# Matematica | LABORATORY OF PROGRAMMING AND CALCULUS

## Matematica LABORATORY OF PROGRAMMING AND CALCULUS

 0512300006 DIPARTIMENTO DI MATEMATICA EQF6 MATHEMATICS 2020/2021

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2018 SECONDO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/08 6 48 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS

 ANGELAMARIA CARDONE T
Objectives
CONOSCENZA E CAPACITÀ DI COMPRENSIONE (KNOWLEDGE AND UNDERSTANDING):
IL CORSO È FINALIZZATO AD ACQUISIRE LA CONOSCENZA TEORICA E AD ANALIZZARE CRITICAMENTE I PRINCIPALI METODI NUMERICI RELATIVI AD ARGOMENTI DI BASE DI ANALISI NUMERICA: RISOLUZIONE DI SISTEMI LINEARI E DI EQUAZIONI NON LINEARI. PARTICOLARE ATTENZIONE SARÀ DATA AI PRINCIPI SU CUI SI BASA LO SVILUPPO DI SOFTWARE MATEMATICO EFFICIENTE NEL LINGUAGGIO DI PROGRAMMAZIONE MATLAB, CON RIFERIMENTO ALLA STIMA DELL'ATTENDIBILITÀ DEI RISULTATI OTTENUTI ED ALLA VALUTAZIONE DELLE PRESTAZIONI DEL SOFTWARE SVILUPPATO.

CAPACITÀ DI APPLICARE CONOSCENZA E COMPRENSIONE (APPLYING KNOWLEDGE AND UNDERSTANDING):
IL CORSO HA L'OBIETTIVO DI RENDERE LO STUDENTE CAPACE DI
•RISOLVERE SISTEMI DI EQUAZIONI LINEARI ED EQUAZIONI NON LINEARI MEDIANTE L'UTILIZZO DI METODI NUMERICI E DEL RELATIVO SOFTWARE MATEMATICO
•SCEGLIERE IL METODO NUMERICO PIÙ IDONEO AL PROBLEMA IN ESAME ATTRAVERSO L’ANALISI DELLE CARATTERISTICHE DEL PROBLEMA STESSO
•STUDIARE LA CONVERGENZA DI UN METODO ITERATIVO
•RICONOSCERE ERRORI DERIVANTI DA OPERAZIONI MACCHINA (IN ARITMETICA A VIRGOLA MOBILE)
Prerequisites
BASIC LINEAR ALGEBRA (VECTOR AND MATRIX COMPUTATION, LINEAR SYSTEMS ...) AND MATHEMATICAL ANALYSIS (LIMITS, DERIVATIVES).
Contents
SOLVING A PROBLEM ON A COMPUTER: FROM THE REAL PROBLEM TO THE METHOD, TO THE ALGORITHM, TO THE PROGRAM, TO THE ANALYSIS OF RESULTS. ERROR SOURCES AND ERROR PROPAGATION. CONDITIONING OF A PROBLEM AND STABILITY OF AN ALGORITHM, SOURCES OF ERROR IN COMPUTATIONAL MODELS, MACHINES REPRESENTATION OF NUMBERS, THE FLOATING POINT NUMBER SYSTEM AND ARITHMETIC.

EVALUATION OF AN ALGORITHM, SPACE AND TIME COMPLEXITY. EXAMPLES: COMPUTATION OF A DETERMINANT.

VECTOR SPACES, NORMS. SYMMETRIC DEFINITE POSITIVE MATRICES, SYLVESTER CRITERION.

CONDITIONING OF LINEAR SYSTEMS. DIRECT METHODS FOR SOLVING LINEAR SYSTEMS. SOLUTION OF TRIANGULAR SYSTEMS, BACK AND FORWARD SUBSTITUTION, COMPUTATIONAL COST. GAUSSIAN ELIMINATION METHOD. PIVOTING. LU FACTORIZAZION. CHOLESKY FACTORIZAZION.

ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS: FORMULATIONS, CONVERGENCE, JACOBI, GAUSS SEIDEL AND SOR RELAXATION METHODS. ALGORITHMS BASED ON ITERATIVE METHODS: ERROR ESTIMATION AND STOP CRITERIA.
SOLUTION OF NONLINEAR EQUATIONS. BISECTION METHOD. LOCAL LINEARIZATION METHODS. SECANT AND TANGENT (NEWTON) METHODS. CONVERGENCE. NEWTON'S METHOD FOR MULTIPLE ROOTS. FIXED POINT ITERATION. COMPUTATIONAL ASPECTS.
CONDITIONING IN THE COMPUTATION OF THE ROOTS OF A POLYNOMIAL.

DEVELOPMENT OF ALGORITHMS AND OF MATLAB PROGRAMS BASED ON THE MAIN STUDIED METHODS.
Teaching Methods
THE LECTURES (6 CFU, 48 CLASS HOURS) INCLUDE FRONTAL LESSONS AND EXERCISES.

THE FRONTAL LESSONS ILLUSTRATE THE METHODOLOGIES AND THE ALGORITHMS. DURING THE EXERCISES, THE ALGORITHMS WILL BE IMPLEMENTED IN SCIENTIFIC COMPUTING ENVIRONMENTS AND TESTED ON SIGNIFICANT TEST EXAMPLES. THE STUDENTS WILL BE GUIDED IN THE VERIFICATION OF THE ACCURACY, STABILITY AND EFFICIENCY OF THE NUMERICAL METHODS ADOPTED.

MOREOVER, THE TEACHING WILL ALSO USE THE SPECIAL TOOLS AVALILABLE IN THE E-LEARNING PLATFORM PROVIDED BY THE COURSE OF STUDIES (IN PARTICULAR RESOURCES, QUIZZES, FORUMS).
Verification of learning
THE EXAM TEST EVALUATES THE ACQUIRED KNOWLEDGE AND THE ABILITY TO APPLY IT TO SOLVING TYPICAL PROBLEMS OF NUMERICAL ANALYSIS, ALSO THROUGH MATHEMATICAL SOFTWARE WRITTEN IN MATLAB LANGUAGE.

IT IS DIVIDED INTO TWO TRIALS: A PRACTICAL TEST IN WHICH THE MATHEMATICAL SOFTWARE DESIGNED AND CONSTRUCTED DURING THE COURSE IS USED TO SOLVE SYSTEMS OF LINEAR EQUATIONS BY DIRECT AND ITERATIVE METHODS, AS WELL AS NONLINEAR EQUATIONS BY MEANS OF ITERATIVE LINEAR LOCALIZATION METHODS ; AN ORAL INTERVIEW, WITH THE PURPOSE OF ASSESSING THE THEORETICAL KNOWLEDGE PRESENTED IN THE LESSONS.
Texts
G. MONEGATO – FONDAMENTI DI CALCOLO NUMERICO – ED. CLUT
A. MURLI, G. GIUNTA, G. LACCETTI, M. RIZZARDI - LABORATORIO DI PROGRAMMAZIONE I, LIGUORI EDITORE
A. QUARTERONI, F. SALERI, CALCOLO SCIENTIFICO: ESERCIZI E PROBLEMI RISOLTI CON MATLAB E OCTAVE, SPRINGER.
V. COMINCIOLI - ANALISI NUMERICA: METODI, MODELLI, APPLICAZIONI - ED. MC GRAW HILLG. MONEGATO – FONDAMENTI DI CALCOLO NUMERICO – ED. CLUT
A. MURLI, G. GIUNTA, G. LACCETTI, M. RIZZARDI - LABORATORIO DI PROGRAMMAZIONE I, LIGUORI EDITORE
A. QUARTERONI, F. SALERI, CALCOLO SCIENTIFICO: ESERCIZI E PROBLEMI RISOLTI CON MATLAB E OCTAVE, SPRINGER
V. COMINCIOLI - ANALISI NUMERICA: METODI, MODELLI, APPLICAZIONI - ED. MC GRAW HILL