
Course unit
NUMERICAL ANALYSIS
SCM0014413, A.A. 2018/19
Information concerning the students who enrolled in A.Y. 2016/17
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Core courses 
MAT/08 
Numerical Analysis 
7.0 
Course unit organization
Period 
Second semester 
Year 
3rd Year 
Teaching method 
frontal 
Type of hours 
Credits 
Teaching hours 
Hours of Individual study 
Shifts 
Practice 
3.0 
24 
51.0 
No turn 
Laboratory 
1.0 
16 
9.0 
No turn 
Lecture 
3.0 
24 
51.0 
No turn 
Start of activities 
25/02/2019 
End of activities 
14/06/2019 
Prerequisites:

Numerical Analysis 
Target skills and knowledge:

Advanced knowledge of Numerical Analysis and its applications in Applied Mathematics. 
Examination methods:

Written exam (theory) and oral exam (laboratory). 
Assessment criteria:

In the written examination we understand the theoretical knowledge obtained by the student, with particular attention to the comprehension of the algorithms, their properties, and proofs of major theorems.
In the part of lab, we evaluate how much the students have programming skills as well as understanding of the performed numerical experiments. 
Course unit contents:

Interpolation.
Orthogonal polynomials.
Numerical quadrature.
Iterative methods for linear algebra.
Nonlinear systems.
Eigenvalues.
Finite differences methods for ODEs and PDEs. 
Planned learning activities and teaching methods:

Interpolation.
The general problem of interpolation, unisolvent sets and determinantal formula of Lagrange, the univariate and multivariate case, Lebesgue constant, fundamental estimate for interpolation error, stability, brief introduction to tensorial product interpolation and Fekete points.
Orthogonal polynomials.
Orthogonalization of the monomial basis, threeterms recurrence, the theorem of the zeros, classical orthogonal polynomials, Chebyshev polynomials.
Numerical quadrature.
Algebraic and composite rules, Gaussian rules, PolyaSteklov theorem, stability, Stieltjes theorem, brief introduction to product rules.
Numerical linear algebra.
Fundamental theorem of matrix inversion and applications (Gershgorin theorem of eigenvalues localization), iterative methods for linear systems, successive approximation theorem, preconditioning, gradient method, step and residual stop criteria, methods for the computation of eigenvalues and eigenvectors, Rayleigh quotient, power method and variants, QR method.
Numerical nonlinear algebra.
Solution of nonlinear systems of equations, fixed point iterations and Banach theorem, convergence estimates and stability, Newton method, local convergence and speed of convergence, step criterion, Newton method as fixed point iteration.
Finite difference methods for ODEs and PDEs.
Initial value problem: Euler method (explicit and implicit), convergence and stability in the Lipschitzian and dissipative case, trapezoidal method (CrankNicolson), stiff problems, conditional and unconditional stability; boundary problems: finite difference methods for the Poisson equations in 1D and 2D, structure of the linear system and convergence, computational issues; the lines method for the heat equation in the 1D and 2D case, relationships with the stiff problems. 
Additional notes about suggested reading:

Slides (as PDF files). 
Textbooks (and optional supplementary readings) 

Quarteroni, Alfio; Saleri, Fausto, Introduzione al Calcolo Scientifico. Springer Milan: , 2006.

Epperson, James F., Introduzione all'analisi numericateoria, metodi, algoritmiJames F. Epperson. Milano [etc.]: McGrawHill, 2003.

Innovative teaching methods: Teaching and learning strategies
 Lecturing
 Laboratory
 Problem based learning
 Working in group
 Problem solving
 Loading of files and pages (web pages, Moodle, ...)
 Slides
Innovative teaching methods: Software or applications used
Sustainable Development Goals (SDGs)

