# Applied Mathematics

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🔬Range of Mathematical topics with practical approach related to various CS fields.

## 📃 Topics:

### 1. Introduction

* Introduction to MatLab and GNU Octave.
* [Basic code](https://github.com/MoElaSec/Applied_Math/blob/main/lab_code.m)

### 2. Floating point numbers

* Floating point systems (properties, rounding, errors).

### 3. SLE + Norms

* Systems of linear equation with error in the right-hand side.
* Matrix
* Norms.
  * 1'st Norm.
  * 2'nd Norm.
  * ♾ Norm
* condition number.

### 4. Numerical solution of linear systems

* Cholesky Decomposition
* LU decomposition
* PLU factorization

### 5. Least square approximation

* Collect the data.
* Select the most appropriate model. 
* Compute the best instance of the chosen model
* Use the model (predicting)
* Model Types:
  * linear model
  * polynomial model
  * trigonometric model
* [Gaussian normal-equation](https://github.com/MoElaSec/Applied_Math/blob/main/Scripts/Gussian_normal_equation.m) ![](https://latex.codecogs.com/svg.image?A^T.Ax\&space;=\&space;A^T.f)

### 6. Polynomial interpolation

* [Lagrangian interpolation](https://github.com/MoElaSec/Applied_Math/blob/main/Scripts/lagrangeRelated.m).
* Defining the Lagrange-polynomial in [Newton form](https://github.com/MoElaSec/Applied_Math/blob/main/Scripts/Newton_method_roots.m).
* Horner’s algorithm.
* Computing with `polyval` & `polyfit`.
* Hermite-interpolation.
* Piecewise interpolation.
* Piecewise Hermite-interpolation.
* Cubic spline interpolation.
* Using your [own spline function](https://github.com/MoElaSec/Applied_Math/blob/main/Scripts/myspline.m).

### 7. Numerical integration

* Approximating definite integrals.
* Interpolational quadrature formulas.
* Simple midpoint (tangent) rule.
* Simple [trapesoidal](https://github.com/MoElaSec/Applied_Math/blob/main/Scripts/mytrap.m) rule.
* Simpson’s simple rule.
* Compound rules.
* Compound mid-point rule.
* Compound trapesoidal rule.
* Simpson’s compund rule.
* Adaptive methods.
* [MatLab: Improper & Unknown Integrals calc + Anonymous Functions](https://github.com/MoElaSec/Applied_Math/blob/main/Scripts/integralex.m).

### 8. Eigenvalue & Eigenvectors + sparse systems

* Introduction to Complex Numbers.
* Defining Eigenvalues and eigenvectors.
* The stronger Gersgorin theorem.
* Power Iteration method.
* Defining `Rayleigh` of a matrix.  ![](https://latex.codecogs.com/svg.image?\mathit{\lambda\&space;\&space;=\&space;\frac{\(Av,\&space;v\)}{\(v,\&space;v\)}})
* Inverse-iteration & with shifting.
* Solving Examples like ( Page ranking & Leslie-model).

### 9. Numerical solution of nonlinear equations

* Defining Non-Learning equations.
* [Method of bisection (mid-point)](https://github.com/MoElaSec/Applied_Math/blob/main/Scripts/mybisect.m).
* Newton-Raphson method.
* Secant method
* Fixed point iteration Algorithm.

### 10. Systems minimization (optimization)

* Intro to Optimization
* Finding [local max](https://github.com/MoElaSec/Applied_Math/blob/main/Scripts/local_maximizer.m) & [local min](https://github.com/MoElaSec/Applied_Math/blob/main/Scripts/local_min_optimization.m). 
* fsolve for multivariate vector-function.
* fsolve for multivariate real-function.
* optimization with built-in functions.
* Intro to fibonacci sequence & golden ratio.
* [Golden section search](https://github.com/MoElaSec/Applied_Math/blob/main/Scripts/gold.m) & implementing an Algorithm.
* Using built-in MatLab function for optimization and 3D ploting.

### 11. Linear programming (LP)

* Intro to Linear Programming.
* Graphical Method.
* LP Normal and Canonical form.
* Simplex method.
* 2 phase Simplex method.
* Duality in linear programming.
* sensitivity analysis.
* Implementing Algorithms to solve real world problems (eg. transportation probelm).