Applied Mathematics
Applied Mathematics curriculum
🔬Range of Mathematical topics with practical approach related to various CS fields.
📃 Topics:
1. Introduction
Introduction to MatLab and GNU Octave.
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
6. Polynomial 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.
8. Eigenvalue & Eigenvectors + sparse systems
Introduction to Complex Numbers.
Defining Eigenvalues and eigenvectors.
The stronger Gersgorin theorem.
Power Iteration method.
Inverse-iteration & with shifting.
Solving Examples like ( Page ranking & Leslie-model).
9. Numerical solution of nonlinear equations
Defining Non-Learning equations.
Newton-Raphson method.
Secant method
Fixed point iteration Algorithm.
10. Systems minimization (optimization)
Intro to Optimization
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).
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