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 - Floating point representation of real numbers. Machine or floating point 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 and 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 square and 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

For exam preparation, students may refer to the course materials uploaded on the Virtuale platform.

+ Recommended reference books for further reading are:

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

[2] A. Quarteroni: Matematica Numerica - Esercizi, Laboratori e Progetti (2a edizione), Springer, 2013.

[3] A. Quarteroni, F. Saleri, P. Gervasio: Calcolo Scientifico - Esercizi e problemi risolti con MATLAB e Octave (6a edizione), Springer 2017.

+ For those who desire a text in English:

[4] G. Stoyan, A. Baran: Elementary Numerical Mathematics for Programmers and Engineers (2nd edition), Springer/Birkhäuser 2024.

[5] E. E. Mikhailov: Programming with MATLAB for Scientists: A Beginner’s Introduction, CRC Press, Inc., 2017.

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

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

Teaching methods

The course is structured in lectures and exercises in the computer laboratory. More precisely, the lectures are followed by laboratory exercises aimed at implementing the illustrated numerical methods in MATLAB and developing an adequate sensitivity and awareness of their use.

Lecture slides and other course materials, including practice exercises, application examples, and sample exams, will be made available on the Virtuale platform.

Although attendance is neither mandatory nor taken into account in the final assessment, it is strongly recommended, as it facilitates the learning process and contributes to a deeper understanding of the course topics.

In consideration of the type of activity and of the adopted teaching methods, the attendance of this training activity requires the prior participation of all students in Modules 1 and 2 of Health and Safety training courses in e-learning mode.

Assessment methods

The final examination is designed to assess the achievement of the following learning outcomes:

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

• ability to exploit basic numerical methods to solve mathematical problems of usual interest in engineering by using a computer.

The examination will take place in the computer laboratory. Students will have 90 minutes to complete two exercises which require both the development of MATLAB code and written answers to theoretical questions covering the topics discussed during the lectures.

For assessment purposes, only the files uploaded to the Esami On Line (EOL) platform before the end of the examination will be considered.

During the examination, students are not permitted to consult textbooks, notes, or any computer-based support tools. Access to online course materials is also prohibited, as is the use of generative Artificial Intelligence (AI) tools.

The examination is considered passed if the student achieves at least 18 points out of a maximum of 32. A score above 30 results in the award of “30 cum laude”, in recognition of an outstanding level of achievement.

Instructions and deadlines for declining a grade will be provided by email upon publication of the examination results.

To participate in the exam the student must register (at least 2 days in advance) in the lists available on the AlmaEsami web platform. On the day of the exam the student will be able to access the laboratory and take the test only after showing an identification document.

The dates of the exams can be consulted on the AlmaEsami web platform and are visible several months in advance.

Students with special educational needs related to learning disorders and/or disabilities are invited to contact the responsible office (see https://site.unibo.it/studenti-con-disabilita-e-dsa/en) as soon as possible, so that they can propose suitable adaptations to be submitted to the lecturer at least 15 days before the exam date.

Teaching tools

LCD projector and PC are used in addition to the traditional blackboard.

The course includes also a laboratory activity in which the MATLAB software will be used.

PDF files of slideshows, examples and solved exercises will be made available on the online platform Virtuale.

Students will also benefit from the support of a tutor who will help them to solve the exercises assigned by the lecturer during the course.

Office hours

See the website of Lucia Romani