NAG Library Routine Document
s21bff (ellipint_legendre_2)
1
Purpose
s21bff returns a value of the classical (Legendre) form of the incomplete elliptic integral of the second kind, via the function name.
2
Specification
Fortran Interface
Real (Kind=nag_wp)  ::  s21bff  Integer, Intent (Inout)  ::  ifail  Real (Kind=nag_wp), Intent (In)  ::  phi, dm 

C Header Interface
#include <nagmk26.h>
double 
s21bff_ (const double *phi, const double *dm, Integer *ifail) 

3
Description
s21bff calculates an approximation to the integral
where
$0\le \varphi \le \frac{\pi}{2}$ and
$m{\mathrm{sin}}^{2}\varphi \le 1$.
The integral is computed using the symmetrised elliptic integrals of Carlson (
Carlson (1979) and
Carlson (1988)). The relevant identity is
where
$q={\mathrm{cos}}^{2}\varphi $,
$r=1m{\mathrm{sin}}^{2}\varphi $,
${R}_{F}$ is the Carlson symmetrised incomplete elliptic integral of the first kind (see
s21bbf) and
${R}_{D}$ is the Carlson symmetrised incomplete elliptic integral of the second kind (see
s21bcf).
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: $\mathbf{phi}$ – Real (Kind=nag_wp)Input
 2: $\mathbf{dm}$ – Real (Kind=nag_wp)Input

On entry: the arguments $\varphi $ and $m$ of the function.
Constraints:
 $0.0\le {\mathbf{phi}}\le \frac{\pi}{2}$;
 ${\mathbf{dm}}\times {\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)\le 1.0$.
 3: $\mathbf{ifail}$ – IntegerInput/Output

On entry:
ifail must be set to
$0$,
$1\text{or}1$. 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
$1\text{or}1$ 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 $\mathbf{1}\text{or}\mathbf{1}$ 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).
6
Error 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, ${\mathbf{phi}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: $0\le {\mathbf{phi}}\le \frac{\pi}{2}$.
 ${\mathbf{ifail}}=2$

On entry, ${\mathbf{phi}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{dm}}=\u2329\mathit{\text{value}}\u232a$; the integral is undefined.
Constraint: ${\mathbf{dm}}\times {\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)\le 1.0$.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
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.
7
Accuracy
In principle s21bff is capable of producing full machine precision. However, roundoff 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 roundoff error. It is reasonable to assume that the result is accurate to within a small multiple of the machine precision.
8
Parallelism and Performance
s21bff 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 argumentconstraints for both forms becomes clear.
For more information on the algorithms used to compute
${R}_{F}$ and
${R}_{D}$, see the routine documents for
s21bbf and
s21bcf, respectively.
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,
$E\left(\varphi m\right)=E\left(\varphi m\right)$. A parameter
$m>1$ can be replaced by one less than unity using
$E\left(\varphi m\right)=\sqrt{m}E\left(\varphi \sqrt{m}\frac{1}{m}\right)\left(m1\right)\varphi $.
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 (s21bffe.f90)
10.2
Program Data
None.
10.3
Program Results
Program Results (s21bffe.r)