# NUMERICAL MATHEMATICS

## Learning outcomes of the course unit

- Knowledge and understanding: through the lectures and provided teaching materials, the student acquires knowledge related to most of the fundamental methods and algorithms of Numerical Mathematics, useful for continuing the deepening of the subject and certainly adequate to a teaching of the basic principles of this discipline in secondary schools.

- Applying knowledge and understanding: during the course, through the carrying out of some theoretical and practical exercises, with the aid of the calculator and the use of the Matlab programming language, the student develops the ability to understand new numerical problems and to apply the knowledge acquired for their resolution, using suitable computational tools.

- Making judgments: at the end of the course, through the drafting of an essay including a theoretical introduction to the method chosen for the approximation of the solution of an assigned numerical problem and the discussion of the results obtained, the student achieves a good autonomy in dealing with the resolution of simple model problems and acquires the ability to autonomously judge the numerical results provided by a code personally developed or already at disposal in numerical libraries.

- Communication skills: through the presentation of the results of the exercises assigned during the course and through the presentation during the examination of the prepared essay, the student develops the ability to verbally illustrate clearly problems and solution strategies and to discuss in a convincing way the obtained numerical results.

- Learning skills: at the end of the course, the student is able to learn the advantages and limitations of numerical models and algorithms and to face new and more complex numerical problems in the subsequent courses of the various areas of Applied Mathematics.

## Prerequisites

Basic notions of Numerical Analysis; knowledge of Matlab or of an equivalent programming language

## Course contents summary

The primary goal of this teaching course is to give the complete basis of Numerical Mathematics, both from a theoretical and algorithmic point of view. It has to be considered as the natural prosecution of the Numerical Analysis course held for undergraduated students. Shortly, the course will deal with the following topics:

- Approximation of Functions and Data.

- Numerical Integration.

- Numerical linear algebra.

- Solution of Nonlinear Equations.

- Numerical Solution of Ordinary Differential Equations

- Numerical Solution of boundary value problems.

## Course contents

- Approximation of Functions and Data. Trigonometric interpolation. Rational interpolation. Least-squares approximation: the continuous and the discrete cases.

- Numerical Integration. Orthogonal polynomials. Gaussian quadrature on bounded and unbounded intervals. Error estimates. Multiple integrals. Adaptive algorithms.

- Numerical linear algebra. Basic iterative methods. Jacobi, Gauss-Seidel and SOR methods. Richardson methods. Coniugate gradient method. GMRES and Bi_CGStab. Convergence results. Stop tests.

- Eigenvalue and eigenvector problem. Localization of eigenvalues. Stability analysis. The power method. The inverse power method. Eigenvalues and eigenvectors of a symmetric matrix: Sturm technique. Householder transformations. Reduction of a general matrix to Hessemberg form. The LR and QR algorithms.

- Solution of Nonlinear Equations: Fixed-point methods.Convergence results. Stop tests. Newton’s method in several variables.

- Numerical Solution of Ordinary Differential Equations: Linear multistep methods for Cauchy problems. Order, convergence and stability analysis. Adams methods. Predictor-corrector methods.

- Boundary valure problems: shooting method, finite difference method, Galerkin method.

## Recommended readings

A.Quarteroni, R.Sacco, F.Saleri, Matematica Numerica, SPRINGER, (2008).

G.Naldi, L.Pareschi, G.Russo, Introduzione al Calcolo Scientifico, McGraw-Hill, (2001)

## Teaching methods

During frontal lessons, the course topics will be analysed, underlying theoretical and algorithmic aspects. Various results obtained by the application of the presented numerical methods will be illustrated and discussed.

Slides shown during lectures will be loaded on Elly e-learning platform at the end of each topic.

## Assessment methods and criteria

Through the preparation of an essay with a theoretical introduction to the chosen numerical method for the approximation of the searched solution and the presentation of related numerical results, students will reach a good autonomy in facing the numerical solution of simple model problems. Results of the knowledge level reached by the students will be verified by an oral exam, where each of them will illustrate what done during the preparation of the above mentioned essay given in advance, and will answer to some questions related to the course program.

## Other informations

During the teaching course, students are asked to resolve some theorical and practical exercises, with the help of computing machines and using Matlab programming language, already introduced in Numerical Analysis course in the previous years.