Skip to content

Add "augmented" Chebyshev nodes of the first kind #54

@pasabanov

Description

@pasabanov

Goal

Add "Chebyshev first kind augmented" point distribution {-1} ∪ {Chebyshev first kind points} ∪ {1}:

"Augmented" Chebyshev nodes of the first kind are the Chebyshev nodes of the first kind with additional points at the ends of the segment:

$$ x_i = \begin{cases} -1, & i = 0 \\ \cos{\frac{(2n-2i-1)\pi}{2n-2}}, & i = 1, ..., n - 1 \\ 1, & i = n \end{cases} $$

The points are translated from the segment $[-1, 1]$ to the segment $[a, b]$, so we can say that $x_1$ is equal to $a$, and $x_n$ is equal to $b$:
{a} ∪ {Chebyshev first kind points} ∪ {b}.

The barycentric weights for this nodes are given as (calculated them myself):

$$ w_i = \begin{cases} 1/2, & i = 0 \\ \displaystyle \frac{(-1)^i}{(n - 1) \sin{\frac{(2i - 1)\pi}{2n-2}}}, & i = 1, ..., n - 1 \\ (-1)^{n+1}/2, & i = n \end{cases} $$

Tasks

Metadata

Metadata

Assignees

Labels

featureNew feature or request

Type

Projects

No projects

Relationships

None yet

Development

No branches or pull requests

Issue actions