NAG Library Function Document

nag_elliptic_integral_complete_K (s21bhc)

▸▿ 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_K (s21bhc) returns a value of the classical (Legendre) form of the complete elliptic integral of the first kind.

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_elliptic_integral_complete_K (double dm, NagError *fail)

3 Description

nag_elliptic_integral_complete_K (s21bhc) calculates an approximation to the integral

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

where

m < 1

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

K (m) = R_{F} (0, 1 - m, 1),

where

R_{F}

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

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: $dm < 1.0$ .
2: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction 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 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NE_REAL: On entry, $dm = 〈value〉$ ; the integral is undefined.
Constraint: $dm < 1.0$ .
On failure, the function returns zero.
NW_INTEGRAL_INFINITE: On entry, $dm = 1.0$ ; the integral is infinite.
On failure, the function returns the largest machine number (see nag_real_largest_number (X02ALC)).

7 Accuracy

In principle nag_elliptic_integral_complete_K (s21bhc) 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 algorithm used to compute