NAG FL Interface
s21bhf (ellipint_​complete_​1)

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1 Purpose

s21bhf returns a value of the classical (Legendre) form of the complete elliptic integral of the first kind, via the function name.

2 Specification

Fortran Interface
Function s21bhf ( dm, ifail)
Real (Kind=nag_wp) :: s21bhf
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: dm
C Header Interface
#include <nag.h>
double  s21bhf_ (const double *dm, Integer *ifail)
The routine may be called by the names s21bhf or nagf_specfun_ellipint_complete_1.

3 Description

s21bhf calculates an approximation to the integral
K(m) = 0 π2 (1-msin2θ) -12 dθ ,  
where m<1 .
The integral is computed using the symmetrised elliptic integrals of Carlson (Carlson (1979) and Carlson (1988)). The relevant identity is
K(m) = RF (0,1-m,1) ,  
where RF is the Carlson symmetrised incomplete elliptic integral of the first kind (see s21bbf).

4 References

NIST Digital Library of Mathematical Functions
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: dm Real (Kind=nag_wp) Input
On entry: the argument m of the function.
Constraint: dm<1.0.
2: 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, dm=value; the integral is undefined.
Constraint: dm<1.0.
On soft failure, the routine returns zero.
On entry, 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 s21bhf 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.
s21bhf 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.

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 (s21bhfe.f90)

10.2 Program Data


10.3 Program Results

Program Results (s21bhfe.r)