Educational offer - University of Padova

Syllabus

Prerequisites:	Basic knowledge of Mathematical analysis, Linear Algebra and Geometry (vector spaces, vectors, matrices, operations, determinants, inverse matrix and particular matrices, scalar product, norms).
Target skills and knowledge:	The student will have the opportunity to acquire basic computer skills and be able to build the model and the numerical solution algorithm for simple problems. At the end of the course he/she will be able to program with the language reference and produce the results in graphic form. The student will acquire knowledge of some basic methods of Numerical Analysis in view of scientific and technological applications, with special attention to the concepts of error, discretization, approximation, convergence, stability, computational cost.
Examination methods:	Written examination and laboratory programming work (related to Numerical Analysis problems). Optional oral examination
Assessment criteria:	Students must demonstrate that they have acquired the knowledge of the various methods from both theoretical and algorithmic point of view, from the point of view of the applications through simple exercises. In lab tests, will need to have purchased a relative familiarity in the use and in writing simple programs in Matlab.
Course unit contents:	Computer Arithmetic: Floating-Point numbers and arithmetic. Errors in computation. Stability of algorithms. Condition number. Nonlinear Equations: Iterative methods. Convergent sequences. Existence and unicity theorems. Bisection algorithm. Fixed point iteration. Newton's methods. Methods for multiple roots. Stopping criteria. Numerical linear algebra: Direct Methods for linear systems: Gauss and matrix factorizations. Cholesky method. Householder (hints). Matrix Inversion. Preconditioning. Iterative Methods for linear systems: Jacobi, Gauss-Seidel, SOR. Convergence theorems. Stopping criteria Polynomial Approximation: Interpolation (Lagrange, Newton, Chebyshev). Convergence. Least squares: linear and polynomial regression. Numerical Integration: Interpolatory formulae: Lagrange, Newton-Cotes. Gauss (hints). Computer Arithmetic: Floating-Point numbers and arithmetic. Errors in computation. Stability of algorithms. Condition number. Nonlinear Equations: Iterative methods. Convergent sequences. Existence and unicity theorems. Bisection algorithm. Fixed point iteration. Newton's methods. Methods for multiple roots. Stopping criteria. Numerical linear algebra: Direct Methods for linear systems: Gauss and matrix factorizations. Cholesky method. Householder (hints). Matrix Inversion. Preconditioning. Iterative Methods for linear systems: Jacobi, Gauss-Seidel, SOR. Convergence theorems. Stopping criteria Polynomial Approximation: Interpolation (Lagrange, Newton, Chebyshev). Convergence. Least squares: linear and polynomial regression. Numerical Integration: Interpolatory formulae: Lagrange, Newton-Cotes. Gauss (hints). Ordinary differential equations: Initial Value Problems. Implicit and Explicit one step methods (Taylor, Euler). Eigenvalue and Eigenvectors (hints)
Planned learning activities and teaching methods:	The course consists of lectures and exercises in the classroom (about 50 hours) and lessons in the computer lab (about 22 hours) with exercises on the computer in Matlab. Many of the basic methods of numerical analysis presented during the lectures, will gradually be used in the laboratory in order to show their actual use and their potential. Gradually the student will also become familiar with a programming environment for numerical problems and at the end of the course should be able to pass a test that is an integral part of the final exam.
Additional notes about suggested reading:	There are numerous tutorials and manuals recoverable in the network also related to the programming environment Matlab. See also the teacher's web site www.math.unipd.it/~michela in the Teaching section.
Textbooks (and optional supplementary readings)	Michela Redivo Zaglia, Calcolo Numerico: Metodi ed Algoritmi. Padova: Libreria Progetto, 2011. Quarta Edizione riveduta Michela Redivo Zaglia, Calcolo Numerico: Esercizi. Padova: Libreria Progetto, 2015. Terza Edizione Michela Redivo Zaglia, Quaderno MATLAB. Padova: Libreria Progetto, 2016.