Computer science | NUMERICAL ANALYSIS

cod. 0512100016

NUMERICAL ANALYSIS

	0512100016
	DIPARTIMENTO DI INFORMATICA

	COMPUTER SCIENCE
	2013/2014

CURRICULUM METODOLOGICO

	OBBLIGATORIO
	YEAR OF COURSE 3
	YEAR OF DIDACTIC SYSTEM 2008
	PRIMO SEMESTRE

TEACHING UNITS

SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
MAT/08	4	32	LESSONS	SUPPLEMENTARY COMPULSORY SUBJECTS
MAT/08	2	24	LAB	SUPPLEMENTARY COMPULSORY SUBJECTS

	Objectives
	1. KNOWLEDGE AND UNDERSTANDING THE AIM OF THE COURSE, SCHEDULED IN TEACHER LESSONS AND LABORATORY ACTIVITIES, IS THE THEORETICAL KNOWLEDGE AND CRITICAL ANALYSIS OF THE MAIN NUMERICAL METHODS CONCERNING THE BASIC TOPICS OF NUMERICAL ANALYSIS: FLOATING-POINT ARITHMETIC, NUMERICAL SOLUTION OF LINEAR SYSTEMS, COMPUTATION OF EIGENVALUES, APPROXIMATIONS OF DATA AND FUNCTIONS, CALCULUS OF DEFINITE INTEGRALS. PARTICULAR ATTENTION WILL BE PAID TO THE PRINCIPLES ABOUT THE DEVELOPMENT OF EFFICIENT MATHEMATICAL SOFTWARE, WITH REGARD TO THE ESTIMATE OF ACCURACY OF THE OBTAINED RESULTS, THE EVALUATION OF THE PERFORMANCE OF THE DEVELOPED SOFTWARE, THE COMPARISON AMONG DIFFERENT METHODS, THE EXTERNAL AND INTERNAL DOCUMENTATION. 2. APPLYING KNOWLEDGE AND UNDERSTANDING THE STUDENT SHOULD BE ABLE TO SOLVE PROBLEMS OF SCIENTIFIC CALCULUS, BY MEANS OF THE DEVELOPMENT AND THE USE OF MATHEMATICAL SOFTWARE AND SUITABLE COMPUTING ENVIRONMENT. HE SHOULD BE ABLE TO CHOOSE THE MOST SUITABLE NUMERICAL METHOD WITH RESPECT TO THE PROBLEM TO SOLVE, BY ANALYZING THE FEATURES OF THE PROBLEM, LIKE THE STRUCTURE OF THE DATA, THE REQUESTED ACCURACY AND THE STABILITY. FOR ANY TOPIC, IT WILL BE FURNISHED AN EXAMPLE WHICH REQUIRES THE APPLICATION OF THE CONSIDERED METHODS. 3. MAKING JUDGEMENTS STUDENTS ARE GUIDED TO HARD LOOK TO WHAT THEY LEARN AND TO ENRICH THEIR MAKING JUDGMENTS SKILL BY STUDYING THE EDUCATIONAL MATERIAL. THE EVALUATION OF THE MATHEMATICAL SOFTWARE, BOTH DONE BY THEMSELVES OR JUST USED, AND THE COMPARISON AMONG DIFFERENT NUMERICAL METHODS, AIM TO DEVELOP THEIR MAKING JUDGMENTS SKILL AND THEIR CRITICAL SENSIBILITY. 4. COMMUNICATION SKILLS THANKS TO THE LABORATORY ACTIVITIES, THE COURSE WILL HELP TO DEVELOP THE STUDENTS’ SKILL TO JUSTIFY AND TO CLAIM THE CHOICES MADE IN THE SOLUTION OF A COMPUTATIONAL PROBLEM, AND THE TEAMWORK SKILL. 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, SCHEDULED IN TEACHER LESSONS AND LABORATORY ACTIVITIES, IS THE THEORETICAL KNOWLEDGE AND CRITICAL ANALYSIS OF THE MAIN NUMERICAL METHODS CONCERNING THE BASIC TOPICS OF NUMERICAL ANALYSIS: FLOATING-POINT ARITHMETIC, NUMERICAL SOLUTION OF LINEAR SYSTEMS, COMPUTATION OF EIGENVALUES, APPROXIMATIONS OF DATA AND FUNCTIONS, CALCULUS OF DEFINITE INTEGRALS. PARTICULAR ATTENTION WILL BE PAID TO THE PRINCIPLES ABOUT THE DEVELOPMENT OF EFFICIENT MATHEMATICAL SOFTWARE, WITH REGARD TO THE ESTIMATE OF ACCURACY OF THE OBTAINED RESULTS, THE EVALUATION OF THE PERFORMANCE OF THE DEVELOPED SOFTWARE, THE COMPARISON AMONG DIFFERENT METHODS, THE EXTERNAL AND INTERNAL DOCUMENTATION.
2. APPLYING KNOWLEDGE AND UNDERSTANDING
THE STUDENT SHOULD BE ABLE TO SOLVE PROBLEMS OF SCIENTIFIC CALCULUS, BY MEANS OF THE DEVELOPMENT AND THE USE OF MATHEMATICAL SOFTWARE AND SUITABLE COMPUTING ENVIRONMENT. HE SHOULD BE ABLE TO CHOOSE THE MOST SUITABLE NUMERICAL METHOD WITH RESPECT TO THE PROBLEM TO SOLVE, BY ANALYZING THE FEATURES OF THE PROBLEM, LIKE THE STRUCTURE OF THE DATA, THE REQUESTED ACCURACY AND THE STABILITY. FOR ANY TOPIC, IT WILL BE FURNISHED AN EXAMPLE WHICH REQUIRES THE APPLICATION OF THE CONSIDERED METHODS.
3. MAKING JUDGEMENTS
STUDENTS ARE GUIDED TO HARD LOOK TO WHAT THEY LEARN AND TO ENRICH THEIR MAKING JUDGMENTS SKILL BY STUDYING THE EDUCATIONAL MATERIAL. THE EVALUATION OF THE MATHEMATICAL SOFTWARE, BOTH DONE BY THEMSELVES OR JUST USED, AND THE COMPARISON AMONG DIFFERENT NUMERICAL METHODS, AIM TO DEVELOP THEIR MAKING JUDGMENTS SKILL AND THEIR CRITICAL SENSIBILITY.
4. COMMUNICATION SKILLS
THANKS TO THE LABORATORY ACTIVITIES, THE COURSE WILL HELP TO DEVELOP THE STUDENTS’ SKILL TO JUSTIFY AND TO CLAIM THE CHOICES MADE IN THE SOLUTION OF A COMPUTATIONAL PROBLEM, AND THE TEAMWORK SKILL.
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.

	Contents
	ERRORS ANALYSIS, PROPAGATION. ILL-CONDITIONING. SYSTEMS OF LINEAR EQUAIONS. DIRECT METHODS: GAUSS AND LU FACTORIZATION. ITERATIVE METHODS: JACOBI, GAUSS-SEIDEL, RELAXATION. CONVERGENCE. SISTEMI DI EQUAZIONI LINEARI. METODI DIRETTI: GAUSS E FATTORIZZAZIONE LU. METODI ITERATIVI: JACOBI, GAUSS – SEIDEL, RILASSAMENTO. CONVERGENZA. EIGENVALUES OF MATRICES. ITERATIVE METHODS. QR METHOD. APPROXIMATION. POLYNOMIAL INTERPOLATION: ERROR, STABILITY, CONVERGNECE. LINEAR PIECEWISE INTERPOLATION AND SPLINE. LEAST SQUARES. CALCULUS OF INTEGRALS. QUADRATURE FORMULAS OF NEWTON-COTES. PRECISION DEGREE. ERROR. COMPOSITION OF FORMULAS, ERROR. ADAPTATIVE QUADRATURE. MATLAB. DEVELOPMENT OF MATHEMATICAL ELEMENTS RELATED TO THE MAIN ALGORITHMS.

Contents

ERRORS ANALYSIS, PROPAGATION. ILL-CONDITIONING.
SYSTEMS OF LINEAR EQUAIONS. DIRECT METHODS: GAUSS AND LU FACTORIZATION. ITERATIVE METHODS: JACOBI, GAUSS-SEIDEL, RELAXATION. CONVERGENCE.
SISTEMI DI EQUAZIONI LINEARI. METODI DIRETTI: GAUSS E FATTORIZZAZIONE LU. METODI ITERATIVI: JACOBI, GAUSS – SEIDEL, RILASSAMENTO. CONVERGENZA.
EIGENVALUES OF MATRICES. ITERATIVE METHODS. QR METHOD.
APPROXIMATION. POLYNOMIAL INTERPOLATION: ERROR, STABILITY, CONVERGNECE. LINEAR PIECEWISE INTERPOLATION AND SPLINE.
LEAST SQUARES.
CALCULUS OF INTEGRALS. QUADRATURE FORMULAS OF NEWTON-COTES. PRECISION DEGREE. ERROR. COMPOSITION OF FORMULAS, ERROR. ADAPTATIVE QUADRATURE.
MATLAB.
DEVELOPMENT OF MATHEMATICAL ELEMENTS RELATED TO THE MAIN ALGORITHMS.

	Teaching Methods
	TEACHER LESSONS; USE OF LABORATORIES;

	Verification of learning
	DEVELOPMENT, TESTING AND EVALUATION OF MATHEMATICAL SOFTWARE. ORAL EXAMINATION

	Texts
	• A. QUARTERONI AND F. SALERI, SCIENTIFIC COMPUTING WITH MATLAB AND OCTAVE, SPRINGER-VERLAG BERLIN, 2006, (FIRST ED. 2003). • MATLAB USER’S GUIDE.