Math.Polynomial.Chebyshev

Documentation

ts :: [Poly Integer]

The Chebyshev polynomials of the first kind with Integer coefficients.

us :: [Poly Integer]

t :: (Num a, Eq a) => Int -> Poly a

Compute the coefficients of the n'th Chebyshev polynomial of the first kind.

u :: (Num a, Eq a) => Int -> Poly a

Compute the coefficients of the n'th Chebyshev polynomial of the second kind.

evalT :: Num a => Int -> a -> a

Evaluate the n'th Chebyshev polynomial of the first kind at a point X. Both more efficient and more numerically stable than computing the coefficients and evaluating the polynomial.

evalTs :: Num a => a -> [a]

Evaluate all the Chebyshev polynomials of the first kind at a point X.

evalU :: Num a => Int -> a -> a

Evaluate the n'th Chebyshev polynomial of the second kind at a point X. Both more efficient and more numerically stable than computing the coefficients and evaluating the polynomial.

evalUs :: Num a => a -> [a]

Evaluate all the Chebyshev polynomials of the second kind at a point X.

evalTU :: Num a => Int -> a -> (a, a)

Evaluate the n'th Chebyshev polynomials of both kinds at a point X.

evalTsUs :: Num a => a -> ([a], [a])

Evaluate all the Chebyshev polynomials of both kinds at a point X.

tRoots :: Floating a => Int -> [a]

Compute the roots of the n'th Chebyshev polynomial of the first kind.

tExtrema :: Floating a => Int -> [a]

Compute the extreme points of the n'th Chebyshev polynomial of the first kind.

chebyshevFit :: Floating a => Int -> (a -> a) -> [a]

chebyshevFit n f returns a list of N coefficients cs such that f x ~= sum (zipWith (*) cs (evalTs x)) on the interval -1 < x < 1.

The N roots of the N'th Chebyshev polynomial are the fitting points at which the function will be evaluated and at which the approximation will be exact. These points always lie within the interval -1 < x < 1. Outside this interval, the approximation will diverge quickly.

This function deviates from most chebyshev-fit implementations in that it returns the first coefficient pre-scaled so that the series evaluation operation is a simple inner product, since in most other algorithms operating on chebyshev series, that factor is almost always a nuissance.

evalChebyshevSeries :: Num a => [a] -> a -> a

Evaluate a Chebyshev series expansion with a finite number of terms.

Note that this function expects the first coefficient to be pre-scaled by 1/2, which is what is produced by chebyshevFit. Thus, this computes a simple inner product of the given list with a matching-length sequence of chebyshev polynomials.