NAG Library Routine Document

S21BGF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 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

FUNCTION S21BGF (

DN, PHI, DM, IFAIL)

REAL (KIND=nag_wp) S21BGF

INTEGER	IFAIL
REAL (KIND=nag_wp)	DN, PHI, DM

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 Parameters

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 – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. 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:

$IFAIL = 1$: PHI lies outside the range $[0, \frac{π}{2}]$ . On soft failure, the routine returns zero.

$IFAIL = 2$: On entry, $DM \times \sin^{2} (PHI) > 1.0$ ; the function is undefined. On soft failure, the routine returns zero.

$IFAIL = 3$: On entry, $\sin (PHI) = 1.0$ and $DM = 1.0$ ; the function is infinite. On soft failure, the routine returns the largest machine number (see X02ALF).

$IFAIL = 4$: On entry, the product $DN \times \sin^{2} (PHI) = 1.0$ ; the function is infinite. On soft failure, the routine returns the largest machine number (see X02ALF).

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 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.