Academic Year 2021-2022

COMPUTATIONAL MATHEMATICS LABORATORY

Teachers

Davide Liessi
Dimitri Breda
Unit Credits
6
Teaching Period
Second Period
Course Type
Characterizing
Prerequisites. Advisable a basic preparation in Mathematics:

– analysis: real and complex numbers, functions, limits, derivatives, integrals

– algebra: vectors, matrices, linear systems

– numerical analysis: methods for linear systems, Newton method, interpolation.

The presentation modalities and the course structure itself allow attendance and learning also for those who possibly do not possess some of the above requisites.

Teaching Methods. – theoretical lectures

– laboratory activities and exercises on learning and using mathematical software

– possible seminars on specific arguments.

Verification of Learning. The exam consists in:

– laboratory project of computational character, on a subject to be arranged with the lecturer, accompanied by a brief written essay including a brief theoretical description of the problem, of the targets of the project and of what done and obtained;

– oral discussion on the developed activities and on the course program, with questions on both theory and applications, and exercises on computational problems.

More Information. – the credits are valid for Percorso Formativo 24 CFU – DM 616/2017 (A-28 “Matematica e Scienze”)

– teaching language: Italian (the course can be taught in English on proposal of the competent didactic structure)

– the arguments proposed for the project can be related to the final thesis for what concerns the possible computational aspects

– the course notes, written by the lecturer in English, are complete and self-contained relatively to the program, and include laboratory activities and relevant codes.

Objectives
The course aims at getting the student adequately used to the utilization of computing facilities as effective and helpful tools towards the theoretical study of mathematics and of its related teaching, application and research activities. It therefore consists in solving experimentally mathematical problems arising in diverse application contexts, driving the student from modelling to coding.

The student will have to:

Knowledge and comprehension:

– know the basic aspects of the mutual interaction between mathematics and computer

– understand the class of mathematical problems in which the model resides

– learn the guidelines to translate the mathematical problem into a computable one

Capacity of applying knowledge and comprehension:

– develop skills in self-learning general mathematical software

– know how to select the mathematical software best suited to the solution of the problem and to program the relevant codes for obtaining the solution itself

Autonomy of judgement:

– be able to analyze in a critical and autonomous manner computer results in relation with theoretical expectation

Communication skills:

– know how to illustrate the computational processes in a clear and comprehensible fashion

– know how to represent effectively the computational results

Learning skills:

– know how to tackle critically and autonomously mathematical problems with computational techniques

Contents
After a general introduction on computational mathematics, machine numbers and analysis of the errors that can be generated in the path from the modelling of the mathematical problem to the computation of its solution, the plan is to divide the course in separate units, distinguished by argument, each of which includes a part of theoretical introduction to the treated subject and to its present implications, followed by a part of development, implementation and testing on a computing system. For every unit, the use of different mathematical software and environments (e.g., GeoGebra, Mathematica, MATLAB) is programmed, as tools for both numerical and symbolic computation, and graphical visualization and data management. Among the potential themes are:

– nonlinear equations and Newton method (application proposal: Newton fractals);

– matrices and linear systems (application proposal: Google’s PageRank);

– differential equations and dynamical systems (application proposal: bifurcation analysis and chaos);

– approximation of data and functions (application proposal: FFT and JPEG compression).

The treated arguments will thus furnish an occasion to tackle some of the issues concerning the interaction with a computing system and its potentialities/limitations with respect to the study of mathematical problems: representation of mathematical objects and their manipulation, problem solving and simulation of models, data management.

Texts
Materials given by the lecturer. References will be given during the course depending on the specific argument. Course notes will be given, written by the lecturer in English, including laboratory activities, relevant codes and specific references with respect to the treated arguments, made available on the e-learning webpage of the course subject to registration (https://elearning.uniud.it/moodle/).