chebyshevT

Chebyshev polynomial of the first kind.

Category: Mathematics > Polynomials

Syntax

P = chebyshevT(n, x)

Description

T_n(x) is defined by T_n(cos(theta)) = cos(n*theta). It satisfies the recurrence T_(n+1) = 2*x*T_n - T_(n-1) with T_0 = 1 and T_1 = x. On [-1, 1] it oscillates between -1 and 1 with n + 1 extrema, which makes it the basis of minimax polynomial approximation.

The zeros of T_n are the Chebyshev nodes, a good choice of interpolation points.

Input arguments

n

Degree, a nonnegative integer or a symbolic variable.

x

Point(s) of evaluation, or a symbolic variable to get the polynomial.

Output arguments

P

Numbers with the size of x, or a symbolic polynomial.

Example

Polynomials and values

syms x
chebyshevT(0:3, x)
chebyshevT(3, [0 0.5 1])

ans = 
[1, x, 2*x^2 - 1, 4*x^3 - 3*x]
ans = 1×3

     0    -1     1

See also

chebyshevU legendreP gegenbauerC hermiteH