First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Science
MATHEMATICS
Course unit
NUMERICAL ANALYSIS
SCM0014413, A.A. 2018/19

Information concerning the students who enrolled in A.Y. 2016/17

Information on the course unit
Degree course First cycle degree in
MATHEMATICS
SC1159, Degree course structure A.Y. 2008/09, A.Y. 2018/19
N0
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Number of ECTS credits allocated 7.0
Type of assessment Mark
Course unit English denomination NUMERICAL ANALYSIS
Website of the academic structure http://matematica.scienze.unipd.it/2018/laurea
Department of reference Department of Mathematics
Mandatory attendance No
Language of instruction Italian
Branch PADOVA
Single Course unit The Course unit can be attended under the option Single Course unit attendance
Optional Course unit The Course unit can be chosen as Optional Course unit

Lecturers
Teacher in charge ALVISE SOMMARIVA MAT/08

ECTS: details
Type Scientific-Disciplinary 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

Calendar
Start of activities 25/02/2019
End of activities 14/06/2019

Syllabus
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, three-terms recurrence, the theorem of the zeros, classical orthogonal polynomials, Chebyshev polynomials.

Numerical quadrature.
Algebraic and composite rules, Gaussian rules, Polya-Steklov 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 (Crank-Nicolson), 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. Cerca nel catalogo
  • Epperson, James F., Introduzione all'analisi numericateoria, metodi, algoritmiJames F. Epperson. Milano [etc.]: McGraw-Hill, 2003. Cerca nel catalogo

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
  • Latex
  • Matlab

Sustainable Development Goals (SDGs)
Quality Education Gender Equality