Numerical Analysis 1

A.Y. 2021/2022
Overall hours
Learning objectives
The aim of the course is to provide students with the basic methods of the Numerical Analysis with examples from the scientific computing.
Expected learning outcomes
Learning of the basic methods and algorithms for solving some mathematical problems including: data and function approximation, linear system resolution, computation of the zeros of nonlinear functions, quadrature formula, approximation of eigenvalues. Students will also be able to implement the learned algorithms using the MATLAB software.
Course syllabus and organization

Single session

Lesson period
First semester
Live lectures (on line) in the form of interactive lessons recorded and made available to students. MATLAB online laboratory.
Asynchronous lectures for deepen some topics. Tutoring for teaching support where possible in presence or synchronously.
Course syllabus
Why numerical analysis. Floating-Point Representation and errors,
stability of computations. Condition Number and ll-Conditioning, stability of algorithms and problems. Examples from scientific computing. Interpolation and approximations of functions and data. e di dati. Polynomial Interpolation: Lagrange form, Newton form. Algorithm for interpolation. Errors in Polynomial Interpolation. Chebyshev polynomial Interpolation. Introduction of Spline functions, linear and cubic case. Smoothing of Data and
the Method of Least Squares. Numerical Integration and interpolation. Newton-Cotes quadrature formula. Error Analysis.
Composite rules. Gaussian Quadrature Formulas.
Locating Roots of Equations: bisection, secant, Newton. Fixed point
iteration. Convergence analysis, end test. Numerical solution of systems of linear equations, error analysis and condition. A) Direct method. Tridiagonal and Banded Systems. Gaussian Elimination, LU factorization, Pivoting. Other factorization0. B) Iterative methods. Convergence, analysis and errors. Splitting, Jacobi and Gauss-Seidel method, SOR, Richardson.
Calculating Eigenvalues and Eigenvectors, Localization (Gershgorin's Theorem). Power method.
Prerequisites for admission
Basics of Linear Algebra and Mathematical Analysis
Teaching methods
Lessons and praticals, computer sessions.
Teaching Resources
V. Comincioli "Analisi numerica: metodi, modelli, applicazioni", e-book Apogeonline, 2005
G. Naldi, L. Pareschi "Matlab. Concetti e progetti" Maggioli Editore, 2020
Assessment methods and Criteria
The final examination consists of three parts: a written exam, a lab exam, and an non-compulsory oral exam.

- During the written exam, the student must solve some exercises in the format of open-ended, with the aim of assessing the student's ability to solve problems in Numerical Analysis. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). In place of a single written exam given during the first examination session, the student may choose instead to take two midterm exams. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal.
- The oral exam is optional and it can be taken only if the written and lab components have been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course and will be required to solve problems regarding numerical analysis in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
-The lab exam consists in developing some (MATLAB) codes and to solve some exercises in the format of open-ended. The lab portion of the final examination serves to assess the capability of the student to put a problem of computational/numerical into context, find a solution and to give a report on the results obtained.

The complete final examination is passed if written and lab parts (and eventually oral part) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the passed exam
MAT/08 - NUMERICAL ANALYSIS - University credits: 9
Practicals: 36 hours
Laboratories: 24 hours
Lessons: 36 hours
Turno A
Professor: Causin Paola
Turno B
Professor: Zampieri Elena
Turno C
Professor: Zampieri Elena