Chebyshev Root Definitions

This file contains the definitions of Chebyshev polynomial roots. The roots of T_N are exactly {cos((2k+1)π/(2N)) | k = 0,...,N-1}.

Definitions

• chebyshevRoot: The k-th root of T_N(x), given by cos((2k+1)π/(2N))
• chebyshevRootsList: List of all N roots of T_N

Tags

Chebyshev polynomials, roots, trigonometry, Kim Morrison standard

def chebyshevRoot (N k : ℕ) : ℝ

The k-th Chebyshev root for polynomial T_N. This is cos((2k+1)π/(2N)), which satisfies T_N(chebyshevRoot N k) = 0 for k < N.

Equations

• chebyshevRoot N k = Real.cos ((2 * ↑k + 1) * Real.pi / (2 * ↑N))

def chebyshevRootsList (N : ℕ) : List ℝ

List of all Chebyshev roots for T_N. Contains the N values cos((2k+1)π/(2N)) for k = 0, 1, ..., N-1.

Equations

• chebyshevRootsList N = List.ofFn fun (k : Fin N) => chebyshevRoot N ↑k