Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts.

$$l_k(x)= \prod_{i=0,, i\neq k}^{n} \frac{x-x_i}{x_k-x_i}=\frac{x-x_0}{x_k-x_0} \cdots \frac{x-x_{k-1}}{x_k-x_{k-1}} \frac{x-x_{k+1}}{x_k-x_{k+1}} \cdots \frac{x-x_{n}}{x_k-x_{n}}$$

$$l_2(1.4)=0.9854$$

$$\left[\begin{array}{ccccc} x_0=0 & f[x_0]=1 & & & \cr x_1=2 & f[x_1]=5 & f[x_0,x_1]=\displaystyle\frac{5-1}{2-0} = 2& & \cr x_2=4 & f[x_2]=17 & f[x_1,x_2]=\displaystyle\frac{17-5}{4-2}=6 & f[x_0,x_1,x_2]= \displaystyle\frac{6-2}{4-0}=1 & \end{array}\right]$$

Then

$$\begin{array}{rcl} P_2(x)&=&f[x_0]+f[x_0,x_1]x+f[x_0,x_1,x_2]x(x-2)\\ &=&1+2x+x(x-2)\\ &=&1+x^2 \end{array}$$

$$f^ \prime(x) = \frac{f(x + \Delta x) - f(x)}{\Delta x}$$

$$f^{\prime \prime}(x) = \frac{f(x + 2\Delta x) - 2f(x + \Delta x) + f(x)}{(\Delta x)^2}$$

$$f^ \prime(x) = \frac{f(x) - f(x - \Delta x)}{\Delta x}$$

$$f^ \prime(x) = \frac{f(x + \Delta x) - f(x - \Delta x)}{2\Delta x}$$

$$f^{\prime \prime}(x) = \frac{f(x + \Delta x) + f(x - \Delta x) - 2f(x)}{(\Delta x)^2}$$

$$\int_{a}^b f(x)dx=\frac{h}{2}(f(a)+f(b)+2\sum_{i=0}^{n-1}f(a + ih))$$ Where:

$$h = \frac{b - a}{numberOfSegments}$$

$$\int_{a}^b f(x)dx=\frac{h}{3}(f(x)+f(x_n)+2\sum_{i(even)=0}^{n-2}f(x_i)+4\sum_{i(odd)=0}^{n-1}f(x_i))$$ Where:

$$h = \frac{b - a}{numberOfSegments}$$

$$\int_{a}^b f(x)dx=\frac{b - a}{2}(f(\frac{b - a}{2} * \frac{1}{\sqrt{3}} + \frac{b + a}{2}) + f(\frac{b - a}{2} * \frac{-1}{\sqrt{3}} + \frac{b + a}{2}))$$

$$\int_{a}^b f(x)dx=(b - a)f(\frac{b + a}{2})$$

$$\begin{bmatrix} \sum_{1}^{n} 1 & \sum_{1}^{n} x_i \\ \sum_{1}^{n} x_i & \sum_{1}^{n} (x_i)^2 \end{bmatrix} \begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} \sum_{1}^{n} y_i \\ \sum_{1}^{n} x_i y_i \end{bmatrix}$$

$$y_{i+1} = y_i + hf(x_i, y_i)$$

$$y_{i+1} = y_i + hf(x_i, y_i) + \frac{h^2}{2}f^\prime(x_i, y_i)$$

$$k_1 = f(x_i, y_i)$$

$$k_2 = f(x_i + p_1h, y_i + q_1 k_1 h)$$

$$y_{i+1} = y_i + h(a_1 k_1 + a_2 k_2)$$

$$k_1 = f(x_i, y_i)$$

$$k_2 = f(x_i + \frac{h}{2}, y_i + \frac{k_1 h}{2})$$

$$k_3 = f(x_i + \frac{h}{2}, y_i + \frac{k_2 h}{2})$$

$$k_4 = f(x_i + h, y_i + k_3 h)$$

$$y_{i+1} = y_i + \frac{h}{6}(k_1 + 2k_2 + 2k_3 + k_4)$$

• Let $x_0$ be an initial approximation to the eigenvector.
• For $k=1,2,3,\ldots$ do
  • Compute $x_k=Ax_{k-1}$,
  • Normalize $x_k=x_k/|x_k|_\infty$
• Stop when tolerance is $t$

$$F(x) = 0$$

$$x = G(x)$$

$$x^{(k)} = G(x^{(k -1)})$$

$$x^{(k)} = G(x^{(k)})$$

$$-F(x^{(k - 1)}) = J(x^{(k - 1)})y^{(k - 1)}$$

$$x^{(k)} = x^{(k - 1)} + y^{(k - 1)}$$

where

$$ J = \begin{bmatrix} \frac{\partial F_1}{\partial x_1} & \frac{\partial F_1}{\partial x_2} & \cdots & \frac{\partial F_1}{\partial x_n} \\ \frac{\partial F_2}{\partial x_1} & \frac{\partial F_2}{\partial x_2} & \cdots & \frac{\partial F_2}{\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial F_m}{\partial x_1} & \frac{\partial F_m}{\partial x_2} & \cdots & \frac{\partial F_m}{\partial x_n} \\ \end{bmatrix} $$

* To get the version Excute: $ python --version Python 3.9.6

if not installed watch the following video

* To get the version Excute: $ git --version git version 2.28.0.windows.1

if not installed watch the following video

$ mkdir Numerical-Methods $ python -m venv venv```

For Windows Users:

$ venv\Scripts\activate.bat 

For Linux Users:

$ source venv/bin/activate 

Then

$ git clone https://github.com/michaelehab/Numerical-Methods && cd Numerical-Methods $ pip install -r requirements.txt