chebyshev_polynomial_t

→Tensor

chebyshev_polynomial_t(x: Tensor, n: int)

source

Chebyshev polynomial of the first kind, $T_n(x)$ .

The Chebyshev T polynomials are orthogonal on $[-1, 1]$ with weight $1/\sqrt{1 - x^2}$ and underlie spectral methods, minimax polynomial approximation, and Clenshaw-Curtis quadrature.

Parameters

xTensor

Argument tensor; any floating-point dtype. Customarily evaluated on

[-1, 1]

but defined for all real x.

nint

Non-negative polynomial degree.

Returns

Tensor

$T_n(x)$ element-wise, same shape and dtype as x.

Notes

Defined by $T_n(\cos\theta) = \cos(n\theta)$ and built by the three-term recurrence

T_0(x) = 1, \quad T_1(x) = x, \quad T_{k+1}(x) = 2x\, T_k(x) - T_{k-1}(x).

On $[-1, 1]$ , $|T_n(x)| \le 1$ ; the extrema lie at the Chebyshev nodes $\cos(j\pi/n)$ for $j = 0,\ldots,n$ . Raises ValueError for n < 0.

Examples

>>> import lucid
>>> from lucid.special import chebyshev_polynomial_t
>>> chebyshev_polynomial_t(lucid.tensor([-1.0, 0.0, 0.5, 1.0]), n=3)
Tensor([-1.0000, 0.0000, -0.5000, 1.0000])