# NAG FL Interfaces21bgf (ellipint_​legendre_​3)

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## 1Purpose

s21bgf returns a value of the classical (Legendre) form of the incomplete elliptic integral of the third kind, via the function name.

## 2Specification

Fortran Interface
 Function s21bgf ( dn, phi, dm,
 Real (Kind=nag_wp) :: s21bgf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: dn, phi, dm
#include <nag.h>
 double s21bgf_ (const double *dn, const double *phi, const double *dm, Integer *ifail)
The routine may be called by the names s21bgf or nagf_specfun_ellipint_legendre_3.

## 3Description

s21bgf calculates an approximation to the integral
 $Π (n;ϕ∣m) = ∫0ϕ (1-nsin2⁡θ) -1 (1-msin2⁡θ) -12 dθ ,$
where $0\le \varphi \le \frac{\pi }{2}$, $m{\mathrm{sin}}^{2}\varphi \le 1$, $m$ and $\mathrm{sin}\varphi$ may not both equal one, and $n{\mathrm{sin}}^{2}\varphi \ne 1$.
The integral is computed using the symmetrised elliptic integrals of Carlson (Carlson (1979) and Carlson (1988)). The relevant identity is
 $Π (n;ϕ∣m) = sin⁡ϕ RF (q,r,1) + 13 n sin3⁡ϕ RJ (q,r,1,s) ,$
where $q={\mathrm{cos}}^{2}\varphi$, $r=1-m{\mathrm{sin}}^{2}\varphi$, $s=1-n{\mathrm{sin}}^{2}\varphi$, ${R}_{F}$ is the Carlson symmetrised incomplete elliptic integral of the first kind (see s21bbf) and ${R}_{J}$ is the Carlson symmetrised incomplete elliptic integral of the third kind (see s21bdf).

## 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: $\mathbf{dn}$Real (Kind=nag_wp) Input
2: $\mathbf{phi}$Real (Kind=nag_wp) Input
3: $\mathbf{dm}$Real (Kind=nag_wp) Input
On entry: the arguments $n$, $\varphi$ 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 $1.0$;
• ${\mathbf{dn}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)\ne 1.0$.
Note that ${\mathbf{dm}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)=1.0$ is allowable, as long as ${\mathbf{dm}}\ne 1.0$.
4: $\mathbf{ifail}$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, ${\mathbf{phi}}=⟨\mathit{\text{value}}⟩$.
Constraint: $0\le {\mathbf{phi}}\le \left(\pi /2\right)$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{phi}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{dm}}=⟨\mathit{\text{value}}⟩$; the integral is undefined.
Constraint: ${\mathbf{dm}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)\le 1.0$.
${\mathbf{ifail}}=3$
On entry, $\mathrm{sin}\left({\mathbf{phi}}\right)=1$ and ${\mathbf{dm}}=1.0$; the integral is infinite.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{phi}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{dn}}=⟨\mathit{\text{value}}⟩$; the integral is infinite.
Constraint: ${\mathbf{dn}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)\ne 1.0$.
${\mathbf{ifail}}=-99$
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.
${\mathbf{ifail}}=-399$
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 s21bgf 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

Background information to multithreading can be found in the Multithreading documentation.
s21bgf 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 algorithms used to compute ${R}_{F}$ and ${R}_{J}$, see the routine documents for s21bbf and s21bdf, 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.

## 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 (s21bgfe.f90)

None.

### 10.3Program Results

Program Results (s21bgfe.r)