NAG FL Interface
s21bef (ellipint_​legendre_​1)

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1 Purpose

s21bef returns a value of the classical (Legendre) form of the incomplete elliptic integral of the first kind, via the function name.

2 Specification

Fortran Interface
Function s21bef ( phi, dm, ifail)
Real (Kind=nag_wp) :: s21bef
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: phi, dm
C Header Interface
#include <nag.h>
double  s21bef_ (const double *phi, const double *dm, Integer *ifail)
The routine may be called by the names s21bef or nagf_specfun_ellipint_legendre_1.

3 Description

s21bef calculates an approximation to the integral
F(ϕm) = 0ϕ (1-msin2θ) -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
F(ϕm) = 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 s21bbf).

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 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× sin2(phi) 1.0 ;
  • Only one of sin(phi) and dm may be 1.0.
Note that dm × sin2(phi) = 1.0 is allowable, as long as dm1.0 .
3: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. 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).

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 soft failure, the routine returns zero.
On entry, phi=value and dm=value; the integral is undefined.
Constraint: dm×sin2(phi)1.0.
On soft failure, the routine returns zero.
On entry, sin(phi)=1 and dm=1.0; the integral is infinite.
On soft failure, the routine returns the largest machine number (see x02alf).
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL 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 FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

In principle s21bef 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.
s21bef is not threaded in any implementation.

9 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 algorithm used to compute RF , see the routine document for s21bbf.
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, F(-ϕ|m)=-F(ϕ|m) and F(sπ±ϕ|m)=2sK(m)±F(ϕ|m) where s is an integer and K(m) is the complete elliptic integral given by s21bhf.
A parameter m>1 can be replaced by one less than unity using F(ϕ|m)=1mF(θ|1m), sinθ=msinϕ.

10 Example

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

10.1 Program Text

Program Text (s21befe.f90)

10.2 Program Data


10.3 Program Results

Program Results (s21befe.r)
GnuplotProduced by GNUPLOT 5.4 patchlevel 6 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 gnuplot_plot_1 f m Example Program Classical (Legendre) Form of the Incomplete Elliptic Integral of the First Kind