shifted_chebyshev_polynomial_u

→Tensor

shifted_chebyshev_polynomial_u(x: Tensor, n: int)

source

Shifted Chebyshev U polynomial $U^*_n(x)$ on $[0, 1]$ .

Standard Chebyshev U polynomials shifted from $[-1, 1]$ onto the unit interval by $x \mapsto 2x - 1$ .

Parameters

xTensor

Argument on

[0, 1]

; any floating-point dtype.

nint

Non-negative polynomial degree.

Returns

Tensor

$U^*_n(x) = U_n(2x - 1)$ , element-wise.

Notes

$U^*_n(x) = U_n(2x - 1)$ . Orthogonal on $[0, 1]$ with weight $\sqrt{x - x^2}$ .

Examples

>>> import lucid
>>> from lucid.special import shifted_chebyshev_polynomial_u
>>> shifted_chebyshev_polynomial_u(lucid.tensor([0.0, 0.5, 1.0]), n=2)
Tensor([1.0000, -1.0000, 1.0000])