# NUMERICAL ANALYSIS

Degree course:
Corso di First cycle degree in MATHEMATICS
Academic year when starting the degree:
2017/2018
Year:
2
Academic year in which the course will be held:
2018/2019
Course type:
Compulsory subjects, characteristic of the class
Credits:
8
Period:
Second semester
Standard lectures hours:
68
Detail of lecture’s hours:
Lesson (56 hours), Exercise (12 hours)
Requirements:

Programming, Comput. Math., Linear Algebra, Calculus.

Final Examination:
Orale

Written and oral exam

Assessment:
Voto Finale

Analysis of algorithms and of their complexity and numerical stability

Training in the direction of constructive proofs and of an algorithmic vision of mathematics

Matrix theory. Unitary, Hermitian, positive definite, normal matrices.

Normal forms: Schur, Jordan

Spectral characterization of unitary, Hermitian, positive definite, normal matrices via the Schur form

Eigenvalues: localization (Theorems by Gerschgorin I, II, III)

Vector norms, matrix norms, induced norms (relation between spectral radius and induced norms)

Theorem of topological equivalence in finite-dimensional vector spaces

Elementary matrices (spectral analysis, inverse): Gauss, Householder

Numerical solution of linear systems: coefficient matrices in special form (unitary, triangular etc)

Conditioning and the problem and stability of the algorithms

Numerical solution of linear systems: Gaussian elimination, pivoting, QR factorization

Choleski algorithm for positive definite matrices

Shermann-Morrison-Woodbury formula (updating efficient techniques)

Numerical stability of the direct algorithms

Iterative methods: general theory, Jacobi and Gauss-Seidel (methods and convergence analysis)

Evaluation of a polynomial at a point. Interpolation. Vandermonde Matrix

“Metodi Numerici per l’Algebra Lineare”, by D. Bini, M. Capovani, O. Menchi, Zanichelli

“Metodi Numerici ”, by R. Bevilacqua, D. Bini, M. Capovani, O. Menchi, Zanichelli

Notes by the Professor

Classroom teaching; practical exercises (on blackboard)

for meeting with students please use email: stefano.serrac@uninsubria.it

GARONI CARLO