Beatrice PATERNOSTER | NUMERICAL CALCULUS I

cod. 0512300012

NUMERICAL CALCULUS I

	0512300012
	DEPARTMENT OF MATHEMATICS
	EQF6
	MATHEMATICS
	2023/2024

PERCORSO SHARED STUDY PATHWAY

	OBBLIGATORIO
	YEAR OF COURSE 2
	YEAR OF DIDACTIC SYSTEM 2018
	SPRING SEMESTER

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/08	7	56	LESSONS	COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS

PROFESSORS

	BEATRICE PATERNOSTER T Curriculum
	ANGELAMARIA CARDONE Curriculum

	Objectives
	1.KNOWLEDGE AND UNDERSTANDING THE AIM OF THE COURSE IS TO GET THEORETICAL KNOWLEDGE AND TO PERFORM 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. THE COURSE IS ALSO DEVOTED TO THE ANALISYS OF THE MAIN PROBLEMS IN THE DEVELOPMENT OF EFFICIENT MATHEMATICAL SOFTWARE. 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.

1.KNOWLEDGE AND UNDERSTANDING
THE AIM OF THE COURSE IS TO GET THEORETICAL KNOWLEDGE AND TO PERFORM 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. THE COURSE IS ALSO DEVOTED TO THE ANALISYS OF THE MAIN PROBLEMS IN THE DEVELOPMENT OF EFFICIENT MATHEMATICAL SOFTWARE. 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.

	Prerequisites
	ELEMENTS OF DISCRETE MATHEMATICS AND LINEAR ALGEBRA. ELEMENTS OF MATHEMATICAL ANALYSIS: CONTINUOUS FUNCTIONS AND MAIN THEOREMS, DERIVATIVE AND INTEGRALS.

	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. EULER METHOD FOR NUMERIAL SOLUTION OF CHAUCHY INITIAL VALUE PROBLEM. 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. EULER METHOD FOR NUMERIAL SOLUTION OF CHAUCHY INITIAL VALUE PROBLEM. THE MATLAB PROGRAMMING LANGUAGE. DEVELOPMENT OF CODES RELATED TO THE MAIN ALGORITHMS.

	Teaching Methods
	LECTURES, PRACTICES,LABORATORY, PROJECTS ON DEVELOPMENT AND EVALUATION OF MATHEMATICAL SOFTWARE

	Verification of learning
	THE FINAL EXAM EVALUATES THE ACQUIRED KNOWLEDGE AND THE ABILITY TO APPLY IT TO SOLVING TYPICAL PROBLEMS OF SCIENTIFIC COMPUTING. IT CONSISTS IN TWO PARTS: A PRACTICAL TEST, IN WHICH THE MATHEMATICAL SOFTWARE DESIGNED DURING THE COURSE IS USED TO SOLVE SOME QUADRATURE PROBLEMS, PROBLEMS OF APPROXIMATION OF FUNCTIONS AND DATA BY POLYNOMIAL INTERPOLATION, APPROXIMATION IN THE SENSE OF THE LEAST SQUARES AND SPLINES, AND NUMERICAL APPROXIMATION OF EIGENVALUES OF MATRICES; AN ORAL EXAM, BASED ON THE THEORETICAL ITEMS PRESENTED DURING THE LESSONS. 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. THERE MAY BE TWO PRACTICAL TESTS DURING THE COURSE (OPTIONAL): ONE ON THE APPROXIMATION OF DATA AND FUNCTIONS IN MATLAB; ONE ON NUMERICAL QUADRATURE, WHICH CONSISTS IN THE DISCUSSION OF A DESIGN AND COMPUTATION OF INTEGRALS DEFINED IN MATLAB, USING THE CODES DEVELOPED DURING THE COURSE. THEY CONSTITUTE PARTIAL EXEMPTION FROM THE PRACTICAL TEST DURING THE FINAL EXAM.

Verification of learning

THE FINAL EXAM EVALUATES THE ACQUIRED KNOWLEDGE AND THE ABILITY TO APPLY IT TO SOLVING TYPICAL PROBLEMS OF SCIENTIFIC COMPUTING.

IT CONSISTS IN TWO PARTS: A PRACTICAL TEST, IN WHICH THE MATHEMATICAL SOFTWARE DESIGNED DURING THE COURSE IS USED TO SOLVE SOME QUADRATURE PROBLEMS, PROBLEMS OF APPROXIMATION OF FUNCTIONS AND DATA BY POLYNOMIAL INTERPOLATION, APPROXIMATION IN THE SENSE OF THE LEAST SQUARES AND SPLINES, AND NUMERICAL APPROXIMATION OF EIGENVALUES OF MATRICES; AN ORAL EXAM, BASED ON THE THEORETICAL ITEMS PRESENTED DURING THE LESSONS.

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.

THERE MAY BE TWO PRACTICAL TESTS DURING THE COURSE (OPTIONAL): ONE ON THE APPROXIMATION OF DATA AND FUNCTIONS IN MATLAB; ONE ON NUMERICAL QUADRATURE, WHICH CONSISTS IN THE DISCUSSION OF A DESIGN AND COMPUTATION OF INTEGRALS DEFINED IN MATLAB, USING THE CODES DEVELOPED DURING THE COURSE. THEY CONSTITUTE PARTIAL EXEMPTION FROM THE PRACTICAL TEST DURING THE FINAL EXAM.

	Texts
	G.MONEGATO, FONDAMENTI DI CALCOLO NUMERICO, CLUT V. COMINCIOLI - ANALISI NUMERICA - ED. MC GRAW HILL

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Beatrice PATERNOSTER | NUMERICAL CALCULUS I