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## NUMERICAL ANALYSIS AND LINEAR ALGEBRA

CODE 61804 2020/2021 9 cfu during the 2nd year of 8759 INFORMATICA (L-31) - GENOVA MAT/08 Italian GENOVA 1° Semester AULAWEB

## OVERVIEW

The course introduces to the basic concepts on using the computer to solve applied mathematical problems.

## AIMS AND CONTENT

### LEARNING OUTCOMES

The course aims to introduce fundamental concepts of numerical analysis (complexity, errors);  we present the main computational methods for solving the most relevant problems of numerical linear algebra and some interpolation and minimization problems.

### TEACHING METHODS

Lectures are mainly given in classroom, except for 2 lab sessions in the official timetable.

In addition, in the second half of semester 2 hours a week are scheduled outside the official timetable, under the assistance of a tutor.

### SYLLABUS/CONTENT

• Error analysis
• Floating-point numbers and machine precision.
• Inerent error. Estimate for rational functions.
• Algorithmic error.
• Total error.
• Solution of nonsingular linear systems
• Numerical solution of linear systems (direct method of Gaussian elimination).
• Conditioning of matrices.
• Complexity and algorithmic error for the solution of linear systems.
• Other topics in linear algebra: geometric interpretation of vectors and matrices
• Scalar product and orthonormal bases.
• Matrices as geometric linear transformations.
• Null space, range and rank.
• Orthogonal matrices: rotations, reflections, QR factorization.
• Approximated solution of linear systems in the least-squares sense
• Geometric formulation of the problem.
• Normal equations.
• Solution through orthogonalization.
• Interpolation by spline functions
• Definition of interpolating spline.
• Computational procedure.
• Survey of mathematical and numerical properties.
• Other topics in linear algebra: eigenvalues
• Eigenvalues, eigenvectors, eigenspaces.
• Characteristic polynomial.
• Similarity relations e diagonalization.
• Applications.
• SVD and applications to least-squares
• Singular values decomposition (SVD) and relations with eigenvalues.
• Geometric properties of SVD and numerical rank.
• Generalized inverse and conditioning.
• Solution of the least-squares problem via SVD.
• Application to discrete data approximation (smoothing).
• Numerical treatment of eigenvalues
• Numerical properties: conditioning and localization.
• Iterative power method and variants.
• Other numerical methods: similarity reduction to a simplified form, QR method.

Computer experiences in C and Matlab languages are planned.

For the parts of the program concerning linear algebra complements, any classic textbook of linear algebra and geometry can help; for instance,

Serge Lang, Linear Algebra, Third Edition. Springer-Verlag New York, 1987.

Concerning the numerical analysis content, the use of lesson afternotes is recommended. Also available on Aulaweb are the notes of the course (in italian) taken by student Stefano Sabatini in the academic year 2010-11 and supervised by the teacher. Common textbooks are generally oversized with respect to the course. Just for reference, we suggest

J. Stoer, R. Bulirsch, Introduction to Numerical Analysis. Springer-Verlag New York, 2002.

## TEACHERS AND EXAM BOARD

### Exam Board

FABIO DI BENEDETTO (President)

MARIA EVELINA ROSSI

FEDERICO BENVENUTO (Substitute)

SABRINA GUASTAVINO (Substitute)

## LESSONS

### Class schedule

All class schedules are posted on the EasyAcademy portal.

## EXAMS

### EXAM DESCRIPTION

The final mark is the sum of the scores of

- laboratory exam

- the written exam

which can be passed in any order.

### ASSESSMENT METHODS

LABORATORY EXAM

4 sheets exercises will take place during the course. For each sheet, each group must deliver the product code, the output results, and a report describing (and possibly explaining) them. 2 sheets must be solved in C and are mandatory to pass the exam, giving a score from 0 to 3 points; other 2 sheets must be solved in Matlab and are optional, giving a score from 0 to 2 points. The deliveries will be evaluated taking into account the following aspects in descending order of relevance:

1. Working code that produces reasonable results (minimum requirement for passing the exam);
2. Efficiency, clarity and readability in presenting the results in the report;
3. Explanation of the results, in the light of the theory;
4. Style and readability of programs;
5. Program Computing Efficiency.

WRITTEN EXAM

It lasts between 2 and 3 hours (depending on the overall difficulty) and is based on theoretical questions and exercises. A maximum score of 26 is given; In the case of a score less than 18, the written exam is not passed.

### Exam schedule

Date Time Location Type Notes
22/01/2021 11:00 GENOVA Scritto I parte ore 11 (solo per 9 CFU) + II parte ore 14 (per tutti) Gli studenti che hanno nel piano 9 CFU devono partecipare ad entrambe le parti della prova (compresa quella delle ore 11). Gli studenti che hanno nel piano 6 CFU devono partecipare solo alla seconda prova (ore 14). Consultare le istruzioni e le modalità di esame visibili su Aulaweb
11/02/2021 11:00 GENOVA Scritto I parte ore 11 (solo per 9 CFU) + II parte ore 14 (per tutti) Gli studenti che hanno nel piano 9 CFU devono partecipare ad entrambe le parti della prova (compresa quella delle ore 11). Gli studenti che hanno nel piano 6 CFU devono partecipare solo alla seconda prova (ore 14). Consultare le istruzioni e le modalità di esame visibili su Aulaweb
23/06/2021 11:00 GENOVA Scritto I parte ore 11 (solo per 9 CFU) + II parte ore 14 (per tutti) Gli studenti che hanno nel piano 9 CFU devono partecipare ad entrambe le parti della prova (compresa quella delle ore 11). Gli studenti che hanno nel piano 6 CFU devono partecipare solo alla seconda prova (ore 14). Consultare le istruzioni e le modalità di esame visibili su Aulaweb
22/07/2021 11:00 GENOVA Scritto I parte ore 11 (solo per 9 CFU) + II parte ore 14 (per tutti) Gli studenti che hanno nel piano 9 CFU devono partecipare ad entrambe le parti della prova (compresa quella delle ore 11). Gli studenti che hanno nel piano 6 CFU devono partecipare solo alla seconda prova (ore 14). Consultare le istruzioni e le modalità di esame visibili su Aulaweb
16/09/2021 11:00 GENOVA Scritto I parte ore 11 (solo per 9 CFU) + II parte ore 14 (per tutti) Gli studenti che hanno nel piano 9 CFU devono partecipare ad entrambe le parti della prova (compresa quella delle ore 11). Gli studenti che hanno nel piano 6 CFU devono partecipare solo alla seconda prova (ore 14). Consultare le istruzioni e le modalità di esame visibili su Aulaweb

### FURTHER INFORMATION

PREVIOUS KNOWLEDGE

It is assumed that the student is familiar with linear algebra operations, differential and integral calculus in one variable and programming in C or C++.