Applied Mathematics

Applied Mathematics curriculum

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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

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

6. Polynomial interpolation

• Lagrangian interpolation.
• Defining the Lagrange-polynomial in Newton form.
• Horner’s algorithm.

• Computing with polyval & polyfit.

• Hermite-interpolation.

• Piecewise interpolation.

• Piecewise Hermite-interpolation.

• Cubic spline interpolation.

• Using your own spline function.

7. Numerical integration

• Approximating definite integrals.

• Interpolational quadrature formulas.

• Simple midpoint (tangent) rule.

• Simple trapesoidal 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.

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.
• 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).
• Newton-Raphson method.

• Secant method

• Fixed point iteration Algorithm.

10. Systems minimization (optimization)

• Intro to Optimization

• Finding local max & local min.
• fsolve for multivariate vector-function.

• fsolve for multivariate real-function.

• optimization with built-in functions.

• Intro to fibonacci sequence & golden ratio.

• Golden section search & 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).