Numerical methods for the approximation of eigenvalues and eigenvectors; singular value decomposition; numerical methods for ordinary differential equations (initial value problems and boundary value problems); gaussian quadrature rules; implementation and numerical tests of the considered methods.
R. L. Burden, J. D. Faires, Numerical Analysis, Brooks/Cole, 2010.
D. Bini, M. Capovani, O. Menchi: Metodi Numerici per l'algebra lineare,Zanichelli, 1988.
G. Monegato, Fondamenti di calcolo numerico, CLUT Editore, 1998.
A. Quateroni, R. Sacco, F. Saleri, Matematica Numerica, Springer, 2000
Learning Objectives
The course provides the students with the basic notions to theoretically understand and apply numerical methods for the approximation of eigenvalues and eigenvectors, for the singular value decomposition, for the computation of integrals (guassian rules) and for the solution of initial value and boundary value problems.
At the end of the course, the student will be able to:
- understand and present the mathematical formulation of the proposed problems and the relation with the corresponding numerical solution;
- understand and present the theoretical arguments guaranteeing the efficiency and accuracy of the numerical methods;
- solve some test problems by writing in Matlab programs implementing the studied methods.
Prerequisites
Essentials of numerical analysis, calculus and linear algebra.
Teaching Methods
Lectures and training sessions with the computer.
Lectures: presentation of the theory described in the course program, with teacher-student direct interaction, to facilitate and ensure a full understanding of the subject.
Training sessions with the computer: training sessions to learn how to numerically solve mathematical problems in the Matlab envinronment.
The training has the goal of:
- helping the students to develop skills to apply the theoretical knowledge;
- encouraging criticism in the students, particularly in assessing the numerical results obtained.
Moodle learning platform: online teacher-student interaction, posting of additional material.
Further information
Frequency of lectures: not mandatory, but strongly recommended
Teaching Tools: textbooks, additional material on UniFi E-Learning: http://e-l.unifi.it
Type of Assessment
The assessment of the learning consists of an oral exam, when it will be verified the knowledge both of the mathematical aspects of the numerical methods (with questions about the theoretical resutls presented during the course), and of their implementation and test in Python (with questions about some exercises given during the course to be done in Python by each student).
Course program
Numerical methods for the approximation of of eigenvalues and eigenvectors: introduction, localization and conditioning theorems; power methods and its variants; QR method.
Singular value decomposition: existence and properties; application to the computation of the condition number, of the pseudo-inverse matrix, and of the solution of the least squares problem.
Gaussian quadrature rules: introduction and properties.
Numerical methods for the solution of ordinary differential equations: Cauchy problem; introduction to one-step methods; consistency, zero-stability and convergence of explicit one-step methods; Runge-Kutta and Runge-Kutta Fehlberg methods; absolute stability; introduction to methods for boundary value problems.
Numerical solution of the considered problems by Python programming.