66736 - Numerical Methods

Learning outcomes

At the end of the course, the student knows the basic concepts for solving a real problem using a computer, with particular attention to the problem of error propagation.

Course contents

1. Numerical Calculus - Goals and problems in solving practical computer problems.

2. Finite numbers - Representation of real numbers. Finite numbers. Representation errors. Floating point arithmetic. Error analysis in elementary arithmetic operations. Error propagation: stability and conditioning.

3. Zeros of Functions - Problem formulation. Resolution Techniques. Iterative methods, convergence and order of methods. Local convergence and global convergence methods. Bisection method and other methods of the first order with global convergence. The Fixed Point iteration method. Convergence theorem. A second-order method: the Newton method. Quasi-Newton methods: the secant method.

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

5. Numerical solution of Linear Systems - The Gauss elimination algorithm. LR factorization of a matrix. Stability of LR factorization. Condition number of a matrix and well-conditioned problems. Cholesky factorization of symmetric and positive definite matrices. Variants of the Gauss elimination algorithm for factorization of tridiagonal matrices. Iterative methods for the solution of linear systems: a necessary and sufficient condition for convergence. Jacobi method and Gauss-Seidel method.

6. Interpolation - Polynomial interpolation. Existence and uniqueness of the interpolating polynomial. Evaluation of the interpolating polynomial: Lagrange form and Newton form. Error expression in polynomial interpolation. Convergence problems. Overview of interpolation with spline functions.

7. Least Squares approximation. Normal equations, QRLS method.

8. Numerical Integration - Newton-Cotes quadrature formulas. Simple formulas and composite formulas. Error of simple and composite integration formulas. Adaptive methods.

9. Fourier analysis: Fourier series, continuous Fourier transform, discrete Fourier transform; properties; convolution and correlation theorems; Fast Fourier Transform (FFT). Signal filtering techniques in the Fourier domain: lowpass filters, highpass filters.

Readings/Bibliography

It will be fundamental to use the notes taken during the lectures and the computer material made available on the web. For further study we recommend:

[1] A. Quarteroni, F. Saleri, Introduzione al Calcolo Scientifico - Esercizi e problemi risolti con MATLAB, Springer Verlag, 2006

[2] A. Quarteroni, R. Sacco, F. Saleri, Matematica Numerica, Springer Verlag, 2000

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

[4] E. O. Brigham, The Fast Fourier Transform and its applications, Prentice-Hall, New Jersey, USA, 1988.

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 problems of mathematics 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.

Assessment methods

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

- knowledge of the fundamental elements of the 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 a single test which includes 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 test, the use of support material such as textbooks, notes, computer supports is not allowed.

Teaching tools

The course includes a laboratory activity in which the MATLAB software will be used. The corresponding teaching material will be made available to the student in electronic format and will be available on the lecturer's website.

Office hours

See the website of Lucia Romani

See the website of Damiana Lazzaro

See the website of