# NAG CL Interfaces21bgc (ellipint_​legendre_​3)

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## 1Purpose

s21bgc returns a value of the classical (Legendre) form of the incomplete elliptic integral of the third kind.

## 2Specification

 #include
 double s21bgc (double dn, double phi, double dm, NagError *fail)
The function may be called by the names: s21bgc, nag_specfun_ellipint_legendre_3 or nag_elliptic_integral_pi.

## 3Description

s21bgc 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 s21bbc) and ${R}_{J}$ is the Carlson symmetrised incomplete elliptic integral of the third kind (see s21bdc).

## 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: $\mathbf{dn}$double Input
2: $\mathbf{phi}$double Input
3: $\mathbf{dm}$double 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: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL
On entry, ${\mathbf{phi}}=⟨\mathit{\text{value}}⟩$.
Constraint: $0\le {\mathbf{phi}}\le \left(\pi /2\right)$.
NE_REAL_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 entry, ${\mathbf{phi}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{dn}}=⟨\mathit{\text{value}}⟩$; the integral is infinite.
Constraint: ${\mathbf{dn}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)\ne 1.0$.
NW_INTEGRAL_INFINITE
On entry, $\mathrm{sin}\left({\mathbf{phi}}\right)=1$ and ${\mathbf{dm}}=1.0$; the integral is infinite.

## 7Accuracy

In principle s21bgc 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

s21bgc is not threaded in any implementation.

You should consult the S Chapter Introduction, which shows the relationship between this function 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 function documents for s21bbc and s21bdc, respectively.
If you wish to input a value of phi outside the range allowed by this function 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 function to produce the table of results.

### 10.1Program Text

Program Text (s21bgce.c)

None.

### 10.3Program Results

Program Results (s21bgce.r)