I have very clear question. I'm searching about polynomial approximation.
In this formula, it say's $$f = (z-z_2)(z-z_3).F_1 + (z-z_3)(z-z_1).F_2 + (z-z_1)(z-z_2)F_3$$
where $F_1 = \frac{f_1}{(z_1-z_2)(z_1-z_3)}$, etc.
I searched it and when we write quadratic approximation we can use taylor series and we can write f like this:
$$f(x) = f(a) + (x-a)\frac{f'(a)}{1!} + (x-a)^2\frac{f''(a)}{2!} +\ldots $$
How it converts $f = (z-z_2)(z-z_3)\cdot F_1 + (z-z_3)(z-z_1)\cdot F_2 + (z-z_1)(z-z_2)\cdot F_3$. What is the source of this formula? It doesn't similar to taylor series. I'm searching it for $3$ days but I can't understand how can we write it? What's the relation between this f expression and taylor series? Can anybody help me please?
That is the formula for Langrange polynomial:
Given a set of $k+1$ data points :$(x_0, y_0),\ldots,(x_j, y_j),\ldots,(x_k, y_k)$ where no two $x_j$ are the same, the '''interpolation polynomial in the Lagrange form''' is a linear combination $$L(x) := \sum_{j=0}^{k} y_j \ell_j(x)$$ of Lagrange basis polynomials :$$\ell_j(x) := \prod_{\begin{smallmatrix}0\le m\le k\\ m\neq j\end{smallmatrix}} \frac{x-x_m}{x_j-x_m} = \frac{(x-x_0)}{(x_j-x_0)} \cdots \frac{(x-x_{j-1})}{(x_j-x_{j - 1})} \frac{(x-x_{j+1})}{(x_j-x_{j+1})} \cdots \frac{(x-x_k)}{(x_j-x_k)},$$ where $0\le j\le k$.
In your context, you have $3$ points, hence we get a quadratic interpolation polynomial. It is the quadratic polynomial that passes through the $3$ points.