NUMERICAL ANALYSIS II
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- Versione italiana
- Academic year
- 2022/2023
- Teacher
- LORENZO PARESCHI
- Credits
- 6
- Didactic period
- Secondo Semestre
- SSD
- MAT/08
Training objectives
- The aim of the Numerical Analysis II course is to complete the knowledge and skills acquired in the Numerical Analysis I course, addressing advanced topics and methods.
The knowledge that the course aims to provide concerns the numerical solution of non-linear systems, optimization problems, numerical integration, methods for solving systems of differential equations with initial conditions, and boundary conditions.
The course also aims to develop students' ability to address and solve scientific computational problems related to the topics covered by the course, up to the creation of Matlab scripts in a scientific computing and visualization environment and the subsequent analysis of the results obtained. Prerequisites
- It is supposed that the students have well acquired the knowledge from the course Numerical Analysis I; in particular, the student has to know the topics related to finite arithmetic, numerical methods for solving linear systems, 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
- The course foresees 48 hours of teaching, during which the theory on numerical methods will alternate with the realization in the laboratory of numerical simulations based on the implementation of the methods and the related analysis and evaluation in terms of efficiency and effectiveness.
- Nonlinear systems (fixed point methods, local and global convergence, Newton and quasi-Newton classes, optimization methods, gradient free methods, stochastic techniques). [18 hours].
- Numerical derivation and integration. Richardson extrapolation. Numerical integration formulas. Interpolation formulas (Newton-Cotes); compound quadrature formulas and their convergence. Gauss formulas. Multidimensional integration. [12 hours]
- Numerical methods for ordinary differential equations. One-step methods: Taylor and Runge Kutta methods; Multi-step methods (linear explicit or implicit and predictor-corrector methods); Absolute stability and stiffness. BDF methods. IMEX methods. Boundary value problems, shooting method. [18 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
- G. Naldi, L. Pareschi, Matlab: concetti e progetti, 3a edizione, Maggioli Editore, 2020
- V.Comincioli - Analisi numerica - McGraw Hill, 1990;
- E. Isaacson, H.B. Keller, Analysis of numerical methods, Dover Publications, 1994
- J. Stoer, L. Bulirsch, Introduction to numerical analysis, Springer, 1993
- L.W.Johnson, R.D. Riess: Numerical Analysis, second edition, Addison Wesley 1982; - Burden R. L., Faires J.D., Numerical Analysis, Prindle Weber & Schmidt, Boston MA. 1985.
