e3x.nn.functions.chebyshev.basic_chebyshev

e3x.nn.functions.chebyshev.basic_chebyshev(x, num, limit=1.0)[source]

Basic Chebyshev polynomial basis functions.

Computes the basis functions

\[\begin{split}\mathrm{chebyshev}_k(x) = \begin{cases} 0 & x < 0 \\ \cos\left(k \arccos\left(2\frac{x}{l}-1 \right)\right), & 0 \le x \le l \\ 0 & x > l \,, \end{cases}\end{split}\]

where \(k=0 \dots K-1\) with \(K\) = num and \(l\) = limit.

Plot for \(K = 5\) and \(l = 1\):

../_images/e3x.nn.functions.chebyshev.basic_chebyshev_0_0.svg

Parameters:

x (<class 'Float[Array, '...']'>) – Input array.
num (int) – Number of basis functions \(K\).
limit (Union[Float[Array, ''], float], default: 1.0) – Basis functions are distributed between 0 and limit.

Return type:

<class 'Float[Array, '... num']'>

Returns:

Value of all basis functions for all values in x. The output shape follows the input, with an additional dimension of size num appended.