20410413 - AN410 - NUMERICAL ANALYSIS 1

Provide the basic elements (including implementation in a programming language) of elementary numerical approximation techniques, in particular those related to solution of linear systems and nonlinear scalar equations, interpolation and approximate integration.
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Linear Systems
Direst methods: Gaussian elimination. Pivoting strategies. Gaussian elimination as a factorization. Doolittle and Choleshy factorizations. Iterative methods: Jacobi, Gauss-Seidel, SOR, Richardson, and related convergence results. Comparison of direct vs iterative solvers. Stability of algorithms for the solution of linear systems.

Iterative Methods for Scalar Nonlinear Equations
The intermediate zero theorem. The algorithms of bisection, Newton, secants, chords, and related convergence results. (Reference: Chapter 1 excluding Section 1.2.3, and Appendices A.1, A.2)

Approximation of Functions
General approximation strategies. Interpolating polynomial in Lagrange and Newton form. Representation of the interpolation error. Convergence of the interpolating polynomial for analytic functions. Refinement strategies in interpolation: Chebyshev nodes, composite approximations. Error estimates. Hermite polynomial, construction and representation of the error. Least Squares approximations. (Reference: Chapter 5 excluding Section 5.2, and Appendix A.4)

Numerical Integration
General principles of numerical integration. Polya's Theorem on the convergence of interpolatory quadrature formulae. Closed and open Newton-Cotes formulae. Stability results and error estimation. Generalized Newton-Cotes formulae and their convergence. Gaussian quadratures and their convergence. (Reference: Chapter 6)

Laboratory Activity
C language coding of some of the major algorithms, and in particular: Gaussian elimination, iterative methods for linear systems and scalar equations, Lagrange/Newton interpolation with a refinement strategy.

N.B.: References are provided with respect to the course notes.

Core Documentation

Roberto Ferretti, "Appunti del corso di Analisi Numerica", available from the course page

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available from the course page

Slides of the lessons, available from the course page

Reference Bibliography

Quarteroni, Sacco, Saleri, Gervasio: Numerical Mathematics (Springer)

Type of delivery of the course

The course is made of classroom activity (pledged to the theoretical part), and laboratory activity (for the coding aspects).



Type of evaluation

The theoretical part consists in a written 2h30 examination; a collection of the exam themes can be found on the course page. The laboratory part requires an additional test, consisting in: - a detailed discussion of the laboratory work (for students attending the lab classes); - a short (2h) C coding exam, on a topic close to what has been treated in the lab classes (for students not attending the lab classes).