# NAG FL Interfaces21bhf (ellipint_​complete_​1)

## 1Purpose

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

## 2Specification

Fortran Interface
 Function s21bhf ( dm,
 Real (Kind=nag_wp) :: s21bhf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: dm
#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.

## 3Description

s21bhf calculates an approximation to the integral
 $Km = ∫0 π2 1-m sin2⁡θ -12 dθ ,$
where $m<1$.
The integral is computed using the symmetrised elliptic integrals of Carlson (Carlson (1979) and Carlson (1988)). The relevant identity is
 $Km = RF 0,1-m,1 ,$
where ${R}_{F}$ is the Carlson symmetrised incomplete elliptic integral of the first kind (see s21bbf).
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

## 5Arguments

1: $\mathbf{dm}$Real (Kind=nag_wp) Input
On entry: the argument $m$ of the function.
Constraint: ${\mathbf{dm}}<1.0$.
2: $\mathbf{ifail}$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 $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{dm}}=〈\mathit{\text{value}}〉$; the integral is undefined.
Constraint: ${\mathbf{dm}}<1.0$.
On soft failure, the routine returns zero.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{dm}}=1.0$; the integral is infinite.
On soft failure, the routine returns the largest machine number (see x02alf).
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

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.

## 8Parallelism and Performance

s21bhf is not threaded in any implementation.

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 ${R}_{F}$, see the routine document for s21bbf.

## 10Example

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

### 10.1Program Text

Program Text (s21bhfe.f90)

None.

### 10.3Program Results

Program Results (s21bhfe.r)