NAG C Library Manual, Mark 26
NAG Library Manual, Mark 26
NAG AD Library Manual, Mark 26
s Chapter Contents
s Chapter Introduction

NAG C Library Function Document

nag_elliptic_integral_complete_K (s21bhc)

NAG C Library Manual, Mark 26
NAG Library Manual, Mark 26
NAG AD Library Manual, Mark 26
s Chapter Contents
s Chapter Introduction

▸▿ Contents

© The Numerical Algorithms Group Ltd. 2018

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
Km = ∫0 π2 1-m sin2⁡θ -12 dθ ,  
where m<1 .
The integral is computed using the symmetrised elliptic integrals of Carlson (Carlson (1979) and Carlson (1988)). The relevant identity is
Km = RF 0,1-m,1 ,  
where RF  is the Carlson symmetrised incomplete elliptic integral of the first kind (see nag_elliptic_integral_rf (s21bbc)).

4
References

NIST Digital Library of Mathematical Functions
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 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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

nag_elliptic_integral_complete_K (s21bhc) 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 algorithm used to compute RF , see the function document for nag_elliptic_integral_rf (s21bbc).

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 (s21bhce.c)

10.2
Program Data

None.

10.3
Program Results

Program Results (s21bhce.r)

NAG C Library Manual, Mark 26
NAG Library Manual, Mark 26
NAG AD Library Manual, Mark 26
s Chapter Contents
s Chapter Introduction
© The Numerical Algorithms Group Ltd. 2018