# LABORATORY FOR COMPUTATIONAL METHODS

## Learning outcomes of the course unit

The objective of the course is to familiarize with implementation of numerical methods for scientific calculus in mathand physic field using Matlab as programming languages.

At the end of the course the student should be able to understand and develop some proposed numerical algorithms, proving his ability with the acquired notions in understanding and solving some problems in physics. Students are required to have the capability to evaluate, give emphasis and communicate their own results through short and schematic written presentations including numerical codes, images and meaningful graphs.

## Prerequisites

Basic notions of mathematical calculus and of classical physics.

## Course contents summary

The basic contents of the course concern the elements of numerical analysis aimed to solve elementary problems in Physics in both experimental and theoretical fields. Numerical codes are developed in Matlab language.

The first part of the course is devoted to the Matlab language notions and to the basic numerical algorithms; several examples are explained during laboratory lessons. As example, integral calculus by numerical quadrature, and by Montecarlo method.

The main subjects in experimental physics are: non-linear interpolation of experimental data, spectral signal analysis. The pendulum motion is deeply analyzed in order to compare real data with simulated ones.

Numerical algorithms to solve ordinary differential equations are also treated to solve physical problems such as the motion of coupled non-linear oscillators, the circular restricted three body problem, the planetary orbits, the stability and caos in Hamiltonian systems.

## Course contents

Elements of Programming in Matlab.

Numerical algoritms: root finding, solution of linear algebraic equations, polynomial interpolation, least squares problem, formulas of numerical integration, random numbers, Monte Carlo method, integration of ordinary differential equations.

Numerical codes: data analyses and least squares method, numerical calculus in one or more dimensions and comparison between different algorithms, Monte Carlo simulation of a physics experiment, solution of ordinary differential equations and comparison between different algorithms in the case of easy problems of classical physics: simple pendulum with friction; variable length pendulum; non-linear coupled oscillators; gravitational two -body problem; three-body problem (Sun,Earth,Moon); n-body problem; basic concepts on the molecular dynamics; basic concepts on stability and chaos in hamiltonian systems.

## Recommended readings

At the end of each lesson, the teacher gives slides, some lecture notes, numerical codes and suggests some readings on web. The students have to sign lto the course on line, on Elly platform in order to keep all the material.

## Teaching methods

Both lectures and computer training in laboratory in order to develop numerical codes in Matlab. The teacher will supply all explained numerical codes on Elly platform weekly.

## Assessment methods and criteria

During the course, the students are required to produce and present their works weekly. They can provide their work putting it on an assigned computer area where the teacher will correct and discuss iit with the students.

Final oral evaluation relies on developing and discussing numerical algorithms introduced during the course; moreover, the student is required to develop an original code that solve a physics problem weekly discussed with the teacher during laboratory activities. For the final examination the student is required to prepare a brief presentation (in latex, word, power point) concerning the numerical codes developed during laboratory activity and the main results obtained. In order to reach the sufficiency is necessary to produce all the numerical codes developed during laboratory activity. During lessons, the teacher indicates the minimum results in order to reach the sufficiency and any further insights.

The final evaluation will be communicate at the end of the oral examination.

In order to attend the exam, it is COMPULSORY to register online.