CODE  61804 

ACADEMIC YEAR  2023/2024 
CREDITS 

SCIENTIFIC DISCIPLINARY SECTOR  MAT/08 
LANGUAGE  Italian 
TEACHING LOCATION 

SEMESTER  1° Semester 
TEACHING MATERIALS  AULAWEB 
OVERVIEW
The course introduces to the basic concepts on using the computer to solve applied mathematical problems (in particular, solution of linear systems and data approximation) and provides basic notions of linear algebra with particular regard to matrix calculus, vector spaces, solution of linear systems and canonical form of matrices.
AIMS AND CONTENT
LEARNING OUTCOMES
Acquiring the basic notions of linear algebra (vectors, matrices, linear transformations and eigenvalues) and of numerical analysis (complexity and error). Knowing the main computational methods for solving numerical linear algebra problems and some approximation problems.
AIMS AND LEARNING OUTCOMES
At the end of the course, the student will be able to:
 Know the fundamentals of numerical computation and know how to evaluate the conditioning of simple mathematical problems and computational cost and stability for some basic algorithms, in particular in the case of linear systems solution.
 Apply matrix theory and vector calculus to numerical analysis problems.
 Understand the fundamental relationships between linear algebra and geometry, know the tool of orthogonal matrices and how to use them for reduction algorithms, understand the concept of eigenvalues and know how to compute them for small matrices.
 Understand the concept of approximation in its various forms, know some techniques and how to solve linear least squares problems.
 Implement some numerical algorithms on the computer and evaluate the reliability of the results.
PREREQUISITES
 Basics of algebraic structures
 Differential and integral calculus
 Programming in C or C++
TEACHING METHODS
Traditional.
Lectures are mainly given in classroom (apart pandemics restrictions), 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
 Floatingpoint numbers and machine precision.
 Inerent error. Estimate for rational functions.
 Algorithmic error.
 Total error.
 Basics of linear algebra and solution of nonsingular linear systems
 Matrix operations and inversion.
 Solution of linear systems by Gaussian elimination.
 Determinants and rank of matrices. Theorems of Laplace, Cramer and RouchéCapelli.
 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 leastsquares 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 leastsquares
 Singular values decomposition (SVD) and relations with eigenvalues.
 Geometric properties of SVD and numerical rank.
 Generalized inverse and conditioning.
 Solution of the leastsquares 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.
RECOMMENDED READING/BIBLIOGRAPHY
For the parts of the program concerning linear algebra basics, any classic textbook of linear algebra and geometry can help; for instance,
Serge Lang, Linear Algebra, Third Edition. SpringerVerlag 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 201011 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. SpringerVerlag New York, 2002.
TEACHERS AND EXAM BOARD
Ricevimento: Reception hours: 1314 on lesson days, prior to email confirmation.
Exam Board
FABIO DI BENEDETTO (President)
FEDERICO BENVENUTO
MATTEO VARBARO (President Substitute)
DAVIDE PARODI (Substitute)
LESSONS
LESSONS START
According to the academic calendar approved for the whole Undergraduate Programme.
Class schedule
L'orario di tutti gli insegnamenti è consultabile all'indirizzo EasyAcademy.
EXAMS
EXAM DESCRIPTION
The course is divided into two parts (one theoretical on basic linear algebra; one more numerical with some complements of theory), which contribute to the final grade with 1/3 and 2/3 weights respectively. To take the exam students must pass (in any order)
 the written test (carried out in a single day and divided into two subtests, referring to the two parts of the course);
 the laboratory test on the second part.
The grade of the first part coincides with the evaluation of the relative written test; the grade of the second part is represented by the sum of the scores of the relative written test and of the laboratory.
ASSESSMENT METHODS
WRITTEN TEST
It includes theoretical questions and exercises to verify the achievement of the learning outcomes described in the appropriate section; it is divided into two parts with separate deliveries.
The first part of the test, concerning the basic concepts of linear algebra, takes place in the morning and lasts 1 hour; if its score is lower than 16 (out of a maximum of 32), the entire written test is not passed. The second part of the test (held in the afternoon, lasting 2 hours and 30 minutes) is assigned a maximum score of 27; if its score is less than 18 (after rounding off), the entire written test is not passed. It is not allowed to keep the grade of one part by repeating the other.
If it is necessary to carry out the written online, consult the specific instructions that will be made available on Aulaweb.
LABORATORY TEST
4 sheets of 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 or 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:
 Working code that produces reasonable results (minimum requirement for passing the exam);
 Efficiency, clarity and readability in presenting the results in the report;
 Explanation of the results, in the light of the theory;
 Style and readability of codes;
 Code computational efficiency.