NUMERICAL ANALYSIS I
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- Versione italiana
- Academic year
- 2021/2022
- Teacher
- VALERIA RUGGIERO
- Credits
- 9
- Didactic period
- Primo Semestre
- SSD
- MAT/08
Training objectives
- This course is the first one dealing with numerical mathematics, that the students face during their bachelor in Mathematics.
The goal here is to provide the students with the basics of the main Numerical Analysis problems, the main methods to face them and the basics of the Matlab computing environment, a software tool designed to solve instances of such problems.
The main knowledges provided to the students will be:
- computer's finite arithmetic and algorithms' computational complexity, both temporal and spatial;
- some of the main numerical methods for the solution of algebraic linear systems and for algebraic nonlinear equations;
- some methods for polynomial interpolation;
- basics for data approximation via the least squares approach;
- basics of the Matlab computing environment, its functions and its graphics support for data visualization.
The main skills the students are expected to acquire (that is to say, the abilities to apply the provided knowledges) will be:
- to operate with finite arithmetic, in both the fixed and the floating point settings, with different numeric basis;
- perform the first-order error analysis of simple algorithms;
- to estimate the conditioning of simple problems as well as the stability of simple algorithms;
- solve algebraic linear systems by both direct and iterative methods;
- to face algebraic nonlinear equations;
- compute polynomial interpolants;
- find the best approximated solution of a linear problem via the least squares approach;
- to program Matlab scripts and functions to solve simple problems and to suitably plot the results. Prerequisites
- To fruitfully attend the lectures, the following knowledges and abilities are needed, which are provided by the courses of Geometry 1, Mathematical Analysis 1 and Computer Programming:
- basic knowledge of the linear algebra;
- basic knowledges of mathematical analysis;
- basic knowledge of structured programming principles and of its structures. Course programme
- The classes will last 72 hours, with about 54 hours dedicated to theoretical concepts and the remaining dedicated to laboratory activities with Matlab. The time scheduling (indicated in parentheses) may vary, even significantly, depending on the difficulties the students have in the different segments of the program.
The classes will face the following topics:
Computer arithmetic (16): finite numbers representation and machine arithmetic operations; problem conditioning;algorithm stability;error propagation.
Solution of algebraic linear systems (20): direct methods (LDU, Cholesky, QR); iterative methods (Jacobi, Gauss-Seidel, SOR).
Solution of algebraic nonlinear equations (8):
conditions for the solution of algebraic nonlinear equations;main methods of order zero: bisection, regula falsi, secants; first and second order methods: Newton-Raphson's method (also known as tangents method).
Polynomial and piecewise polynomial interpolation (8): conditions for the polynomial interpolations; interpolation polynomial in both the Lagrange and the Newton forms; Hermite interpolation; piecewise polynomial interpolants: linear splines.
Data approximation (2): conditions for the solution of overdetermined algebraic linear systems;approximate solution via the least squares approach.
Laboratory activity with Matlab programming (18). Didactic methods
- The classes will have frontal lectures on all of the mentioned topics, as well as laboratory lectures for the implementation of the algorithms in the Matlab environment and their test on simple problems.
Learning assessment procedures
- The goal of the examination tests is to assess the learning level of the training objectives, with respect to both the knowledges as well as the skills, also including the laboratory activity in Matlab.
The examination is partitioned in a written test and an oral test. The written test will be dedicated to the laboratory topics in Matlab. Those who attain a mark not less than 18/30 in the written test are allowed to give the oral test.
The final mark is the arithmetic mean of the marks of the written and the oral tests.
The mark of the written test expires within the academic year. Reference texts
- [1] L.W. Johnson, R.D. Riess, "Numerical Analysis", second edition, Addison Wesley, 1982.
[2] V. Comincioli, "Analisi numerica", McGraw Hill, 1990.
[3] R.L. Burden, J.D. Faires, "Numerical Analysis", 10-th edition, Brooks/Cole Pub. Co., 2015.
[4] A. Quarteroni, F. Saleri, R. Sacco, Matematica numerica, Springer Verlag, 2008
[5] I. Galligani, Elementi di Analisi Numerica, Calderini Editrice Bologna, 1986
[6] Myron B. Allen, Eli L. Isaacson, Numerical Analysis for Applied Science, J. Wiley, 2019, e-book.
