nag_elliptic_integral_F (s21bec) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_elliptic_integral_F (s21bec)


    1  Purpose
    7  Accuracy

1  Purpose

nag_elliptic_integral_F (s21bec) returns a value of the classical (Legendre) form of the incomplete elliptic integral of the first kind.

2  Specification

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

3  Description

nag_elliptic_integral_F (s21bec) calculates an approximation to the integral
Fϕm = 0ϕ 1-m sin2θ -12 dθ ,  
where 0ϕ π2 , msin2ϕ1  and m  and sinϕ  may not both equal one.
The integral is computed using the symmetrised elliptic integrals of Carlson (Carlson (1979) and Carlson (1988)). The relevant identity is
Fϕm = RF q,r,1 sinϕ ,  
where q=cos2ϕ , r=1-m sin2ϕ  and RF  is the Carlson symmetrised incomplete elliptic integral of the first kind (see nag_elliptic_integral_rf (s21bbc)).

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:     phi doubleInput
2:     dm doubleInput
On entry: the arguments ϕ and m of the function.
  • 0.0phi π2;
  • dm× sin2phi 1.0 ;
  • Only one of sinphi  and dm may be 1.0.
Note that dm × sin2phi = 1.0  is allowable, as long as dm1.0 .
3:     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, phi=value.
Constraint: 0phiπ2.
On failure, the function returns zero.
On entry, phi=value and dm=value; the integral is undefined.
Constraint: dm×sin2phi1.0.
On failure, the function returns zero.
On entry, sinphi=1 and dm=1.0; the integral is infinite.
On failure, the function returns the largest machine number (see nag_real_largest_number (X02ALC)).

7  Accuracy

In principle nag_elliptic_integral_F (s21bec) 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_F (s21bec) is not threaded in any implementation.

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 algorithm used to compute RF , see the function document for nag_elliptic_integral_rf (s21bbc).
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. For example, F-ϕ|m=-Fϕ|m and Fsπ±ϕ|m=2sKm±Fϕ|m where s is an integer and Km is the complete elliptic integral given by nag_elliptic_integral_complete_K (s21bhc).
A parameter m>1 can be replaced by one less than unity using Fϕ|m=1mFθ|1m, sinθ=msinϕ.

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 (s21bece.c)

10.2  Program Data


10.3  Program Results

Program Results (s21bece.r)

GnuplotProduced by GNUPLOT 4.6 patchlevel 3 0.8 0.9 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 2 4 6 8 10 Example Program Classical (Legendre) Form of the Incomplete Elliptic Integral of the First Kind gnuplot_plot_1 f m

nag_elliptic_integral_F (s21bec) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG Library Manual

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