The objective of this course is to illustrate the basic principles, techniques, and programming of the most common numerical methods for linear algebra and functional analysis. Particular effort will be devoted to the implementation aspects with practical exercise sessions performed using the Python programming language.

0 Foundations of Linear Algebra and Matrix Analysis
0.1 Vector Spaces
0.2 Matrices and operations with matrices
0.3 Rank and determinant of a matrix
0.4 Eigenvalues and eigenvectors
0.5 Scalar product and norms in vector spaces
0.6 Matrix norms

1 I Principles of Numerical Mathematics
1.1 Well-posedness and Condition Number of a Problem
1.2 Stability of Numerical Methods
1.3 A Priori And A Posteriori Analysis
1.4 Sources of Error in Computational Models

2 Solution of Linear Systems
2.1 Direct Methods
2.1.1 The Gaussian Elimination Method
2.1.2 LU Factorization
2.2 Computing the Inverse of a Matrix
2.3 Iterative Methods for Solving Linear Systems
2.3.1 Jacobi Method
2.3.2 Gauss-Seidel Method
2.3.3 Stationary and Nonstationary Iterative Methods
2.3.3 Convergence Analysis of the Richardson Method
2.3.4 Relaxation Method
2.3.5 The Gradient Method
2.3.6 The Conjugate Gradient Method
2.3.7 Stopping Criteria For Iterative Methods
2.4 Undetermined Systems
2.5 Regression
2.5.1 Least-squares method

3 Rootfinding for Nonlinear Equations and systems of nonlinear Equations
3.1 The Bisection Method
3.2 Metodi sul termine lineare e/o sulla sua derivata
3.2.1 The Methods of Chord
3.2.2 The Methods of Secant
3.2.3 The Newton's method
3.2.4 Fixed Point Methods
3.3 Stopping Criteria
3.4 Aitken’s Acceleration
3.3 Rootfinding for Systems of Nonlinear Equations
3.3.1 The Newton's Method
3.3.2 The Quasi Newton's Method
3.3.3 The Broyden's Method
3.3.4 Fixed Point Methods

4 Function Interpolation
4.1 Polynomial Interpolation
4.1.1 Interpolation with Lagrance Polynomials
4.2 Stability of Polynomial Interpolation

5 Numerical Integration
5.1 Interpolatory Quadratures
5.1.1 The Midpoint or Rectangle Formula
5.1.2 The Trapezoidal Formula
5.1.3 The Cavalieri-Simpson Formula
5.2 Newton-Cotes Formulae
5.3 Error Analysis of Numerical Integration
5.4 Gauss Points

6 Approximation of Eigenvalues and Eigenvectors
6.1 The Eigenvalue Problem 
6.2 The Power Method
6.2.1 Convergence of Power Method
6.2.2 Stopping Criteria
6.3 Inverse Power Method
6.4 Geometrical Location of the Eigenvalues
6.4.1 Theorem of the Gershgorin circles
6.5 The QR Iteration

7 Numerical Solution of Ordinary Differential Equations
7.1 The Cauchy Problem
7.2 Concept of Stability 
7.3 Numerical Approximation of Cauchy Problem
7.3.1 Euler Explicit
7.3.2 Euler Implicit
7.3.3 Crank-Nicholson
7.3.4 Heun's Method
7.4 Analysis of One-Step Methods
7.5 Multistep Methods
7.5.1 Adams Methods
7.6 BDF Methods
7.7 Runge-Kutta Methods

No prerequisites

Knowledge and understanding.  Learn the techniques for the numerical programming of numerical methods for linear algebra and functional analysis. At the end of the course, the student will have acquired a good knowledge of the mathematical topics covered in the classes. 

Applying knowledge and understanding. Acquiring the ability to implement numerical methods for linear algebra and functional analysis. Developing the ability to program, testing interpreting the results correctly. Acquiring the ability to solve mathematical problems using problem solving environment.  

Making judgments. acquiring the ability to find the most suitable numerical method for the solution of linear algebra or differential problems.

Communication skills. acquiring the ability to rigorously define the mathematical problem studied in the course and to expose its numerical methods, outlining its fundamental properties

Learning skills. ability to study and solve problems similar but not necessarily the same as those dealt with during lessons.

The teaching material prepared by the lecturer in addition to recommended textbooks (such as for instance slides, lecture notes, exercises, bibliography) and communications from the lecturer specific to the course can be found inside the Moodle platform › blended.uniurb.it

Teaching

- Frontal lessons

- Examples and exercises using Python in the computer laboratory

Attendance

Not compulsory 

Course books

Quarteroni, A., Sacco, R., & Saleri, F. (2007). Numerical Mathematics. In Texts in Applied Mathematics. Springer New York. https://doi.org/10.1007/b98885

Assessment

The assessment consists of a project in python and an oral examination.

- The project, which has to be developed by groups composed of one or preferably two students on a set of problems that changes each academic year, consists of the implementation of a colab notebook in python. The project has to be submitted at least 10 days before the oral exam. The project is passed if the mark is at least 18/30; the mark is valid, even if the oral exam is taken but not passed.

- The oral examination consists of two questions on the topics covered in the classes. It is passed if the mark is at least 18/30.

Both the project and the oral exam must have a mark of at least 18/30. The final mark is the weighted average of the project and the oral exam (25% project, 75% oral exam).

Disability and Specific Learning Disorders (SLD)

Students who have registered their disability certification or SLD certification with the Inclusion and Right to Study Office can request to use conceptual maps (for keywords) during exams.

To this end, it is necessary to send the maps, two weeks before the exam date, to the course instructor, who will verify their compliance with the university guidelines and may request modifications.