# Matematica | LABORATORY OF PROGRAMMING AND CALCULUS

## Matematica LABORATORY OF PROGRAMMING AND CALCULUS

 0512300006 DIPARTIMENTO DI MATEMATICA MATHEMATICS 2015/2016

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2010 SECONDO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/08 6 48 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
 BEATRICE PATERNOSTER T DAJANA CONTE
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 C 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)

COMMUNICATION SKILLS
THE COURSE WILL DEVELOP IN THE STUDENT THE CAPACITY TO MOTIVATE AND DEFEND HIS CHOICES IN THE SOLUTION OF THE CALCULUS PROBLEM, AND WILL SUPPORT THE DEVELOPMENT OF THE CAPACITY TO WORK IN A TEAM.

MAKING JUDGEMENTS:
THE STUDENT WILL HAVE TO BE ABLE TO CHOOSE THE MOST APPROPRIATE NUMERICAL MATHOD TO SOLVE THE PROBLEM UNDER EXAMINATION.
THE ANALYSIS OF THE USED MATHEMATICAL METHODS AND OF THE OBTAINED RESULTS, AIM TO DEVELOP MATURITY OF JUDGMENT AND CRITICAL SENSE.

LEARNING SKILLS:
THE COURSE WILL GIVE THE BASIC MEANS TO LEARN NEW NUMERICAL METHODS AND TO USE OR DEVELOP NEW MATHEMATICAL SOFTWARE.
Prerequisites
BASIC LINEAR ALGEBRA (VECTOR AND MATRIX COMPUTATION, LINEAR SYSTEMS ...) AND MATHEMATICAL ANALYSIS (LIMITS, DERIVATIVES). CAN BE USEFUL SOME PRELIMINARIES OF PROGRAMMING (VARIABLES AND COSTANTS, LOOPS, LOGICAL CONDITIONS, C LANGUAGE).
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 C PROGRAMS BASED ON THE MAIN STUDIED METHODS.
Teaching Methods
LECTURES,PRACTICES,LABORATORY, PROJECTS
Verification of learning
1) TEST AND EVALUATION OF MATHEMATICAL SOFTWARE BASED ON THE STUDIED NUMERICAL METHOS
2) ORAL EXAM ON THE THEORETICAL CONTENTS OF THE COURSE
Texts
G. MONEGATO – FONDAMENTI DI CALCOLO NUMERICO – ED. CLUT
A. MURLI, G. GIUNTA, G. LACCETTI, M. RIZZARDI - LABORATORIO DI PROGRAMMAZIONE I, LIGUORI EDITORE
A. MURLI - MATEMATICA NUMERICA: METODI, ALGORITMI E SOFTWARE, PARTE PRIMA, LIGUORI EDITORE
V. COMINCIOLI - ANALISI NUMERICA: METODI, MODELLI, APPLICAZIONI - ED. MC GRAW HILL