User:ONE.059/Sandbox

$$ \begin{align} x_0 &\approx \sqrt[2]A \\ x_{k+1} &= \frac{x_k^2 + 3A}{3x_k^2 + A}x_k \\ \lim_{k \to \infty}x_k &= \sqrt[2]A \end{align} $$

$$ \begin{align} x_0 &\approx \sqrt[3]A \\ x_{k+1} &= \frac{x_k^3 + 2A}{2x_k^3 + A}x_k \\ \lim_{k \to \infty}x_k &= \sqrt[3]A \end{align} $$

$$ \begin{align} x_0 &\approx \sqrt[n]A \\ x_{k+1} &= \frac{(n - 1)x_k^n + (n + 1)A}{(n + 1)x_k^n + (n - 1)A}x_k \\ \lim_{k \to \infty}x_k &= \sqrt[n]A \end{align} $$

$$ \begin{cases} a^2 = h^2 + x^2 \\ c^2 = h^2 + (b - x)^2 \end{cases} $$

$$ \begin{align} a^2 - c^2 &= (h^2 + x^2) - (h^2 + (b - x)^2) \\ &= 2bx - b^2 \\ 2bx &= a^2 + b^2 - c^2 \\ x &= \frac{a^2 + b^2 - c^2}{2b} \end{align} $$

$$ \begin{align} a^2 &= h^2 + x^2 \\ &= h^2 + \left(\frac{a^2 + b^2 - c^2}{2b}\right)^2 \\ h^2 &= a^2 - \left(\frac{a^2 + b^2 - c^2}{2b}\right)^2 \\ &= \frac{(2ab)^2 - (a^2 + b^2 - c^2)^2}{4b^2} \\ h &= \sqrt{\frac{(2ab)^2 - (a^2 + b^2 - c^2)^2}{4b^2}} \\ &= \frac{\sqrt{(2ab)^2 - (a^2 + b^2 - c^2)^2}}{2b} \end{align} $$

$$ \begin{align} A &= \frac{b \times h}{2} \\ &= \frac{b \times \frac{\sqrt{(2ab)^2-(a^2 + b^2 - c^2)^2}}{2b}}{2} \\ &= \frac{\sqrt{(2ab)^2 - (a^2 + b^2 - c^2)^2}}{4} \end{align} $$

$$ \begin{bmatrix} a_0 \\ a_1 \\ \vdots \\ a_n \end{bmatrix} = \begin{bmatrix} 1 & x_0 & x_0^2 & \cdots & x_0^n \\ 1 & x_1 & x_1^2 & \cdots & x_1^n \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & x_n & x_n^2 & \cdots & x_n^n \\ \end{bmatrix}^{-1} \times \begin{bmatrix} f(x_0) \\ f(x_1) \\ \vdots \\ f(x_n) \end{bmatrix} $$