Vibe coding: programming through conversation with artificial intelligence

1 What do they do, and what’s the result?

1.1.1

1.1.2

1.1.3

1.2.1

1.2.2

1.2.3

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2.1.2

2.1.3

2.2.1

2.2.2

3.1.1

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4.1.3

At first, we sample $f(x)$ in the $N$ ( $N$ is odd) equidistant points around $x^*$ :

$f_k = f(x_k),\: x_k = x^*+kh,\: k=-\frac{N-1}{2},\dots,\frac{N-1}{2}$

where $h$ is some step.
Then we interpolate points $\{(x_k,f_k)\}$ by polynomial

(1) $\begin{equation*} P_{N-1}(x)=\sum_{j=0}^{N-1}{a_jx^j} \end{equation*}$

Its coefficients $\{a_j\}$ are found as a solution of system of linear equations:

(2) $\begin{equation*} \left\{ P_{N-1}(x_k) = f_k\right\},\quad k=-\frac{N-1}{2},\dots,\frac{N-1}{2} \end{equation*}$

Here are references to existing equations: (1), (2).
Here is reference to non-existing equation (??).

Done