...one of the most highly
regarded and expertly designed C++ library projects in the
world.
— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards
#include <boost/math/special_functions/ellint_1.hpp>
namespace boost { namespace math { template <class T1, class T2> calculatedresulttype ellint_1(T1 k, T2 phi); template <class T1, class T2, class Policy> calculatedresulttype ellint_1(T1 k, T2 phi, const Policy&); template <class T> calculatedresulttype ellint_1(T k); template <class T, class Policy> calculatedresulttype ellint_1(T k, const Policy&); }} // namespaces
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> calculatedresulttype ellint_1(T1 k, T2 phi); template <class T1, class T2, class Policy> calculatedresulttype ellint_1(T1 k, T2 phi, const Policy&);
Returns the incomplete elliptic integral of the first kind F(φ, k):
Requires k^{2}sin^{2}(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> calculatedresulttype ellint_1(T k); template <class T> calculatedresulttype 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.
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.
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 
GNU C++ version 7.1.0 
Sun compiler version 0x5150 
Microsoft Visual C++ version 14.1 


Elliptic Integral F: Mathworld Data 
Max = 0.94ε (Mean = 0.509ε) 
Max = 0ε (Mean = 0ε) 
Max = 0.94ε (Mean = 0.509ε) 
Max = 0.919ε (Mean = 0.542ε) 
Elliptic Integral F: Random Data 
Max = 1.57ε (Mean = 0.56ε) 
Max = 0ε (Mean = 0ε) 
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, MSVC15.5 at double
precision, and GCC7.1/Ubuntu for long
double
and __float128
.
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 1000bit precision and this implementation.
For up to 80bit long double precision the complete versions of these functions are implemented as Taylor series expansions as in: "Fast computation of complete elliptic integrals and Jacobian elliptic functions", Celestial Mechanics and Dynamical Astronomy, April 2012.
Otherwise these functions are implemented in terms of Carlson's integrals using the relations:
and