# NUMERICAL MODELS AND METHODS

## Learning outcomes of the course unit

The primary goal of this teaching course is to give a complete background of Numerical Mathematics, presenting in a balanced way both theoretical and algorithmic aspects, together with some applications. It has to be considered as the natural prosecution of the Numerical Analysis course held in previous years. At the end, students will know a large amount of basic numerical methods useful to go deeper inside this subject and in general to face different contexts of Applied Mathematics.

A sound balancing of theoretical analysis, description of algorithms and discussion of applications is the primary concern.

## Prerequisites

Numerical Analysis

Numerical Analysis 1, Computational Numerical Lab.

## Course contents summary

- 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.

Approximation of Functions and Data: Trigonometric interpolation. Rational interpolation. Least-squares approximationof functions and data.

Numerical Integration:

Orthogonal polynomials. Gaussian quadrature on bounded and unbounded intervals. Error estimates. Multiple integrals.

Numerical linear algebra:

QR-decomposition. Basic iterative methods. Jacobi, Gauss-Seidel and SOR methods. Convergence results. Coniugate gradient method. Stop tests.

Eigenvalue and eigenvector problem. Localization of eigenvalues. Stability analysis. The power method. The inverse power method. Eigenvalues and eigenvectors of a tridiagonal matrix. Householder transformations. Reduction of a general matrix to Hessemberg form. The LR algorithm.

The QR algorithm.

Solution of Nonlinear Equations:

Secant method, False Position method. Convergence results. Fixed-point methods. Rate of convergence. Zeros of polynomials. The Newton-Horner method; the Bairstow method. Stop tests. Newton’s method in several variables.

Numerical Solution of Ordinary Differential Equations:

Linear multistep methods. Order, convergence and stability analysis. Adams methods. Predictor-corrector methods.

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

## 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)

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.

Oral lessons and Lab.

## Assessment methods and criteria

Through the preparation of a work with a theorical 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 work given in advance, and will answer to some questions related to the course program.

Oral exam and Lab test.

## 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.