Matematica | NUMERICAL CALCULUS I

cod. 0512300012

NUMERICAL CALCULUS I

	0512300012
	DIPARTIMENTO DI MATEMATICA

	MATHEMATICS
	2013/2014

CURRICULUM MATEMATICA AD INDIRIZZO APPLICATIVO

	OBBLIGATORIO
	YEAR OF COURSE 2
	YEAR OF DIDACTIC SYSTEM 2010
	SECONDO SEMESTRE

TEACHING UNITS

SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
MAT/08	5	40	LESSONS	COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
MAT/08	2	24	EXERCISES	COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS

	Objectives
	1. KNOWLEDGE AND UNDERSTANDING THE AIM OF THE COURSE IS TO DISCUSS THE MAIN PROBLEMS IN THE DEVELOPMENT OF EFFICIENT MATHEMATICAL SOFTWARE. IT IS DEVOTED TO THE THEORETICAL KNOWLEDGE AND A CRITICAL ANALYSIS OF THE MAIN NUMERICAL METHODS FOR THE BASIC ARGUMENTS, LIKE THE APPROXIMATION OF FUNCTIONS AND OF NUMERICAL DATA, THE COMPUTATION OF DEFINITE INTEGRALS, THE NUMERICAL SOLUTION OF NONLINEAR SYSTEMS AND THE COMPUTATION OF THE EIGENVALUES OF A MATRIX. MOREOVER THE STUDENT WILL LEARN THE METHODOLOGY FOR THE PROJECT OF EFFICIENT ALGORITHM AND THE USAGE OF SUITABLE ENVIRONMENTS FOR NUMERICAL COMPUTATION TO SOLVE PROBLEMS OF SCIENTIFIC COMPUTATION. 2. APPLYING KNOWLEDGE AND UNDERSTANDING BY MEANS OF THE PRACTICE IN THE LABORATORY, THE STUDENTS WILL BE ABLE TO EXPERIMENT NUMERICAL METHODS, TO ESTIMATE THE RELIABILITY OF THE OBTAINED RESULTS, TO DEVELOP ELEMENTS OF MATHEMATICAL SOFTWARE AND EVALUATE PERFORMANCES. 3. MAKING JUDGEMENTS THE COURSE AIMS TO DEVELOP THE STUDENTS’ SKILL TO CHOOSE THE MOST SUITABLE NUMERICAL METHOD FOR ANY SPECIFIC PROBLEM TO SOLVE, AND TO COMPARE THE PERFORMANCES OF THE CODES BASED ON DIFFERENT NUMERICAL METHODS, RESPECT TO EFFICIENCY, ACCURACY AND STABILITY. 4. COMMUNICATION SKILLS THE STUDENT WILL BE ABLE TO ILLUSTRATE THE KNOWN NUMERICAL METHODS, THEIR THEORETICAL PROPERTIES AND THE RELATED ALGORITHMS, AND HE SHOULD TO EXPLAIN THE PERFORMANCES OF THE NUMERICAL METHODS APPLIED TO SPECIFIC PROBLEMS. HE WILL BE ABLE TO WRITE THE INTERNAL AND THE EXTERNAL DOCUMENTATION OF THE DEVELOPED SOFTWARE. 5. LEARNING SKILLS THE COURSE WILL GIVE THE BASIC MEANS TO LEARN NEW NUMERICAL METHODS AND TO USE OR DEVELOP NEW MATHEMATICAL SOFTWARE.

1. KNOWLEDGE AND UNDERSTANDING
THE AIM OF THE COURSE IS TO DISCUSS THE MAIN PROBLEMS IN THE DEVELOPMENT OF EFFICIENT MATHEMATICAL SOFTWARE. IT IS DEVOTED TO THE THEORETICAL KNOWLEDGE AND A CRITICAL ANALYSIS OF THE MAIN NUMERICAL METHODS FOR THE BASIC ARGUMENTS, LIKE THE APPROXIMATION OF FUNCTIONS AND OF NUMERICAL DATA, THE COMPUTATION OF DEFINITE INTEGRALS, THE NUMERICAL SOLUTION OF NONLINEAR SYSTEMS AND THE COMPUTATION OF THE EIGENVALUES OF A MATRIX. MOREOVER THE STUDENT WILL LEARN THE METHODOLOGY FOR THE PROJECT OF EFFICIENT ALGORITHM AND THE USAGE OF SUITABLE ENVIRONMENTS FOR NUMERICAL COMPUTATION TO SOLVE PROBLEMS OF SCIENTIFIC COMPUTATION.
2. APPLYING KNOWLEDGE AND UNDERSTANDING
BY MEANS OF THE PRACTICE IN THE LABORATORY, THE STUDENTS WILL BE ABLE TO EXPERIMENT NUMERICAL METHODS, TO ESTIMATE THE RELIABILITY OF THE OBTAINED RESULTS, TO DEVELOP ELEMENTS OF MATHEMATICAL SOFTWARE AND EVALUATE PERFORMANCES.
3. MAKING JUDGEMENTS
THE COURSE AIMS TO DEVELOP THE STUDENTS’ SKILL TO CHOOSE THE MOST SUITABLE NUMERICAL METHOD FOR ANY SPECIFIC PROBLEM TO SOLVE, AND TO COMPARE THE PERFORMANCES OF THE CODES BASED ON DIFFERENT NUMERICAL METHODS, RESPECT TO EFFICIENCY, ACCURACY AND STABILITY.
4. COMMUNICATION SKILLS
THE STUDENT WILL BE ABLE TO ILLUSTRATE THE KNOWN NUMERICAL METHODS, THEIR THEORETICAL PROPERTIES AND THE RELATED ALGORITHMS, AND HE SHOULD TO EXPLAIN THE PERFORMANCES OF THE NUMERICAL METHODS APPLIED TO SPECIFIC PROBLEMS. HE WILL BE ABLE TO WRITE THE INTERNAL AND THE EXTERNAL DOCUMENTATION OF THE DEVELOPED SOFTWARE.
5. LEARNING SKILLS
THE COURSE WILL GIVE THE BASIC MEANS TO LEARN NEW NUMERICAL METHODS AND TO USE OR DEVELOP NEW MATHEMATICAL SOFTWARE.

	Prerequisites
	ELEMENTS OF DISCRETE MATHEMATICS AND LINEAR ALGEBRA.ELEMENTS OF MATHEMATICAL ANALYSIS: CONTINUOS FUNCTIONS AND MAIN THEOREMS, CONCEPTS OF DERIVATIVE AND INTEGRALS. BASICS ON PROGRAMMING.

	Contents
	ERROR ANALYSIS AND FLOATING - POINT ARITHMETIC. APPROXIMATION OF DATA AND FUNCTIONS. POLYNOMIAL INTERPOLATION AND SPLINES. LEAST SQUARES APPROXIMATION. ITERATIVE METHODS FOR NONLINEAR EQUATIONS. NUMERICAL QUADRATURE: NEWTON - COTES AND GAUSSIAN FORMULAS. AUTOMATIC INTEGRATORS BASED ON FIXED AND ADAPTATIVE SCHEMES. EIGENVALUES OF MATRICES. ITERATIVE METHODS AND METHODS BASED ON SIMILARITY TRANSFORMATIONS. THE MATLAB PROGRAMMING LANGUAGE. DEVELOPMENT OF CODES RELATED TO THE MAIN ALGORITHMS.

Contents

ERROR ANALYSIS AND FLOATING - POINT ARITHMETIC. APPROXIMATION OF DATA AND FUNCTIONS. POLYNOMIAL INTERPOLATION AND SPLINES. LEAST SQUARES APPROXIMATION. ITERATIVE METHODS FOR NONLINEAR EQUATIONS. NUMERICAL QUADRATURE: NEWTON - COTES AND GAUSSIAN FORMULAS. AUTOMATIC INTEGRATORS BASED ON FIXED AND ADAPTATIVE SCHEMES. EIGENVALUES OF MATRICES. ITERATIVE METHODS AND METHODS BASED ON SIMILARITY TRANSFORMATIONS. THE MATLAB PROGRAMMING LANGUAGE. DEVELOPMENT OF CODES RELATED TO THE MAIN ALGORITHMS.

	Teaching Methods
	LECTURES, EXCERCISES IN LABORATORY, DEVELOPMENT OF MATHEMATICAL SOFTWARE

	Verification of learning
	- DEVELOPMENT, TESTING AND EVALUATION OF ELEMENTS OF MATHEMATICAL SOFTWARE, ORAL EXAMINATION ON THE TOPICS OF THE COURSE

	Texts
	G.MONEGATO, FONDAMENTI DI CALCOLO NUMERICO, CLUT V. COMINCIOLI - ANALISI NUMERICA - ED. MC GRAW HILL ALMERICO MURLI, MATEMATICA NUMERICA: METODI, ALGORITMI E SOFTWARE, PARTE SECONDA, LIGUORI EDITORE