NAG Library Function Document

nag_elliptic_integral_complete_E (s21bjc)

+− 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_complete_E (s21bjc) returns a value of the classical (Legendre) form of the complete elliptic integral of the second kind.

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_elliptic_integral_complete_E (double dm, NagError *fail)

3 Description

nag_elliptic_integral_complete_E (s21bjc) calculates an approximation to the integral

E (m) = \int_{0}^{\frac{π}{2}} {(1 - m \sin^{2} θ)}^{\frac{1}{2}} d θ,

where

m \leq 1

The integral is computed using the symmetrised elliptic integrals of Carlson (Carlson (1979) and Carlson (1988)). The relevant identity is

E (m) = R_{F} (0, 1 - m, 1) - \frac{1}{3} m R_{D} (0, 1 - m, 1),

where

R_{F}

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

R_{D}

is the Carlson symmetrised incomplete elliptic integral of the second kind (see nag_elliptic_integral_rd (s21bcc)).

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: dm – doubleInput: On entry: the argument $m$ of the function.
Constraint: $dm \leq 1.0$ .
2: 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, $dm = ⟨value⟩$ ; the integral is undefined.
Constraint: $dm \leq 1.0$ .

7 Accuracy

In principle nag_elliptic_integral_complete_E (s21bjc) 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