Chebyshev Polynomials

Synopsis

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

namespace boost{ namespace math{

template<class Real1, class Real2, class Real3>
calculated-result-type chebyshev_next(Real1 const & x, Real2 const & Tn, Real3 const & Tn_1);

template<class Real>
calculated-result-type chebyshev_t(unsigned n, Real const & x);

template<class Real, class Policy>
calculated-result-type chebyshev_t(unsigned n, Real const & x, const Policy&);

template<class Real>
calculated-result-type chebyshev_u(unsigned n, Real const & x);

template<class Real, class Policy>
calculated-result-type chebyshev_u(unsigned n, Real const & x, const Policy&);

template<class Real>
calculated-result-type chebyshev_t_prime(unsigned n, Real const & x);

template<class Real1, class Real2>
calculated-result-type chebyshev_clenshaw_recurrence(const Real* const c, size_t length, Real2 x);

}} // namespaces

"Real analysts cannot do without Fourier, complex analysts cannot do without Laurent, and numerical analysts cannot do without Chebyshev" --Lloyd N. Trefethen

The Chebyshev polynomials of the first kind are defined by the recurrence T_n+1(x) := 2xT_n(x) - T_n-1(x), n > 0, where T₀(x) := 1 and T₁(x) := x. These can be calculated in Boost using the following simple code

double x = 0.5;
double T12 = boost::math::chebyshev_t(12, x);

Calculation of derivatives is also straightforward:

double T12_prime = boost::math::chebyshev_t_prime(12, x);

The complexity of evaluation of the n-th Chebyshev polynomial by these functions is linear. So they are unsuitable for use in calculation of (say) a Chebyshev series, as a sum of linear scaling functions scales quadratically. Though there are very sophisticated algorithms for the evaluation of Chebyshev series, a linear time algorithm is presented below:

double x = 0.5;
std::vector<double> c{14.2, -13.7, 82.3, 96};
double T0 = 1;
double T1 = x;
double f = c[0]*T0/2;
unsigned l = 1;
while(l < c.size())
{
   f += c[l]*T1;
   std::swap(T0, T1);
   T1 = boost::math::chebyshev_next(x, T0, T1);
   ++l;
}

This uses the chebyshev_next function to evaluate each term of the Chebyshev series in constant time. However, this naive algorithm has a catastrophic loss of precision as x approaches 1. A method to mitigate this way given by Clenshaw, and is implemented in boost as

double x = 0.5;
std::vector<double> c{14.2, -13.7, 82.3, 96};
double f = chebyshev_clenshaw_recurrence(c.data(), c.size(), Real x);

N.B.: There is factor of 2 difference in our definition of the first coefficient in the Chebyshev series from Clenshaw's original work. This is because two traditions exist in notation for the Chebyshev series expansion,

f(x) ≈ ∑_n=0^N-1 a_nT_n(x)

and

f(x) ≈ c₀/2 + ∑_n=1^N-1 c_nT_n(x)

boost math always uses the second convention, with the factor of 1/2 on the first coefficient.

Chebyshev polynomials of the second kind can be evaluated via chebyshev_u:

double x = -0.23;
double U1 = boost::math::chebyshev_u(1, x);

The evaluation of Chebyshev polynomials by a three-term recurrence is known to be mixed forward-backward stable for x ∊ [-1, 1]. However, the author does not know of a similar result for x outside [-1, 1]. For this reason, evaluation of Chebyshev polynomials outside of [-1, 1] is strongly discouraged. That said, small rounding errors in the course of a computation will often lead to this situation, and termination of the computation due to these small problems is very discouraging. For this reason, chebyshev_t and chebyshev_u have code paths for x > 1 and x < -1 which do not use three-term recurrences. These code paths are much slower, and should be avoided if at all possible.

Evaluation of a Chebyshev series is relatively simple. The real challenge is generation of the Chebyshev series. For this purpose, boost provides a Chebyshev transform, a projection operator which projects a function onto a finite-dimensional span of Chebyshev polynomials. But before we discuss the API, let's analyze why we might want to project a function onto a span of Chebyshev polynomials.

We want a numerically stable way to evaluate the function's derivative.
Our function is expensive to evaluate, and we wish to find a less expensive way to estimate its value. An example are the standard library transcendental functions: These functions are guaranteed to evaluate to within 1 ulp of the exact value, but often this accuracy is not needed. A projection onto the Chebyshev polynomials with a low accuracy requirement can vastly accelerate the computation of these functions.
We wish to numerically integrate the function.

The API is given below.

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

namespace boost{ namespace math{

template<class Real>
class chebyshev_transform
{
public:
    template<class F>
    chebyshev_transform(const F& f, Real a, Real b, Real tol=500*std::numeric_limits<Real>::epsilon());

    Real operator()(Real x) const

    Real integrate() const

    const std::vector<Real>& coefficients() const

    Real prime(Real x) const
};

}}// end namespaces

The Chebyshev transform takes a function f and returns a near-minimax approximation to f in terms of Chebyshev polynomials. By near-minimax, we mean that the resulting Chebyshev polynomial is "very close" the polynomial p_n which minimizes the uniform norm of f - p_n. The notion of "very close" can be made rigorous; see Trefethen's "Approximation Theory and Approximation Practice" for details.

The Chebyshev transform works by creating a vector of values by evaluating the input function at the Chebyshev points, and then performing a discrete cosine transform on the resulting vector. In order to do this efficiently, we have used FFTW3. So to compile, you must have FFTW3 installed, and link with -lfftw3 for double precision, -lfftw3f for float precision, -lfftw3l for long double precision, and -lfftwq for quad (__float128) precision. After the coefficients of the Chebyshev series are known, the routine goes back through them and filters out all the coefficients whose absolute ratio to the largest coefficient are less than the tolerance requested in the constructor.