NUMERICAL ANALYSIS II
Academic year and teacher
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- Versione italiana
- Academic year
- 2019/2020
- Teacher
- LORENZO PARESCHI
- Credits
- 6
- Didactic period
- Secondo Semestre
- SSD
- MAT/08
Training objectives
- The aim of this course is to complete the knowledge and the skills acquired in the course “Analisi Numerica I (Numerical Analysis I)”, with advanced topics and methods, with special focus on numerical stability and complexity.
The knowledge provided by this course regards numerical solution of nonlinear systems, numerical methods for derivation and integration, methods for the solution of differential equation systems with initial and boundary conditions.
Moreover, this course aims to develop student’s ability to adress and solve scientific computing problems, related to the topics of the lectures, until to the implementation of scripts in interactive computing and scientific visualization environments and the analysis of the numerical results. Prerequisites
- It is supposed that the students has well acquired the knowledge from the course “Analisi Numerica I (Numerical Analysis I)”; in particular, the student has to know the topics related to finite arithmetic, numerical methods for solving linear system and QR and LU factorization, polynomial interpolating techniques and piecewise polynomial interpolating techniques, numerical methods for nonlinear equations. It is assumed that the student has the basic knowledge for using Matlab. Moreover, the student has to have fully understood the main topics from the course of “Analisi Matematica I (Mathematical Analysis I)” (sequences, series, integrals, O.D.E. Systems) and “Geometria I (Geometry I)” (linear algebra).
Course programme
- 48 hours are scheduled, divided in theory about the numerical methods and in numerical simulation in I.T. Laboratory; these simulations are based on the implementation of the methods studied in class; the analysis regarding efficiency and effectiveness will be performed.
Nonlinear systems (fixed point method, local and global convergence, Newton and quasi-Newton methods, method globalization, inexact method); (10 hours).
Numerical derivation formulae (Richardson's estrapolation technique); stability. Numerical integration formulae: precision and stability. Interpolating formulae (Newton-Cotes); Polya's convergence theorem; composite quadrature formulas and convergence: Romberg's methods and adaptive methods. Gauss' formulae. Mention on multiple integration (15 hours).
Introduction on the Cauchy problems: well-position and Lyapunov stability; one-step methods: Taylor’s series methods andRunge-Kutta methods; consistency, 0-stability, convergence and error analysis; variable-step Runge-Kutta methods. Multistep methods (linear, explicit, implicit, predictor corrector methods): consistency, 0-stability and convergence;Dahlquist barrier. Absolute stability and stiffness. BDF methods. (18 hours).
Boundary problems. Shooting , collocation and finite difference methods (5 hours) Didactic methods
- Lectures on all the topics previously stated are scheduled. During the lessons the theoretical discussion is supported by exercises in I.T. laboratory where applied problems are faced and solved by implementing in MatLab the methods. Several exercises will be assigned during the course: such exercises have to be solved individually.
Learning assessment procedures
- The aim of the final exam consists in verifying the level of knowledge of the formative objectives previously stated.
The final exam consists in an oral test, dedicated to verify the knowledge of the methods explained during the course and to discuss the results obtained in the individual assigned exercise.
This discussion allows understand the level of knowledge and skills acquired by the students on the methods learned.
The oral test is successfully passed if a score of almost 18 is achieved. Reference texts
- - Teacher’s handouts
Reference Texts
- G.Naldi, L.Pareschi, G.Russo, Introduzione al calcolo scientifico, Mc-Graw Hill, 2003
- E. Isaacson, H.B. Keller, Analysis of numerical methods, Dover Publications, 1994
- J. Stoer, L. Bulirsch, Intrpduction to numerical analysis, Springer, 1993
- L.W.Johnson, R.D. Riess: Numerical Analysis, second edition, Addison Wesley 1982;
- V.Comincioli - Analisi numerica - McGraw Hill, 1990;
- Burden R. L., Faires J.D., Numerical Analysis, Prindle Weber & Schmidt, Boston MA. 1985.
