04524 - Numerical Analysis

Learning outcomes

At the end of the course the student knows the numerical-mathematical aspects and the main algorithmic methodologies that allow him to solve problems of interest in engineering. In particular: the methods of numerical linear algebra to solve linear and non-linear systems, least squares linear problems, differential models. The student carries out exercises and projects together with the teacher during the hours of computer laboratory with the help of the MATLAB software.

Course contents

1. Finite numbers - Representation of real numbers. Finite numbers. Representation errors. Floating point arithmetic. Error analysis in elementary arithmetic operations. Error propagation: conditioning of a problem and stability of an algorithm.

2. Linear Algebra recalls on vectors, matrices and vector spaces. Vector norms and matrix norms.

3. Zeros of Functions - Problem formulation. Resolution techniques. Iterative methods, convergence and order of convergence of methods. Global convergence and local convergence methods. Bisection method, Newton's method and the secant method. Fixed-point iteration methods. Convergence theorems.

4. Numerical solution of Linear Systems - Condition number of a matrix and conditioning of the problem. The Gauss elimination algorithm. LU factorization of a matrix. Stability of LU factorization. Pivoting. Cholesky factorization of symmetric and positive definite matrices. Householder method and QR factorization of non-square matrices. Properties of the illustrated methods.

5. Least Squares approximation - Normal equations and QRLS method (least squares solution using QR factorization). Properties of the two methods.

6. Interpolation - Polynomial interpolation. Existence and uniqueness of the interpolating polynomial. Lagrange's form and Newton's form of the interpolating polynomial. Error expression in polynomial interpolation. Convergence problems. Conditioning of the polynomial interpolation problem.

7. Numerical Integration - Newton-Cotes quadrature formulas (Trapezoidal and Simpson's formula). Simple and composite quadrature formulas. Error of simple and composite formulas. Adaptive quadrature formulas.

Readings/Bibliography

It will be fundamental to use the notes taken during the lectures and the computer material made available at https://virtuale.unibo.it/.

For further study we recommend:

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

[2] A. Quarteroni, F. Saleri: Introduzione al Calcolo Scientifico - Esercizi e problemi risolti con MATLAB (3a edizione), Springer Verlag, 2006.

[3] R. Bevilacqua, D. Bini, M. Capovani, O. Menchi: Metodi Numerici, Zanichelli, Bologna, 1992.

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

Alternativey, for those who want a text in English we recommend:

[5] A. Quarteroni, F. Saleri, P. Gervasio: Scientific Computing with MATLAB and Octave (4th edition), Springer, 2014.

[6] U. M. Ascher, C. Greif: A first Course in Numerical Methods, SIAM, 2011.

[7] C. Moler: Numerical Computing with MATLAB, SIAM, 2004.

Teaching methods

The course is structured in lectures and exercises in the computer laboratory. More precisely, the lectures on the basic numerical methods to solve classical mathematical problems through the use of a computer, are followed by laboratory exercises aimed at implementing these methods in MATLAB and developing an adequate sensitivity and awareness of their use.

In consideration of the type of activity and the teaching methods adopted, the attendance of this training activity requires the prior participation of all students in Modules 1 and 2 of Health and Safety training courses [https://elearning-sicurezza.unibo.it/?lang=en] in e-learning mode.

Assessment methods

The end-of-course exam aims to verify the achievement of the following educational objectives:

- knowledge of the fundamental elements of numerical calculus, illustrated during the lectures;

- ability to use basic numerical methods to solve real problems using a computer.

The end-of-course exam (the evaluation of which is in thirtieths) will take place in the computer laboratory. The student has 90 minutes to solve two exercises that include both the development of MATLAB codes for solving numerical problems, and the written answer to theoretical questions on the topics covered in the lessons.

During the exam, the use of support material such as textbooks, notes and computer supports is not allowed.

To access the exam it is necessary to show an identification document.

The dates of the exams can be consulted on the AlmaEsami web platform of the University of Bologna.

Teaching tools

The course includes a laboratory activity in which the MATLAB software is used. The corresponding teaching material will be made available to the student in electronic format, and will be downlodable at https://virtuale.unibo.it/

Office hours

See the website of Lucia Romani