## Weekly outline

• ### Mathematical modelling/ Mathematical modelling

Pri Matematičnem modeliranju študenti (relativno) realne probleme rešujejo s pomočjo matematičnih modelov. Študenti se spoznajo s timskim delom in lahko sami, skupaj s kolegi v ekipi (in ob pomoči učiteljske ekipe) izpeljejo projekt od začetka do konca.

Predmet je razdeljen na tri glavna poglavja:

1. Linearni modeli: sistemi linearnih enačb, posplošeni inverzi matrik
2. Nelinearni modeli: vektorske funkcije vektorske spremenljivke, sistemi nelienarnih enačb, krivulje in ploskve
3. Dinamični modeli: diferencialne enačbe in dinamični sistemi

During the course of Mathematical Modelling the students will be solving (relatively) real life problems with the help of mathematical models. Students will be introduced to team work and will be given the opportunity to carry out a complete relatively complex project from the beginning to the very end  together with their colleagues (and with the help of the teaching team) .

The course consists of three main topics:

1. Linear models: systems of linear equations, generalized inverses of matrices
2. Nonlinear models: vector functions of a vector variable, nonlinear systems, curves and surfaces
3. Dynamic models: differential equations and dynamical systems
• ### 19 February - 25 February

Lectures: Introduction, a linear matematical model, some basics of linear algebra.

• ### 26 February - 3 March

Lectures: Generalized matrix pseudoinverses and their use in solving linear systems, the Moore-Penrose pseudoinverse.
• ### 4 March - 10 March

Lecture: SVD and MP inverse computation, underdetermined systems with many and overdetermined systems with no solutions.

Extra lecture (instead of the next week's one): Vector and matrix norm, PCA, introduction into nonlinear models (see the updated lecture notes) and this pdf.

• ### 11 March - 17 March

No lectures this week.
• ### 18 March - 24 March

Lectures: Vector functions. Derivative/Jacobian matrix. Linear approximation. Tangent method. Newton method. Gauss-Newton method.
• ### 25 March - 31 March

Lecture: Rate of convergence of Newton's method. Newton's optimization. Gradient descent. Broyden's method.

• Podroben opis reševanja 1. naloge iz tedna 5 in 1. naloge iz tedna 6.

• ### 1 April - 7 April

Lectures: Curves - different parametrizations, arc length. (Pages up to 115 in the notes)

• ### 8 April - 14 April

Lectures: Curves - different parametrizations, polar coordinates, arc length, area bounded by the curve, natural parametrization, curvature of parametric curves, plotting plane curves.

• ### 15 April - 21 April

Lectures: Parametrization of surfaces. Surface of revolution. Tangent plane. Differential equations: introduction.

• ### 22 April - 28 April

Lectures: Differential equations - Applications, separation of variables, first order linear ODE, variation of constants,

• ### 6 May - 12 May

Lectures: Orthogonal trajectories. Homogeneous DE, direction field, Euler's method, Runge-Kutta methods. DOPRI5.

• ### 13 May - 19 May

Lectures: Transforming an ODE of higher order into a system of first order ODE. Systems of differential equations. Autonomous linear systems. Numerical methods for systems.

• ### 20 May - 26 May

Lectures: Phase portraits, linearization of a nonlinear system, autonomous linear DE of higher order.

• ### 27 May - 2 June

Lectures: Wronskian determinant, nonhomogeneous second order DEs.