NAG Library Manual, Mark 28.3

Interfaces: FL CL CPP AD

NAG CL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s21bg: FL CL CPP AD

NAG CL Interface
s21bgc (ellipint_legendre_3)

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NAG Library Manual, Mark 28.3

Interfaces: FL CL CPP AD

NAG CL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s21bg: FL CL CPP AD

▸▿ Contents

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

© The Numerical Algorithms Group Ltd. 2022

CL Name Style:
Short (a00aac)
Long (impl_details)
Full (nag_info_impl_details)

1 Purpose

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

2 Specification

#include <nag.h>

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.

3 Description

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 s21bbc) and

R_{J}

is the Carlson symmetrised incomplete elliptic integral of the third kind (see 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: $dn$ – double Input

2: $phi$ – double Input

3: $dm$ – double Input

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 7 in the Introduction to the NAG Library CL Interface).

6 Error 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, $phi = ⟨ value ⟩$ .
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 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

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 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.

10 Example

This example simply generates a small set of nonextreme arguments that are used with the function to produce the table of results.

10.1 Program Text

Program Text (s21bgce.c)

10.2 Program Data

None.

10.3 Program Results

Program Results (s21bgce.r)

NAG Library Manual, Mark 28.3

Interfaces: FL CL CPP AD

NAG CL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s21bg: FL CL CPP AD

© The Numerical Algorithms Group Ltd. 2022