# NAG Library Routine Document

## 1Purpose

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

## 2Specification

Fortran Interface
 Function s21bef ( phi, dm,
 Real (Kind=nag_wp) :: s21bef Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: phi, dm
#include <nagmk26.h>
 double s21bef_ (const double *phi, const double *dm, Integer *ifail)

## 3Description

s21bef calculates an approximation to the integral
 $Fϕ∣m = ∫0ϕ 1-m sin2⁡θ -12 dθ ,$
where $0\le \varphi \le \frac{\pi }{2}$, $m{\mathrm{sin}}^{2}\varphi \le 1$ and $m$ and $\mathrm{sin}\varphi$ 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
 $Fϕ∣m = RF q,r,1 sin⁡ϕ ,$
where $q={\mathrm{cos}}^{2}\varphi$, $r=1-m{\mathrm{sin}}^{2}\varphi$ and ${R}_{F}$ is the Carlson symmetrised incomplete elliptic integral of the first kind (see s21bbf).

## 4References

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

## 5Arguments

1:     $\mathbf{phi}$ – Real (Kind=nag_wp)Input
2:     $\mathbf{dm}$ – Real (Kind=nag_wp)Input
On entry: the arguments $\varphi$ and $m$ of the function.
Constraints:
• $0.0\le {\mathbf{phi}}\le \frac{\pi }{2}$;
• ${\mathbf{dm}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)\le 1.0$;
• Only one of $\mathrm{sin}\left({\mathbf{phi}}\right)$ and dm may be $1.0$.
Note that ${\mathbf{dm}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)=1.0$ is allowable, as long as ${\mathbf{dm}}\ne 1.0$.
3:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . 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  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  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, ${\mathbf{phi}}=〈\mathit{\text{value}}〉$.
Constraint: $0\le {\mathbf{phi}}\le \frac{\pi }{2}$.
On soft failure, the routine returns zero.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{phi}}=〈\mathit{\text{value}}〉$ and ${\mathbf{dm}}=〈\mathit{\text{value}}〉$; the integral is undefined.
Constraint: ${\mathbf{dm}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)\le 1.0$.
On soft failure, the routine returns zero.
${\mathbf{ifail}}=3$
On entry, $\mathrm{sin}\left({\mathbf{phi}}\right)=1$ and ${\mathbf{dm}}=1.0$; the integral is infinite.
On soft failure, the routine returns the largest machine number (see x02alf).
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

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.

## 8Parallelism and Performance

s21bef 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.
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, $F\left(-\varphi |m\right)=-F\left(\varphi |m\right)$ and $F\left(s\pi ±\varphi |m\right)=2sK\left(m\right)±F\left(\varphi |m\right)$ where $s$ is an integer and $K\left(m\right)$ is the complete elliptic integral given by s21bhf.
A parameter $m>1$ can be replaced by one less than unity using $F\left(\varphi |m\right)=\frac{1}{\sqrt{m}}F\left(\theta |\frac{1}{m}\right)$, $\mathrm{sin}\theta =\sqrt{m}\mathrm{sin}\varphi$.

## 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 (s21befe.f90)

None.

### 10.3Program Results

Program Results (s21befe.r)