# NUMERICAL ANALYSIS

## Learning outcomes of the course unit

One of the main aims of the course is to provide the mathematical foundation underlying the different methods or algorithms, recall the main theoretical properties: stability, accuracy, algorithmic complexity, and show examples and counterexamples which illustrate the advantages and weaknesses. It also aims to test the algorithms presented in a simple and fairly universal software such as MATLAB.

## Prerequisites

Basics: Calculus and Linear Algebra.

## Course contents summary

Error Analysis - Approximation of data and functions - Numerical integration: Newton-Cotes formulas - Hint formulas for integrals in multiple dimensions - Systems of linear equations: direct methods, factorization, iterative methods - Nonlinear equations - Ordinary differential equations (discrete methods one step) - Introduction to Matlab

## Course contents

Error Analysis, Representation of numbers in a computer, rounding errors, machine operations, Cancellation numerical conditioning of a problem and stability of an algorithm.

Accuracy of data and functions: polynomial interpolation, Lagrange interpolation formula, Hermite interpolation formula, the formula of Newton divided differences, interpolation of piecewise polynomial functions, spline functions, interpolation of functions of several variables (hint).

Numerical integration: interpolatory quadrature formulas, according to Newton-Cotes Integration, Error estimates, Formule composed, Applications of quadrature formulas.

Numerical linear algebra: direct methods, the method of Gaussian elimination, Gauss decomposition and LU factorization, Iterative Refinement, matrix inverse. Iterative methods: Jacobi method, Gauss-Seidel method, Method of sovrarilassamento (SOR).

Equations and nonlinear systems: real roots of nonlinear equations, bisection method, secant methods, the tangents (Newton-Raphson), Test of convergence, iterative methods in general, Aitken acceleration method.

Ordinary differential equations: methods one-step explicit Runge-Kutta methods, the local behavior of one-step methods, convergence of the methods one-step explicit estimate local truncation error and the choice of the integration step. Multistep methods (hint). Stability of numerical methods.

## Recommended readings

A.Quarteroni, R.Sacco, F.Saleri: Matematica Numerica, Springer.

G.Naldi, Lorenzo Pareschi, G.Russo, Introduzione al Calcolo Scientifico, McGraw-Hill.

G.Monegato, Fondamenti di Calcolo Numerico, CLUT.

## Teaching methods

Lectures: during the lessons as well as the rigorous presentation of the topics of the course program will be proposed and solved exercises, presented numerical applications which can then also be developed in the laboratory of Matlab programming.

In the lessons in the laboratory, as well as presenting Matlab as a programming language, will be proposed and developed exercises with the direct collaboration of the students.

It will also be dedicated to an appropriate time for the correction of exercises assigned individually.

## Assessment methods and criteria

Written test laboratory followed by an oral examination.

The written test consists in solving simple exercises and coding in Matlab language of basic algorithms of Numerical Analysis.

The oral exam in a few questions about the topics developed in the theoretical part of the course in order to assess the understanding of the topics covered in the lessons and the student's ability to express them properly.

The final evaluation is expressed by a single vote for the examination as a whole.