Gianluca FRASCA CACCIA | LABORATORY OF PROGRAMMING AND CALCULUS

cod. 0512300006

LABORATORY OF PROGRAMMING AND CALCULUS

	0512300006
	DEPARTMENT OF MATHEMATICS
	EQF6
	MATHEMATICS
	2024/2025

PERCORSO SHARED STUDY PATHWAY

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2018
	SPRING SEMESTER

TEACHING UNITS

SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
MAT/08	4	32	LESSONS	COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
MAT/08	2	24	LAB	COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS

PROFESSORS

	DAJANA CONTE T Curriculum
	GIANLUCA FRASCA CACCIA Curriculum

EXAMS

	Exam	Date	Session
	LABORATORIO DI PROGRAMMAZIONE E CALCOLO	05/05/2025 - 14:00	SESSIONE ORDINARIA

	Objectives
	COURSE AIM THE COURSE IS AIMED AT LET THE STUDENTS ACQUIRE THEORETICAL KNOWLEDGE OF THE MAIN NUMERICAL METHODS AND MATHEMATICAL SOFTWARE DEVELOPMENT SKILLS FOR THE NUMERICAL RESOLUTION OF LINEAR SYSTEMS AND NON-LINEAR EQUATIONS. KNOWLEDGE AND UNDERSTANDING STUDENTS WILL ACQUIRE BASIC KNOWLEDGE ON: •NUMERICAL METHODS FOR SOLVING LINEAR SYSTEMS WITH DIRECT AND ITERATIVE METHODS, AND NON-LINEAR EQUATIONS; •REPRESENTATION OF REAL NUMBERS ON THE CALCULATOR AND ROUNDING ERRORS; •ALGORITHMIC ASPECTS AND PRINCIPLES ON WHICH THE DEVELOPMENT OF EFFICIENT MATHEMATICAL SOFTWARE IN A MATLAB ENVIRONMENT IS BASED, WITH REFERENCE TO THE ESTIMATION OF THE RELIABILITY OF THE OBTAINED RESULTS, AND THE EVALUATION OF THE PERFORMANCE OF THE DEVELOPED SOFTWARE; •BASIC KNOWLEDGE OF THE MATLAB COMPUTING ENVIRONMENT AND THE RELATED SCIENTIFIC COMPUTING FUNCTIONS. APPLYING KNOWLEDGE AND UNDERSTANDING STUDENTS WILL BE ABLE TO: •SOLVE SOLVE SYSTEMS OF LINEAR EQUATIONS AND NON-LINEAR EQUATIONS THROUGH THE DEVELOPMENT AND USE OF MATHEMATICAL SOFTWARE IN THE MATLAB ENVIRONMENT; •CARRY OUT TESTING AND EVALUATION OF MATHEMATICAL SOFTWARE IN TERMS OF ACCURACY AND EFFICIENCY, ALSO BY COMPARING PERFORMANCE BETWEEN DIFFERENT CODES. MAKING JUDGMENTS STUDENTS WILL BE ABLE TO: •CHOOSE THE MOST SUITABLE NUMERICAL METHOD FOR THE PROBLEM UNDER EXAMINATION THROUGH THE ANALYSIS OF THE CHARACTERISTICS OF THE PROBLEM ITSELF; •ANALYZE THE CONVERGENCE OF AN ITERATIVE METHOD; •ESTIMATE THE ACCURACY OF A NUMERICAL METHOD BY CRITICALLY INTERPRETING THE OBTAINED RESULTS; •PROVIDE THEORETICAL JUSTIFICATIONS FOR THE EFFECTIVENESS OF DIFFERENT METHODS FOR SOLVING THE PROBLEMS STUDIED; •RECOGNIZE ERRORS RESULTING FROM MACHINE OPERATIONS (IN FLOATING POINT ARITHMETIC). COMMUNICATION SKILLS STUDENTS WILL BE ABLE TO: •DESCRIBE THE RESULTS OBTAINED USING GRAPHS AND TABLES; •COMMUNICATE THE KNOWLEDGE ACQUIRED IN WRITTEN AND ORAL FORM WITH CORRECT TECHNICAL-SCIENTIFIC LANGUAGE. LEARNING SKILL STUDENTS WILL BE ABLE TO: •LEARN NEW METHODS FOR DEVELOPING MATHEMATICAL SOFTWARE, APPRECIATING THEIR LIMITS AND ADVANTAGES; •PROCEED WITH THE CONTINUOUS UPDATING OF ONE'S KNOWLEDGE, USING TECHNICAL AND SCIENTIFIC LITERATURE, USING TRADITIONAL BIBLIOGRAPHIC TOOLS AND DIGITAL RESOURCES.

COURSE AIM
THE COURSE IS AIMED AT LET THE STUDENTS ACQUIRE THEORETICAL KNOWLEDGE OF THE MAIN NUMERICAL METHODS AND MATHEMATICAL SOFTWARE DEVELOPMENT SKILLS FOR THE NUMERICAL RESOLUTION OF LINEAR SYSTEMS AND NON-LINEAR EQUATIONS.
KNOWLEDGE AND UNDERSTANDING
STUDENTS WILL ACQUIRE BASIC KNOWLEDGE ON:
•NUMERICAL METHODS FOR SOLVING LINEAR SYSTEMS WITH DIRECT AND ITERATIVE METHODS, AND NON-LINEAR EQUATIONS;
•REPRESENTATION OF REAL NUMBERS ON THE CALCULATOR AND ROUNDING ERRORS;
•ALGORITHMIC ASPECTS AND PRINCIPLES ON WHICH THE DEVELOPMENT OF EFFICIENT MATHEMATICAL SOFTWARE IN A MATLAB ENVIRONMENT IS BASED, WITH REFERENCE TO THE ESTIMATION OF THE RELIABILITY OF THE OBTAINED RESULTS, AND THE EVALUATION OF THE PERFORMANCE OF THE DEVELOPED SOFTWARE;
•BASIC KNOWLEDGE OF THE MATLAB COMPUTING ENVIRONMENT AND THE RELATED SCIENTIFIC COMPUTING FUNCTIONS.
APPLYING KNOWLEDGE AND UNDERSTANDING
STUDENTS WILL BE ABLE TO:
•SOLVE SOLVE SYSTEMS OF LINEAR EQUATIONS AND NON-LINEAR EQUATIONS THROUGH THE DEVELOPMENT AND USE OF MATHEMATICAL SOFTWARE IN THE MATLAB ENVIRONMENT;
•CARRY OUT TESTING AND EVALUATION OF MATHEMATICAL SOFTWARE IN TERMS OF ACCURACY AND EFFICIENCY, ALSO BY COMPARING PERFORMANCE BETWEEN DIFFERENT CODES.
MAKING JUDGMENTS
STUDENTS WILL BE ABLE TO:
•CHOOSE THE MOST SUITABLE NUMERICAL METHOD FOR THE PROBLEM UNDER EXAMINATION THROUGH THE ANALYSIS OF THE CHARACTERISTICS OF THE PROBLEM ITSELF;
•ANALYZE THE CONVERGENCE OF AN ITERATIVE METHOD;
•ESTIMATE THE ACCURACY OF A NUMERICAL METHOD BY CRITICALLY INTERPRETING THE OBTAINED RESULTS;
•PROVIDE THEORETICAL JUSTIFICATIONS FOR THE EFFECTIVENESS OF DIFFERENT METHODS FOR SOLVING THE PROBLEMS STUDIED;
•RECOGNIZE ERRORS RESULTING FROM MACHINE OPERATIONS (IN FLOATING POINT ARITHMETIC).
COMMUNICATION SKILLS
STUDENTS WILL BE ABLE TO:
•DESCRIBE THE RESULTS OBTAINED USING GRAPHS AND TABLES;
•COMMUNICATE THE KNOWLEDGE ACQUIRED IN WRITTEN AND ORAL FORM WITH CORRECT TECHNICAL-SCIENTIFIC LANGUAGE.
LEARNING SKILL
STUDENTS WILL BE ABLE TO:
•LEARN NEW METHODS FOR DEVELOPING MATHEMATICAL SOFTWARE, APPRECIATING THEIR LIMITS AND ADVANTAGES;
•PROCEED WITH THE CONTINUOUS UPDATING OF ONE'S KNOWLEDGE, USING TECHNICAL AND SCIENTIFIC LITERATURE, USING TRADITIONAL BIBLIOGRAPHIC TOOLS AND DIGITAL RESOURCES.

	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. (2 HOURS OF LESSON) MACHINES REPRESENTATION OF NUMBERS, THE FLOATING POINT NUMBER SYSTEM AND ARITHMETIC. EVALUATION OF AN ALGORITHM, SPACE AND TIME COMPLEXITY. EXAMPLES: COMPUTATION OF A DETERMINANT. (6 HOURS OF LESSON) VECTOR SPACES, NORMS. SYMMETRIC DEFINITE POSITIVE MATRICES, SYLVESTER CRITERION. (2 HOURS OF LESSON) 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. (16 HOURS OF LESSON) 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. (6 HOURS OF LESSON) DEVELOPMENT OF ALGORITHMS AND OF MATLAB PROGRAMS BASED ON THE MAIN STUDIED METHODS. (24 HOURS OF LABORATORY )

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. (2 HOURS OF LESSON)

MACHINES REPRESENTATION OF NUMBERS, THE FLOATING POINT NUMBER SYSTEM AND ARITHMETIC.
EVALUATION OF AN ALGORITHM, SPACE AND TIME COMPLEXITY. EXAMPLES: COMPUTATION OF A DETERMINANT. (6 HOURS OF LESSON)

VECTOR SPACES, NORMS. SYMMETRIC DEFINITE POSITIVE MATRICES, SYLVESTER CRITERION. (2 HOURS OF LESSON)

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. (16 HOURS OF LESSON)

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. (6 HOURS OF LESSON)

DEVELOPMENT OF ALGORITHMS AND OF MATLAB PROGRAMS BASED ON THE MAIN STUDIED METHODS. (24 HOURS OF LABORATORY )

	Teaching Methods
	THE LECTURES (6 CFU, 56 CLASS HOURS) INCLUDE FRONTAL LESSONS (4 CFU, 32 CLASS HOURS) AND LABORATORY LESSONS (2 CFU, 24 CLASS HOURS) . 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).

Teaching Methods

THE LECTURES (6 CFU, 56 CLASS HOURS) INCLUDE FRONTAL LESSONS (4 CFU, 32 CLASS HOURS) AND LABORATORY LESSONS (2 CFU, 24 CLASS HOURS) .

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/OCTAVE LANGUAGE. IT IS DIVIDED INTO TWO TEST: A PRACTICAL TEST, IN WHICH THE MATLAB CALCULATION ENVIRONMENT AND THE MATHEMATICAL SOFTWARE DESIGNED AND CREATED DURING THE COURSE, ARE USED TO CARRY OUT EXERCISES RELATED TO: FLOATING POINT ARITHMETIC, RESOLUTION OF SYSTEMS OF LINEAR EQUATIONS THROUGH DIRECT AND ITERATIVE METHODS, SOLUTION OF NON-LINEAR EQUATIONS; AN ORAL INTERVIEW, FOR THE PURPOSE OF ASSESSING THE THEORETICAL KNOWLEDGE PRESENTED IN LESSON. THE PRACTICAL TEST IS PREPARATORY TO THE ORAL INTERVIEW AND LASTS ABOUT AN HOUR. THE PRACTICAL TEST WEIGHS ABOUT 40% ON THE FINAL MARK, THE ORAL INTERVIEW WEIGHS ABOUT 60%. THE INTERVIEW TAKES PLACE IMMEDIATELY AFTER THE PRACTICAL TEST AND LASTS ABOUT 30 MINUTES. HONORS CAN BE AWARDED TO STUDENTS WHO DEMONSTRATE THAT THEY ARE ABLE TO APPLY THE ACQUIRED KNOWLEDGE AND SKILLS WITH A CRITICAL SENSE AND WITH ORIGINALITY.

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/OCTAVE LANGUAGE.

IT IS DIVIDED INTO TWO TEST: A PRACTICAL TEST, IN WHICH THE MATLAB CALCULATION ENVIRONMENT AND THE MATHEMATICAL SOFTWARE DESIGNED AND CREATED DURING THE COURSE, ARE USED TO CARRY OUT EXERCISES RELATED TO: FLOATING POINT ARITHMETIC, RESOLUTION OF SYSTEMS OF LINEAR EQUATIONS THROUGH DIRECT AND ITERATIVE METHODS, SOLUTION OF NON-LINEAR EQUATIONS; AN ORAL INTERVIEW, FOR THE PURPOSE OF ASSESSING THE THEORETICAL KNOWLEDGE PRESENTED IN LESSON.
THE PRACTICAL TEST IS PREPARATORY TO THE ORAL INTERVIEW AND LASTS ABOUT AN HOUR. THE PRACTICAL TEST WEIGHS ABOUT 40% ON THE FINAL MARK, THE ORAL INTERVIEW WEIGHS ABOUT 60%. THE INTERVIEW TAKES PLACE IMMEDIATELY AFTER THE PRACTICAL TEST AND LASTS ABOUT 30 MINUTES. HONORS CAN BE AWARDED TO STUDENTS WHO DEMONSTRATE THAT THEY ARE ABLE TO APPLY THE ACQUIRED KNOWLEDGE AND SKILLS WITH A CRITICAL SENSE AND WITH ORIGINALITY.

	Texts
	G. MONEGATO – METODI E ALGORITMI PER IL 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. COURSE HANDBOOKS: WWW.ELEARNING.UNISA.IT

	More Information
	EMAIL OF THE PROFESSORS: DAJCONTE@UNISA.IT, PDIAZDEALBA@UNISA.IT

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Gianluca FRASCA CACCIA | LABORATORY OF PROGRAMMING AND CALCULUS