Aim of the course is to give students some basic tools and topics in Functional Analysis and Numerical Methods.

1. The foundations of numerical mathematics
1.1 Good location and conditioning number of a problem
1.2 Stability of numerical methods
1.3 A priori and a posteriori analysis
1.4 Sources of errors in computational models
1.5 p-adic representations

2. Foundations of Matrix Analysis 
2.1 Vector Spaces 
2.2 Matrices and linear maps
2.3 Operations with Matrices
2.4 Trace and Determinant of a Matrix 
2.5 Rank and Kernel of a Matrix 
2.6 Special Matrices
2.7 Eigenvalues and Eigenvectors 
2.8 Similarity Transformations 
2.9 The Singular Value Decomposition (SVD) 
2.10 Scalar Product and Norms in Vector Spaces 
2.11 Matrix Norms 
2.12 Positive Definite, Diagonally Dominant and M-Matrices

3 Solving linear systems
3.1 Direct methods
3.1.1 Gauss elimination method
3.1.2 Gauss factorization method
3.1.3 Cholewski factorization method
3.1.4 Accuracy of Results
3.2 Iterative methods
3.2.1 Jacobi method
3.2.2 Gauss-Seidel method
3.2.3 Convergence of iterative methods
3.2.4 Relaxation methods
3.2.5 Shutdown test
3.3 Indeterminate systems
3.4 Regression
3.4.1 Approximation of functions with least squares
3.5 Advanced iterative methods
3.5.1 Methods of descent
3.5.2 Gradient method
3.5.3 Conjugate gradient method

4 Roots of nonlinear equations
4.1 Bisection method
4.2 Methods on the linear term and/or on its derivative
4.2.1 String method
4.2.2 Method of secants
4.2.3 Newton or tangent method
4.2.4 Shutdown test
4.3 Roots of a system of nonlinear equations

5. Non-linear algebraic systems
5.1 Variational formulation
5.2 Principle of weak maximum
5.3 Principle of maximum strong
5.4 The discrete Dirichlet problem
5.5 Difference equations
5.6 Examples and applications

6 Interpolation of functions and numerical integration
6.1 Polynomial interpolation
6.1.1 Piecewise polynomial interpolation
6.2 Approximations with splines functions
6.3 Hermite interpolation
6.4 Interpolation quadrature formulas
6.4.1 Calculation of the error
6.4.2 Composite linear interpolation
6.4.3 Gauss points
6.5 Quadrature formulas in two dimensions

7 Ordinary differential equations: the Cauchy problem
7.1 Consistency
7.2 Stability
7.2.1 Minimum stability and zero-stability
7.3 Convergence
7.4 A-stability
7.4.1 Explicit Euler
7.4.2 Implicit Euler
7.4.3 Crank-Nicholson
7.4.4 Regions of A-stability
7.5 Computational costs of implicit methods

8. Appendix: Normed spaces, metric spaces, topology
8.1 Normed vector spaces
8.2 Euclidean spaces: inner product
8.3 Metric spaces
8.4 Topological spaces
8.5 Limits
8.6 Product of topological spaces
8.7 Linear functions between vector spaces
8.8 Norms. Equivalent Norms
8.9 Continuity of multilinear functions
8.10 Sequences and subsequences
8.11 Metric spaces: completeness
8.12 Normed spaces and completeness: canonical examples
8.13 Equivalent norms in finite vector spaces
8.14 Sequential notion of compactness
8.15 Subspaces of compact spaces
8.16 Continuous functions on compact spaces
8.17 Product of compact spaces
8.18 Applications
8.19 Connected spaces
8.20 The Banach-Caccioppoli theorem

Some basic knowledges of Mathematical Analysis and Linear Algebra are necessary.

Knowledge and understanding. At the end of the course the student must have acquired a good knowledge of the mathematical topics covered in the course. He must be able to argue correctly and with language properties on the topics covered in the program. Examples and working methods are shown in the classroom during the lessons and proposed in the exercises.

Applied knowledge and understanding. At the end of the course the student must have acquired a good ability to use the main tools of basic mathematics. He must be able to correctly apply the formulation studied and must be able to solve general mathematical problems similar to those studied. In particular, he must be able to apply the acquired knowledge even in contexts slightly different from those studied, and have the ability to use the acquired knowledge to independently solve problems that may appear new. Examples of such applications are shown in the classroom during the lessons and proposed in the exercises.

Autonomy of judgment. At the end of the course the student must have acquired a good ability to analyze topics and problems in general mathematics, the ability to critically evaluate any proposed solutions, and a correct interpretation of similar topics.

Communication skills. At the end of the course the student must have acquired a good ability to clearly communicate his / her statements and considerations concerning general mathematics problems. The working method is shown in the classroom during the lessons and proposed in the exercises.

Ability to learn. At the end of the course the student must have acquired a good capacity for autonomy in the study of the discipline, in the reading and interpretation of a qualitative phenomenon, in the search for useful information to deepen the knowledge of the topics covered.

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

There are no supporting activities.

Teaching

Theorical and practical lessons.

Attendance

Although strongly recommended, course attendance is not mandatory.

Course books

A. Quarteroni, F. Saleri, R. Sacco, Matematica Numerica, Springer Verlag, Ed. 3, 2008.

R. Bevilacqua, D. Bini, M. Capovani, O. Menchi, Introduzione alla Matematica Computazionale, Zanichelli, 1987.

G. De Marco, Analisi Due. Teoria ed Esercizi. Zanichelli, Bologna, 1999.

W. Rudin. Principles of Mathematical Analysis. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill, New York-Auckland-Düsseldorf, 1976.

S. Salsa, Partial Differential Equations in Action, From Modelling to Theory  - Springer, 2007. 

Assessment

The expected objectives are verified through the following two tests, both mandatory:

1. A formative assessment test: consisting of a written paper - lasting 2 hours and 30 minutes - divided into five exercises on the following topics:

A theme

Two exercises

on the treated theoretical arguments.
2. An oral interview: consisting of the discussion of the written paper and three open questions on the theoretical topics covered in the course.

For both tests, the evaluation criteria are as follows:
- relevance and effectiveness of the responses in relation to the contents of the program;
- the level of articulation of the response;
- adequacy of the disciplinary language used.

The overall evaluation is expressed with a mark out of thirty taking into account both tests.

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.

Teaching

Theorical and practical lessons.

Attendance

Although strongly recommended, course attendance is not mandatory.

Course books

A. Quarteroni, F. Saleri, R. Sacco, Matematica Numerica, Springer Verlag, Ed. 3, 2008.

R. Bevilacqua, D. Bini, M. Capovani, O. Menchi, Introduzione alla Matematica Computazionale, Zanichelli, 1987.

G. De Marco, Analisi Due. Teoria ed Esercizi. Zanichelli, Bologna, 1999.

W. Rudin. Principles of Mathematical Analysis. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill, New York-Auckland-Düsseldorf, 1976.

S. Salsa, Partial Differential Equations in Action, From Modelling to Theory  - Springer, 2007. 

Assessment

The expected objectives are verified through the following two tests, both mandatory:

1. A formative assessment test: consisting of a written paper - lasting 2 hours and 30 minutes - divided into five exercises on the following topics:

A theme

Two exercises

on the treated theoretical arguments.
2. An oral interview: consisting of the discussion of the written paper and three open questions on the theoretical topics covered in the course.

For both tests, the evaluation criteria are as follows:
- relevance and effectiveness of the responses in relation to the contents of the program;
- the level of articulation of the response;
- adequacy of the disciplinary language used.

The overall evaluation is expressed with a mark out of thirty taking into account both tests.

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.