)

GnuplotProduced by GNUPLOT 5.0 patchlevel 3 1 1.1 1.2 1.3 1.4 1.5 1.6 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0.8 0.9 1 1.1 1.2 1.3 Example Program Classical (Legendre) Form of the Incomplete Elliptic Integral of the Second Kind gnuplot_plot_1 f m
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017