[/
  Copyright 2019, Nick Thompson
  Distributed under the Boost Software License, Version 1.0.
  (See accompanying file LICENSE_1_0.txt or copy at
  http://www.boost.org/LICENSE_1_0.txt).
]

[section:gegenbauer Gegenbauer Polynomials]

[h4 Synopsis]

``
#include <boost/math/special_functions/gegenbauer.hpp>
``

   namespace boost{ namespace math{

   template<typename Real>
   Real gegenbauer(unsigned n, Real lambda, Real x);

   template<typename Real>
   Real gegenbauer_prime(unsigned n, Real lambda, Real x);

   template<typename Real>
   Real gegenbauer_derivative(unsigned n, Real lambda, Real x, unsigned k);

   }} // namespaces

Gegenbauer polynomials are a family of orthogonal polynomials.

A basic usage is as follows:

    using boost::math::gegenbauer;
    double x = 0.5;
    double lambda = 0.5;
    unsigned n = 3;
    double y = gegenbauer(n, lambda, x);

All derivatives of the Gegenbauer polynomials are available.
The /k/-th derivative of the /n/-th Gegenbauer polynomial is given by

    using boost::math::gegenbauer_derivative;
    double x = 0.5;
    double lambda = 0.5;
    unsigned n = 3;
    unsigned k = 2;
    double y = gegenbauer_derivative(n, lambda, x, k);

For consistency with the rest of the library, `gegenbauer_prime` is provided which simply returns `gegenbauer_derivative(n, lambda, x,1 )`.

[$../graphs/gegenbauer.svg]

[h3 Implementation]

The implementation uses the 3-term recurrence for the Gegenbauer polynomials, rising.

[h3 Performance]

Double precision timing on a consumer x86 laptop is shown below.
Included is the time to generate a random number argument in the interval \[-1, 1\] (which takes 11.5ns).

``
Run on (16 X 4300 MHz CPU s)
CPU Caches:
  L1 Data 32K (x8)
  L1 Instruction 32K (x8)
  L2 Unified 1024K (x8)
  L3 Unified 11264K (x1)
Load Average: 0.21, 0.33, 0.29
-----------------------------------------
Benchmark                            Time
-----------------------------------------
Gegenbauer<double>/1              12.5 ns
Gegenbauer<double>/2              13.5 ns
Gegenbauer<double>/3              14.6 ns
Gegenbauer<double>/4              16.0 ns
Gegenbauer<double>/5              17.5 ns
Gegenbauer<double>/6              19.2 ns
Gegenbauer<double>/7              20.7 ns
Gegenbauer<double>/8              22.2 ns
Gegenbauer<double>/9              23.6 ns
Gegenbauer<double>/10             25.2 ns
Gegenbauer<double>/11             26.9 ns
Gegenbauer<double>/12             28.7 ns
Gegenbauer<double>/13             30.5 ns
Gegenbauer<double>/14             32.5 ns
Gegenbauer<double>/15             34.3 ns
Gegenbauer<double>/16             36.3 ns
Gegenbauer<double>/17             38.0 ns
Gegenbauer<double>/18             39.9 ns
Gegenbauer<double>/19             41.8 ns
Gegenbauer<double>/20             43.8 ns
UniformReal<double>               11.5 ns
``

[h3 Accuracy]

Some representative ULP plots are shown below.
The relative accuracy cannot be controlled at the roots of the polynomial, as is to be expected.

[$../graphs/gegenbauer_ulp_3.svg]
[$../graphs/gegenbauer_ulp_5.svg]
[$../graphs/gegenbauer_ulp_9.svg]

[h3 Caveats]

Some programs define the Gegenbauer polynomial with \u03BB = 0 via renormalization (which makes them Chebyshev polynomials).
We do not follow this convention: In this case, only the zeroth Gegenbauer polynomial is nonzero.


[endsect]
