|SCIENTIFIC DISCIPLINARY SECTOR||MAT/08|
The course is an introduction to Numerical Analysis, and consists in the description of strategies and algorithms for the solution of basic mathematical problems. Particular importance is given to the use of the computer and to the study of the problems it entails. The teaching is completed by some exercises, carried out in groups, to solve simple problems with numerical techniques. These exercises are reserved for students of Materials Science, for whom the teaching is worth 6 CFU.
Knowledge and understanding of concepts and fundamentals of numerical computation.
Particular emphasis is given to:
• the understanding of the aspects related to the numerical solution of problems such as conditioning and stability;
• the understanding of the concept of approximate solution as a means to solve real problems.
The main purpose of the teaching of Numerical Calculus and Programming is to provide tools for the calculation or approximation of the solution of basic mathematical problems. The main objective is to shift the point of view, in dealing with mathematical problems, from a completely abstract field to a more applied one, in order to prepare the student to deal with problems deriving from the study of real phenomena. Particular attention is paid to basic numerical concepts, such as the conditioning of a problem and the stability of an algorithm, and to the critical interpretation of the results. Specifically, the student will be able: - to solve linear systems of any size; - to calculate the regression line associated with a set of points; - to analyze errors due to perturbed data and / or floating point arithmetic; - to approximate the zeros of a function; - to find the polynomial that interpolates a set of points.
The basic concepts of analysis, analytical geometry and trigonometry taught in high school.
Theoretical lessons: 4 CFU (32 over two semesters). These lessons are provided both for students of "Scienza dei Materiali" and for students of "Chimica e Tecnologie Chimiche". At the end of each lesson, summary slides are published on Aulaweb. Quizzes on aulaweb allow each student to self-evaluate. Exercises. For students of Materials Science an additional 2 CFU are awarded (16 hours) (1 for each semester). During the first semester, by applying the innovative teaching technique "Team Based Learning" (TBL), specific numerical problems are analyzed and solved. During the second semester an interdisciplinary exercise is proposed, to be solved using notions of Chemistry, Physics and Mathematics. Lectures and exercises are held in-person. Should the health emergency persist, to safeguard everyone's health and safety, the lessons will be delivered online on the Teams platform (and recorded).
The program covers topics from different areas:
Matrix operations, vector and matrix norms.
Solution of linear systems: backward substitution method for triangular systems, the Gauss method.
The condition number of a matrix.
Overdetermined systems: the method of the normal equations. The regression line.
Error Analysis: the use of the floating point arithmetic and algorithmic errors, the cancellation and round-off error. Conditioning of the problem of in the evaluating a real function.
Solution of nonlinear equations: the bisection method, the Newton-like methods.
Interpolation: the interpolating polynomial in the Lagrange form.
Bevilacqua-Bini-Capovani-Menchi: “Introduzione alla Matematica Computazionale”, Zanichelli
Bini-Capovani-Menchi: “Metodi Numerici per l’Algebra Lineare”, Zanichelli
Handouts provided by the teacher and available on AulaWeb at the address
Office hours: By appointment by sending an email to fassino at dima.unige.it
CLAUDIA FASSINO (President)
The course is developed on the first and second semester, following the timetable set out in the "Manifesto"
The exam consists of two parts. 1) Written exam: exercises concerning the whole theory developed during classroom lessons. In order to have access to the written exam, it is necessary to have passed some quizzes, published on Aulaweb during both semesters. 2) Oral exam: questions regarding the theory, with particular attention to theorems and proofs. Final mark: it is given by the average of the mark of the written and of the oral mark. For students of Materials Science, the grade of the written exam is increased with the bonus obtained with the TBL and with the interdisciplinary exercise. The written test and the oral test can be taken in two ways: - finished the lessons and then cover the entire program - or the exam can be divided into two parts. After the end of the first semester, the written and oral tests concerning the program carried out in the first semester can be taken and after the end of the second semester, the written and oral tests concerning the program carried out in the second semester can be taken. The average of the marks in the two parts provides the final mark. After the end of the lessons (from June onwards) the exam can always be taken in two separate sessions, and the grade of the passed part does not expire. It is required to pass the exam related to the first part of the course, to take the exam related to the second part. In the event of the pandemic emergency, the written exam will be replaced by a more consistent set of quizzes and the oral exam will be in remote mode.
Students with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the teacher at the beginning of the course to agree on teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools.
The guided exercises aim to verify the ability to solve, from a numerical point of view, simple mathematical problems. The written test is based on the solution of exercises related to the theory carried out in the classroom, to ascertain the ability to analyze and solve a numerical problem. The oral test aims to verify the understanding of the theory part, with particular attention to the proof of the theorems.