# 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

 DAJANA CONTE T
Objectives
KNOWLEDGE AND UNDERSTANDING:
THE AIM OF THE COURSE IS THE THEORETICAL KNOWLEDGE AND CRITICAL ANALYSIS OF THE MAIN NUMERICAL METHODS CONCERNING THE BASIC TOPICS OF NUMERICAL ANALYSIS: NUMERICAL SOLUTION OF LINEAR SYSTEMS AND OF NONLINEAR EQUATIONS.
PARTICULAR ATTENTION WILL BE PAID TO THE PRINCIPLES ABOUT THE DEVELOPMENT OF EFFICIENT MATHEMATICAL SOFTWARE IN MATLAB PROGRAMMING LANGUAGE, WITH REGARD TO THE ESTIMATE OF ACCURACY OF THE OBTAINED RESULTS AND THE EVALUATION OF THE PERFORMANCE OF THE DEVELOPED SOFTWARE.

APPLYING KNOWLEDGE AND UNDERSTANDING:
THE AIM OF THE COURSE IS TO MAKE THE STUDENT CAPABLE TO
•SOLVE PROBLEMS LINEAR SYSTEMS AND NONLINEAR EQUATIONS BY USING MATHEMATICAL SOFTWARE AND SUITABLE CALCULUS ENVIRONMENTS
•CHOOSE THE MORE APPROPRIATE NUMERICAL METHOD TO SOLVE THE PROBLEM UNDER EXAMINATION, BY MEANS OF THE ANALYSIS OF THE CHARACTERISTICS OF THE PROBLEM ITSELF.
•STUDY THE CONVERGENCE OF AN ITERATIVE METHOD
•RECOGNIZE ERRORS DERIVING FROM MACHINE OPERATIONS (IN FLOATING POINT ARITHMETIC)
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, SCIENTIFIC COMPUTING WITH MATLAB AND OCTAVE, SPRINGER
V. COMINCIOLI - ANALISI NUMERICA: METODI, MODELLI, APPLICAZIONI - ED. MC GRAW HILL