Elliptic Integrals of the First Kind

Elliptic Integrals of the First Kind - Legendre Form

Synopsis

#include <boost/math/special_functions/ellint_1.hpp>

namespace boost { namespace math {

template <class T1, class T2>
calculated-result-type ellint_1(T1 k, T2 phi);

template <class T1, class T2, class Policy>
calculated-result-type ellint_1(T1 k, T2 phi, const Policy&);

template <class T>
calculated-result-type ellint_1(T k);

template <class T, class Policy>
calculated-result-type ellint_1(T k, const Policy&);

}} // namespaces

Description

These two functions evaluate the incomplete elliptic integral of the first kind F(φ, k) and its complete counterpart K(k) = F(π/2, k).

The return type of these functions is computed using the result type calculation rules when T1 and T2 are different types: when they are the same type then the result is the same type as the arguments.

template <class T1, class T2>
calculated-result-type ellint_1(T1 k, T2 phi);

template <class T1, class T2, class Policy>
calculated-result-type ellint_1(T1 k, T2 phi, const Policy&);

Returns the incomplete elliptic integral of the first kind F(φ, k):

Requires k²sin²(phi) < 1, otherwise returns the result of domain_error.

The final Policy argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the policy documentation for more details.

template <class T>
calculated-result-type ellint_1(T k);

template <class T>
calculated-result-type ellint_1(T k, const Policy&);

Returns the complete elliptic integral of the first kind K(k):

Requires |k| < 1, otherwise returns the result of domain_error.

Accuracy

These functions are computed using only basic arithmetic operations, so there isn't much variation in accuracy over differing platforms. Note that only results for the widest floating point type on the system are given as narrower types have effectively zero error. All values are relative errors in units of epsilon.

Table 8.63. Error rates for ellint_1

	GNU C++ version 7.1.0 linux long double	GNU C++ version 7.1.0 linux double	Sun compiler version 0x5150 Sun Solaris long double	Microsoft Visual C++ version 14.1 Win32 double
Elliptic Integral F: Mathworld Data	Max = 0.94ε (Mean = 0.509ε) (<cmath>: Max = +INFε (Mean = +INFε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 2.1: Max = 0.919ε (Mean = 0.544ε))	Max = 0.94ε (Mean = 0.509ε)	Max = 0.919ε (Mean = 0.542ε)
Elliptic Integral F: Random Data	Max = 1.57ε (Mean = 0.56ε) (<cmath>: Max = 2.56ε (Mean = 0.816ε))	Max = 0ε (Mean = 0ε) (GSL 2.1: Max = 2.99ε (Mean = 0.797ε))	Max = 1.57ε (Mean = 0.561ε)	Max = 2.26ε (Mean = 0.631ε)

The following error plot are based on an exhaustive search of the functions domain, MSVC-15.5 at double precision, and GCC-7.1/Ubuntu for long double and __float128.

Testing

The tests use a mixture of spot test values calculated using the online calculator at functions.wolfram.com, and random test data generated using NTL::RR at 1000-bit precision and this implementation.

Implementation

These functions are implemented in terms of Carlson's integrals using the relations:

and