s21bg: FL CL CPP AD PY MB

NAG FL Interface
s21bgf (ellipint_legendre_3)

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NAG Library Manual, Mark 28.5

Interfaces: FL CL CPP AD PY MB

NAG FL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s21bg: FL CL CPP AD PY MB

▸▿ 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

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

2 Specification

Fortran Interface

Function s21bgf (

dn, phi, dm, ifail)

Real (Kind=nag_wp)	::	s21bgf
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	dn, phi, dm

C Header Interface

#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.

3 Description

s21bgf 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 s21bbf) and

R_{J}

is the Carlson symmetrised incomplete elliptic integral of the third kind (see s21bdf).

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: $dn$ – Real (Kind=nag_wp) Input

2: $phi$ – Real (Kind=nag_wp) Input

3: $dm$ – Real (Kind=nag_wp) 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: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

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

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $phi = ⟨ value ⟩$ .
Constraint: $0 \leq phi \leq (π / 2)$ .

$ifail = 2$: On entry, $phi = ⟨ value ⟩$ and $dm = ⟨ value ⟩$ ; the integral is undefined.
Constraint: $dm \times \sin^{2} (phi) \leq 1.0$ .

$ifail = 3$: On entry, $\sin (phi) = 1$ and $dm = 1.0$ ; the integral is infinite.

$ifail = 4$: On entry, $phi = ⟨ value ⟩$ and $dn = ⟨ value ⟩$ ; the integral is infinite.
Constraint: $dn \times \sin^{2} (phi) \neq 1.0$ .

$ifail = - 99$: 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.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

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.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

s21bgf is not threaded in any implementation.

9 Further Comments

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.