# 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. Such skills will also be developed in a direct confrontation with the lecturer in frontal and laboratory lessons. Starting from the basic and fundamental knowledge provided by the course and the laboratory, the student will be able to consult autonomously specialized texts in order to deal with entering the world of work or to enroll in further training courses where they are required use of mathematics.

## 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 - iterative solutions of non-linear equations : Chord Method, Newton's Method, Method of False Position, Methods for polinomials : Bairstow's method, Sturm sequences - Ordinary differential equations (discrete methods one step) Methods based on approximating the derivative: Euler-Cauchy method, Improving the accuracy of the Numerical solution, One-step methods based on quadrature formulae, Multistep methods based on Quadrature Formulae , Runge-Kutta's methods, Boundary value problems - 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.

E. Isaacson, H.B. Keller, Analysis of Numerical Methods, John Wiley and Sons.

## Teaching methods

Lectures and exercises in the classroom. MATLAB numerical exercises in the laboratory. Correction of exercises assigned individually and groups to experience a team collaboration.

## Assessment methods and criteria

Assessment of learning takes place in a traditional way through the evaluation of an elaborate (with some exercises to be developed in the Laboratory) in which the student will have to demonstrate knowledge of the topics taken by solving the exercises also through the use of simple MATLAB programs. As an alternative to the final written exam, the student must deliver, on the agreed dates, the exercises offered during the front lessons. The written exam follows an oral test in which the student must discuss (present) the written test arguments or the exercises delivered