NUMERICAL CALCULUS I

Beatrice PATERNOSTER NUMERICAL CALCULUS I

0512300012
DEPARTMENT OF MATHEMATICS
EQF6
MATHEMATICS
2024/2025

OBBLIGATORIO
YEAR OF COURSE 2
YEAR OF DIDACTIC SYSTEM 2018
SPRING SEMESTER
CFUHOURSACTIVITY
756LESSONS
ExamDate
CALCOLO NUMERICO I10/01/2025 - 09:00
CALCOLO NUMRICO I31/01/2025 - 09:00
CALCOLO NUMERICO I27/02/2025 - 09:00
Objectives
1. KNOWLEDGE AND UNDERSTANDING
THE COURSE IS AIMED AT ACQUIRING THE THEORETICAL KNOWLEDGE AND CRITICAL ANALYSIS OF THE MAIN METHODS RELATED TO BASIC ARGUMENTS, SUCH AS APPROXIMATION OF DATA AND FUNCTIONS, NUMERICAL CALCULATION OF DEFINED INTEGRALS, THE NUMERICAL RESOLUTION OF NONLINEAR SYSTEMS, THE CALCULATION OF MATRIX EIGENVALUES. IT ALSO ADDRESSES THE MAIN PROBLEMS ENCOUNTERED IN MATHEMATICAL SOFTWARE DEVELOPMENT. IN ADDITION, THE STUDENT WILL LEARN EFFICIENT ALGORITHMS DESIGN METHODOLOGIES
AND THE USE OF APPROPRIATE NUMERICAL COMPUTING ENVIRONMENTS FOR SOLVING MATHEMATICAL PROBLEMS.

2. APPLIED KNOWLEDGE AND UNDERSTANDING

THE PURPOSE OF THE LABORATORY EXERCISES IS TO TEST THE ABOVE METHODS AND TO ESTIMATE
THE RELIABILITY OF THE RESULTS OBTAINED, TO DEVELOP MATHEMATICAL SOFTWARE AND TO EVALUATE ITS 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 (2 HOURS).
•APPROXIMATION OF DATA AND FUNCTIONS. POLYNOMIAL INTERPOLATION AND SPLINES. LEAST SQUARES APPROXIMATION. ITERATIVE METHODS FOR NONLINEAR EQUATIONS. (13 HOURS)
•NUMERICAL QUADRATURE: NEWTON - COTES AND GAUSSIAN FORMULAS. AUTOMATIC INTEGRATORS BASED ON FIXED AND ADAPTATIVE SCHEMES. (17 HOURS)
•EIGENVALUES OF MATRICES. ITERATIVE METHODS AND METHODS BASED ON SIMILARITY TRANSFORMATIONS. (4 HOURS)
•EULER METHOD FOR NUMERIAL SOLUTION OF CHAUCHY INITIAL VALUE PROBLEM. (2 HOURS)
•THE MATLAB PROGRAMMING LANGUAGE. DEVELOPMENT OF CODES RELATED TO THE MAIN ALGORITHMS. (16 HOURS)

TEACHING METHODS
THE COURSE CONSISTS OF LECTURES (7 CFU, 56 HOURS). DURING THE LECTURES, EXERCITATIONS, LABORATORY ACTIVITIES AND PROJECTS FOR THE DEVELOPMENT AND EVALUATION OF MATHEMATIC SOFTWARE WILL ALSO BE CARRIED OUT.
Teaching Methods
THE COURSE CONSISTS OF LECTURES (7 CFU, 56 HOURS). DURING THE LECTURES, EXERCITATIONS, LABORATORY ACTIVITIES AND PROJECTS FOR THE DEVELOPMENT AND EVALUATION OF MATHEMATIC SOFTWARE WILL ALSO BE CARRIED OUT.
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
•R. D’AMBROSIO, NUMERICAL APPROXIMATION OF ORDINARY DIFFERENTIAL PROBLEMS: FROM DETERMINISTIC TO STOCHASTIC NUMERICAL METHODS, SPRINGER (ES. 1.4 – FAKE NEWS DIFFUSING AS EPIDEMICS)

- SLIDES AND NOTES OF THE COURSE ARE AVAILABLE ON THE MS-TEAMS PLATFORM AT THE LINK HTTPS://UNISALERNO.SHAREPOINT.COM/:F:/S/UNI23-CALCOLONUMERICOI-051230001273991NESSUNPARTIZIONAMENTOM/EMFGRMZ3HFNMTVZ203SOUPEBNWYO8D-EZ7UH1RGWJUU92G?E=45LHOD
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