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Numerieke Differentiatie Formules

Numerieke Integratie Formules
 * $$V=\begin{bmatrix}

1 & \alpha_1 & \alpha_1^2 & \dots & \alpha_1^{n-1}\\ 1 & \alpha_2 & \alpha_2^2 & \dots & \alpha_2^{n-1}\\ 1 & \alpha_3 & \alpha_3^2 & \dots & \alpha_3^{n-1}\\ \vdots & \vdots & \vdots & \ddots &\vdots \\ 1 & \alpha_m & \alpha_m^2 & \dots & \alpha_m^{n-1} \end{bmatrix}$$
 * $$V_n = \begin{bmatrix}

x_1 & x_2 & \cdots & x_n \\ x_1^2 & x_2^2 & \cdots & x_n^2 \\ \vdots & \vdots & \ddots & \vdots \\ x_1^n & x_2^n & \cdots & x_n^n \end{bmatrix}$$
 * $$V=\begin{bmatrix}

0^0 & 0^1 & 0^2 & \dots & 0^n\\ 1^0 & 1^1 & 1^2 & \dots & 1^n\\ 2^0 & 2^1 & 2^2 & \dots & 2^n\\ \vdots & \vdots & \vdots & \ddots &\vdots \\ n^0 & n^1 & n^2 & \dots & n^n \end{bmatrix}$$
 * $$f(x) = \sum_{n=0}^{\infin} \frac{f^{(n)}(x_0)}{n!} (x-x_0)^{n}.$$
 * $$\begin{bmatrix} f_0 \\ f_1 \\ f_2 \\ \vdots \\ f_n \end{bmatrix}=\begin{bmatrix}

0^0 & 0^1 & 0^2 & \dots & 0^n\\ 1^0 & 1^1 & 1^2 & \dots & 1^n\\ 2^0 & 2^1 & 2^2 & \dots & 2^n\\ \vdots & \vdots & \vdots & \ddots &\vdots \\ n^0 & n^1 & n^2 & \dots & n^n \end{bmatrix}\begin{bmatrix} \frac{f^{(0)}(x_0)}{0!} h^0 \\ \frac{f^{(1)}(x_0)}{1!} h^1 \\ \frac{f^{(2)}(x_0)}{2!} h^2 \\ \vdots \\ \frac{f^{(n)}(x_0)}{n!} h^n \end{bmatrix}\Rightarrow\begin{bmatrix} f^{(0)}(x_0) \frac{h^0}{0!} \\ f^{(1)}(x_0) \frac{h^1}{1!} \\ f^{(2)}(x_0) \frac{h^2}{2!} \\ \vdots \\ f^{(n)}(x_0) \frac{h^n}{n!} \end{bmatrix}=\begin{bmatrix} 0^0 & 0^1 & 0^2 & \dots & 0^n\\ 1^0 & 1^1 & 1^2 & \dots & 1^n\\ 2^0 & 2^1 & 2^2 & \dots & 2^n\\ \vdots & \vdots & \vdots & \ddots &\vdots \\ n^0 & n^1 & n^2 & \dots & n^n \end{bmatrix}^{-1}\begin{bmatrix} f_0 \\ f_1 \\ f_2 \\ \vdots \\ f_n \end{bmatrix}$$
 * $$\begin{bmatrix}

\frac{1}{1!} & \frac{1}{2!} & \frac{1}{3!} & \dots & \frac{1}{n!}\\ 0 & \frac{1}{1!} & \frac{1}{2!} & \dots & \frac{1}{(n-1)!}\\ 0 & 0 & \frac{1}{1!} & \dots & \frac{1}{(n-2)!}\\ \vdots & \vdots & \vdots & \ddots &\vdots \\ 0 & 0 & 0 & \dots & \frac{1}{1!} \end{bmatrix}^{-1}\!\!\!\!=\,\begin{bmatrix} \frac{B_0}{0!} & \frac{B_1}{1!} & \frac{B_2}{2!} & \dots & \frac{B_{n-1}}{(n-1)!}\\ 0 & \frac{B_0}{0!} & \frac{B_1}{1!} & \dots & \frac{B_{n-2}}{(n-2)!}\\ 0 & 0 & \frac{B_0}{0!} & \dots & \frac{B_{n-3}}{(n-3)!}\\ \vdots & \vdots & \vdots & \ddots &\vdots \\ 0 & 0 & 0 & \dots & \frac{B_0}{0!} \end{bmatrix}$$

Constructing the interpolation polynomial
Suppose that the interpolation polynomial is in the form
 * $$p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0. \qquad (1) $$

The statement that p interpolates the data points means that
 * $$p(x_i) = y_i \qquad\mbox{for all } i \in \left\{ 0, 1, \dots, n\right\}.$$

If we substitute equation (1) in here, we get a system of linear equations in the coefficients $$a_k$$. The system in matrix-vector form reads
 * $$\begin{bmatrix}

x_0^n & x_0^{n-1} & x_0^{n-2} & \ldots & x_0 & 1 \\ x_1^n & x_1^{n-1} & x_1^{n-2} & \ldots & x_1 & 1 \\ \vdots & \vdots & \vdots & & \vdots & \vdots \\ x_n^n & x_n^{n-1} & x_n^{n-2} & \ldots & x_n & 1 \end{bmatrix} \begin{bmatrix} a_n \\ a_{n-1} \\ \vdots \\ a_0 \end{bmatrix} = \begin{bmatrix} y_0 \\ y_1 \\ \vdots \\ y_n \end{bmatrix}. $$ We have to solve this system for $$a_k$$ to construct the interpolant $$p(x).$$ The matrix on the left is commonly referred to as a Vandermonde matrix.
 * $$\det(V) = \prod_{i,j=0, i<j}^n (x_i - x_j) = \prod_{0 \le i < j \le n} (x_i - x_j)$$
 * $$L(x) := \sum_{j=0}^{k} y_j \ell_j(x)$$

of Lagrange basis polynomials
 * $$\ell_j(x) := \prod_{\begin{smallmatrix}0\le m\le k\\ m\neq j\end{smallmatrix}} \frac{x-x_m}{x_j-x_m} = \frac{(x-x_0)}{(x_j-x_0)} \cdots \frac{(x-x_{j-1})}{(x_j-x_{j-1})} \frac{(x-x_{j+1})}{(x_j-x_{j+1})} \cdots \frac{(x-x_k)}{(x_j-x_k)}.$$