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- [/
- 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]
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