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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 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
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

nag_elliptic_integral_pi (s21bgc) is not threaded in any implementation.

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.