# NUMERICAL ANALYSIS

**Academic Year 2023/2024**- Teacher:

**SEBASTIANO BOSCARINO**

## Course Structure

To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or compensatory measures, based on the didactic objectives and specific needs. It is also possible to contact the referent teacher CInAP (Center for Active and Participated Integration - Services for Disabilities and / or SLD) of our Department, prof. Filippo Stanco.

## Detailed Course Content

**Introduction to the use of the computer for numerical calculations.**

Use of matlab.

loating point representation. Machine numbers. Truncation and rounding. Machine operations. Digital cancellation. Order of accuracy.

**Numerical linear algebra**: vectors, matrices, determinants, inverse
matrix. Vector and matrix norms. Natural norms and their representation.
Eigenvalues. Spectral radius and its properties. Some special matrices.
Direct methods for solving linear systems: triangular systems, Gaussian
elimination, pivoting. Factorization A = LU and PA = LU. Choleski factorization and their implementation.
Conditioning of a linear system. Condition number. Sparse matrices and
their representation.
Iterative methods for solving linear systems: Jacobi method,
Gauss-Seidel method, SOR method. Stopping criteria. . Localization
of the eigenvalues: the theorems Gershgorin-Hadamard. Eigenvalues
calculation: direct and inverse power methods.

**Approximation of functions and data: **Polynomial interpolation. Lagrange form. Linear interpolation
operator. Calculation of the interpolation polynomials. Newton's formula
of divided differences. The error formula (Lagrange form). Chebyshev
polynomials, the property of minimal norm. The problem of the
convergence of a sequence of interpolators schemes. Interpolation by
piecewise-polynomials. Splines. Calculation of cubic splines. Method of
least squares and applications. Normal equations and their geometric
interpretation.

**Solution of nonlinear equations: **General concepts. Methods of bisection, secant and Newton. General
theory of iterative methods for nonlinear equations and fixed point
problems. Convergence order. Stopping criteria.

**Quadrature formulas: **Weighted integrals. General form of a quadrature formula. Polynomial
order (or degree of precision) of a quadrature formula. Interpolatory
formulas. Convergence theorem. Newton-Cotes formulas. Gaussian formulas.
Composite formulas: trapezoid and Simpson formulae. Romberg method.
Adaptive quadrature.

## Learning Assessment

### Learning Assessment Procedures

The final exam consists of a written test and an oral interview.

Access to the oral interview is granted if the written test is passed with a grade of no less than 18/30.

The exam is considered passed if an oral interview is judged to be at least sufficient (18/30).

Booking for an exam session is mandatory and must be made exclusively via the internet through the student portal within the set period.

**Criteria for assigning marks:**
both for the written and the oral exams, the following will be taken
into account: clarity of presentation, completeness of knowledge,
ability to connect different topics. The student must demonstrate that
they have acquired sufficient knowledge of the main topics covered
during the course and that they are able to carry out at least the
simplest of the assigned exercises.

The following criteria will normally be followed to assign the grade:

Not approved: the student has not acquired the basic concepts and is not able to carry out the exercises.

18-23: the student demonstrates minimal mastery of the basic concepts, their skills in exposition and connection of contents are modest, they are able to solve simple exercises.

24-27: the student demonstrates good mastery of the course contents, their presentation and content connection skills are good, they solve the exercises with few errors.

28-30 cum laude: the student has acquired all the contents of the course and is able to explain them fully and connect them with a critical spirit; they solve the exercises completely and without mistakes.