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Gegenbauer Polynomials

#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 ).


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


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


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


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