Introduce students to numerical analysis, namely to the development and analysis of methods used to solve the problems posed by the continuous mathematics.

Knowledge and understanding: understand the fundamental principles of Numerical Analysis as a tool for solving problems posed by the continuous mathematics; understand in which
way numerical methods are developed and in which way they are analyzed.

Applying knowledge and understanding: be able to apply numerical methods to real problems along with implementations in some programming language.

Making judgments: be able to recognize the strengths and weaknesses of numerical methods.

Communication: know how to expose the resolution of a problem of continuous mathematics using a numerical method.

Lifelong learning skills: know how to gather information from the web or textbooks to solve a problem of continuous mathematics.

Calculus and linear algebra. The students need Analysis I and Geometry to pass Numerical Analysis.

This course provides an introduction to numerical computing for scientific and technological applications.

1. Floating-point system and error propagation.

Floating-point representation of real numbers. Truncation and rounding error, machine precision, error propagation in arithmetic operations with approximate numbers, conditioning of a problem, and the concept of stability of an algorithm.

2. Numerical solution of nonlinear equations.
Introduction to the root-finding problem. Localization intervals of a root. The bisection method. The Newton method. The Newton method for systems.

3. Interpolation of functions and data.
The Lagrange interpolation problem. The Lagrange form of the interpolation polynomial. Function interpolation. Increasing the number of interpolation nodes. Piecewise interpolation.

4. Quadrature.
Interpolatory quadrature formulas. General quadrature formulas. Newton-Cotes quadrature formulas. Polynomial order (degree of precision) and error of the quadrature formulas. Composite quadrature formulas.

5. Elements of numerical linear algebra.

Vector and matrix norms.
Orthogonal matrices, symmetric matrices, diagonalization by orthogonal transformation, symmetric semidefinite and definite positive matrices, SVD, approximations of lower rank, SVD and data compression.

Notes provided by the instructor available at https://moodle2.units.it/

A. Quarteroni, R. Sacco, F. Saleri, P. Gervasio: Matematica Numerica (4a edizione), Springer Verlag, 2014

A. Quarteroni, F. Saleri, Calcolo Scientifico esercizi e problemi risolti con Matlab e Octave. Springer, 2008.

V. Comincioli, Analisi Numerica Metodi Modelli Applicazioni, McGraw-Hill Libri Italia, 1995.

D. Bini, M. Capovani, O. Menchi: Metodi numerici per l'algebra lineare, Zanichelli, Bologna, 1996.

This course provides an introduction to numerical computing for scientific and technological applications.

1. Floating-point system and error propagation.

Floating-point representation of real numbers. Truncation and rounding error, machine precision, error propagation in arithmetic operations with approximate numbers, conditioning of a problem, and the concept of stability of an algorithm.

2. Numerical solution of nonlinear equations.
Introduction to the root-finding problem. Localization intervals of a root. The bisection method. The Newton method. The Newton method for systems.

3. Interpolation of functions and data.
The Lagrange interpolation problem. The Lagrange form of the interpolation polynomial. Function interpolation. Increasing the number of interpolation nodes. Piecewise interpolation.

4. Quadrature.
Interpolatory quadrature formulas. General quadrature formulas. Newton-Cotes quadrature formulas. Polynomial order (degree of precision) and error of the quadrature formulas. Composite quadrature formulas.

5. Elements of numerical linear algebra.

Vector and matrix norms.
Orthogonal matrices, symmetric matrices, diagonalization by orthogonal transformation, symmetric semidefinite and definite positive matrices, SVD, approximations of lower rank, SVD, and data compression.

Lectures with slides and blackboard plus some hands-on sessions using Matlab.
Class attendance is highly recommended for learning and exam preparation.
The teaching material (teacher notes and programs) will be available online through the Moodle platform.
Some textbooks are suggested for further study.

Changing public health circumstances for COVID-19 may cause changes in the procedures here described, even moving to fully online instruction at some point during the semester. These changes will be notified to students through the Department and Course websites if the situation develops.

The final exam consists of a written test composed of two exercises, two theoretical questions, and a MATLAB knowledge test. Exercises, theoretical questions, and MATLAB tests are each graded on a scale from 0 to 10. These scores are summed to obtain a total score ranging from 0 to 50. The final grade is obtained as the upper integer part of the total score multiplied by 34/50. A final grade of 34 or 33 corresponds to 30 cum laude. A final grade of 30, 31, or 32 corresponds to 30. The instructors reserve the right to conduct an additional oral examination if deemed necessary.

This course explores topics closely related to one or more goals of the United Nations 2030 Agenda for Sustainable Development (SDGs).