NAG Library Function Document

nag_elliptic_integral_pi (s21bgc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

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

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_elliptic_integral_pi (double dn, double phi, double dm, NagError *fail)

3 Description

nag_elliptic_integral_pi (s21bgc) calculates an approximation to the integral

Π (n; ϕ ∣ m) = \int_{0}^{ϕ} {(1 - n \sin^{2} θ)}^{- 1} {(1 - m \sin^{2} θ)}^{- \frac{1}{2}} d θ,

where

0 \leq ϕ \leq \frac{π}{2}

m \sin^{2} ϕ \leq 1

m

and

\sin ϕ

may not both equal one, and

n \sin^{2} ϕ \neq 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 R_{F} (q, r, 1) + \frac{1}{3} n \sin^{3} ϕ R_{J} (q, r, 1, s),

where

q = \cos^{2} ϕ

r = 1 - m \sin^{2} ϕ

s = 1 - n \sin^{2} ϕ

R_{F}

is the Carlson symmetrised incomplete elliptic integral of the first kind (see nag_elliptic_integral_rf (s21bbc)) and

R_{J}

is the Carlson symmetrised incomplete elliptic integral of the third kind (see nag_elliptic_integral_rj (s21bdc)).

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: dn – doubleInput

2: phi – doubleInput

3: dm – doubleInput

On entry: the arguments

n

ϕ

and

m

of the function.

Constraints:

$0.0 \leq phi \leq \frac{π}{2}$ ;
$dm \times \sin^{2} (phi) \leq 1.0$ ;
Only one of $\sin (phi)$ and dm may be $1.0$ ;
$dn \times \sin^{2} (phi) \neq 1.0$ .

Note that

dm \times \sin^{2} (phi) = 1.0

is allowable, as long as

dm \neq 1.0

4: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

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.
NE_REAL: On entry, $phi = ⟨value⟩$ .
Constraint: $0 \leq phi \leq (π / 2)$ .
NE_REAL_2: On entry, $phi = ⟨value⟩$ and $dm = ⟨value⟩$ ; the integral is undefined.
Constraint: $dm \times \sin^{2} (phi) \leq 1.0$ .
On entry, $phi = ⟨value⟩$ and $dn = ⟨value⟩$ ; the integral is infinite.
Constraint: $dn \times \sin^{2} (phi) \neq 1.0$ .
NW_INTEGRAL_INFINITE: On entry, $\sin (phi) = 1$ and $dm = 1.0$ ; the integral is infinite.

7 Accuracy

In principle nag_elliptic_integral_pi (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.

8 Parallelism and Performance

Not applicable.

9 Further Comments

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 nag_elliptic_integral_rf (s21bbc) and nag_elliptic_integral_rj (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.