*********************************************** Information for LMSU students interested in coming in TV (Rome "Tor Vergata" university): Here below it is described the Academic Year in TV math bachelor and master degrees: I semester: period of courses: end of September - December or half January period of exams: January and February (the five months of the erasmus+ program may cover the period Sept 15, 2017 - Feb 15, 2018) II semester: courses: March - Beginning of June exams: June, July and September (the five months of the erasmus+ program may cover the period Feb 15, 2018 - July 15, 2018) ------------------------------------------------------ List of Numerical Analysis courses in Math Bachelor and Math Master in TV [ For TV courses on other math topics, see the web-pages: https://www.mat.uniroma2.it/didattica/foreignstudents.php; https://www.mat.uniroma2.it/didattica/For_stud/insegnam-eng1718.php] ---------- First semester: - "Elements of numerical analysis" (for Master in Mathematics) 64 hours. Di Fiore Program: Bernoulli polynomials and numbers, Eulero-Mclaurin formula, Romberg extrapolation tecniques. Methods for the computation of the eigenvalues and eigenvectors of a matrix, Perron-Frobenius theory for non-negative matrices, computing the pagerank eigenvector, the power method. Low complexity matrix algebras and applications. Numerical solution of ordinary and partial differential problems, the finite difference method. For all these subjects both mathematical and algorithmic aspects are investigated. (For a more detailed program see www.mat.uniroma2.it/~difiore) Educational aims: complete some of the basic knowledges on the Numerical Analysis, investigating some particular subjects. Text books: D. Bertaccini, C. Di Fiore, P. Zellini, Complessita' e Iterazione - percorsi, matrici e algoritmi veloci nel calcolo numerico, Bollati Boringhieri, Torino, 2013 Modality of test: written and oral examination. - "Numerical Analysis 1" (for third year students of Math Bachelor) 72 hours. Carla Manni Program: the basics of the main standard subjects of Numerical analysis. - "Complements of Numerical Analysis 1" (for Master Math students) 64 hours. Carla Manni Program: The course provides an introduction to spline functions and to their use in geometric modeling and numerical treatment of partial differential equations. Contents: Bernstein polynomials and Bezier curves. B-splines: definition and analytic properties. Geometric properties of B-splines. NURBS. Approximation properties of splines. Total positivity. Optimal bases. Tensor-product splines. Applications in the context numerical treatment of multivariate elliptic problems. Objectives. The course provides an introduction to construction and main properties of B-splines both from the analytic and geometric point of view. These functions are the key mathematical tools in several application fields ranging from Computer Graphics to the numerical treatment of PDEs (Isogeometric Analysis). Examination. Oral exam. Testi suggested: lecture notes; C. de Boor, A practical Guide to Splines, Springer 2001 - "Numerical Analysis 1" (for Computer Science Bachelor students) 48 hours. Francesca Pelosi Program: the very-basics of the main standard subjects of Numerical analysis. - "Numerical Analysis 2" (for third year students of Bachelor in Sciences and Technologies of Media) 56 hours Francesca Pelosi Program: Short summary about Fourier series and Fourier transform. Wavelet transform and its properties. Haar wavelets: definition and properties. Multiresolution analysis: definition and examples. Decomposition and reconstruction algorithms. Construction of orthonormal wavelets. The family of Daubechies wavelets. Compression and denoising. Tensor product bidimensional wavelets. Applications to image processing. Teaching goals: The course is aimed to provide basic concepts about construction and properties of wavelets and their use in image processing. Exam procedure: at the beginning of the course are tested students' prior knowledge; 2 intermediate tests are assigned. Typically, the final exam is based upon a written test and an oral discussion. - "Numerical Methods in Computer Graphics" (for Math Master students) 64 hours, Hendrick Speleers Program: PROGRAM: Computer graphics is widely used in the video game and movie industry. The goal of this course is to provide some basic techniques in computer graphics, and to give an introduction to the programming language Java. The course consists of two parts. Part 1. Introduction to Java as an object-oriented programming language. Part 2. Principles of computer graphics, the basic rendering pipeline, and photo-realistic rendering by ray-tracing. OBJECTIVES: insight in the basic computer graphics techniques for modelling and visualization applications the ability to implement small to medium-sized problems in an object-oriented programming language as Java EXAMINATION: Written exam, project assignment and oral discussion TEXT BOOKS: Thinking in JAVA, by Bruce Eckel Computer Graphics Using OpenGL, by Francis S. Hill and Stephen M. Kelley ------- Second semester: - "Numerical Analysis - basics of Numerical Linear Algebra with Applications" (for third year students of Bachelor in Sciences and Technologies of Media) 48 hours. Di Fiore Program: Basics of Matrix theory, unitary matrices and the Schur triangularization theorem, normal matrices, spectral radius, A^k->O iff spectralradius(A)<1,..; the discrete Fourier transforms and the algebra of eps-circulant matrices; vector and matrix norms; positive definite matrices and discrete least squares polynomial approximation; conditioning of a linear system, Gershgorin eigenvalue localization theorems; matrix triangularization by Gauss-pivot, Givens and Householder methods and the corresponding PLU and QR factorizations, the Cholesky factorization; stationary iterative methods, Jacobi, Gauss-Seidel, Euler-Richardson; the Southwell non stationary iterative method, the power method in solving the pagerank computation problem (cenni). Written exam, the student can ask to do also an oral examination. - "Numerical Analysis 2" (for third year students of Bachelor in Math) 48 hours. Di Fiore and Bertaccini Program: [ Actually this is the program in Acad.Year 2016/2017; in 2017/2018 it could change a bit ] (Di Fiore and Stefano Cipolla): The set of polynomials in a matrix X and the set of matrices commuting with X, displacement decompositions and Hessenberg algebras, eps-circulant matrices dagonalizable by discrete-Fourier type transforms, inversion Toeplitz formulas in terms of eps-circulants; gradient and conjugate gradient methods for minimizing positive definite quadratic forms, and their preconditioned versions, the best eps-circulant matrix approximation; Newton and Broyden methods to solve systems of nonlinear equations, Gradient and Newton methods to minimize functions, the quasi-Newton BFGS method to minimize functions. (Bertaccini and Fabio Durastante): Krylov spaces and their characterization. Krylov methods as projection methods on subspaces. Gradient, Conjugate Gradient and GMRES derived from Lanczos and Arnoldi orthogonalization procedures, respectively. Givens QR factorization of Hessenberg matrices and the definition fo GMRES with Givens-QR and modified Arnoldi. Convergence analysis reduced to approximation in matrix polynomial spaces; the non normal case, relation with the symmetric case. Spectral cluster and convergence theory, preconditioning to improve convergence. Incomplete LU factorizations by fixed pattern for M-matrices (cenni). Oral examination. - "Complements of Numerical Analysis 2" (for Master Math students) 64 hours. Daniele Bertaccini Program: Introduction to subspace methods and their characterization. Krylov subspace methods as general methods of projection on subpages. The conjugated gradient methods and GMRES from  Lanczos and Arnoldi orthogonalization algorithms. Brief summary of QR factorization with Givens rotations for Hessemberg matrices in order to construct GMRES with modified Givens-QR and Arnoldi. Convergence analysis of GMRES also in presence of spectral clustering. Generalities on preconditioning. Incomplete LU Factorization and M-matrices. Oral examination or projects presentation.