Learning objectives
- Knowledge and ability to understand the language and the typical problems in the transition from continuous mathematics to discrete mathematics.
- Ability to apply knowledge and understanding in critical analysis of obtained numerical results.
- Autonomy of judgment in evaluating the approximation algorithms and the obtained results also through discussion with one's peers.
- Ability to clearly communicate the concepts acquired and to argue the results achieved.
- Ability to learn limits and advantages of numerical methods and to apply them consistently.
Prerequisites
Basic concepts of Mathematical Analysis and Linear Algebra.
Course unit content
- Introduction to MATLAB.
- Error analysis.
- Approximation of data and functions.
- Numerical integration by Newton-Cotes formulas.
- Resolution of linear systems: direct methods, factorizations, iterative methods.
- Numerical resolution of non-linear equations.
Full programme
Floating Point system: representation of machine numbers; error of representation, numerical cancellation, conditioning.
Research of roots of non-linear equations: problem conditioning; method of bisection, of chords, of secants, of Newton; criteria of arrest.
Resolution of linear systems: condition number associated to a matrix; forward and backward substitution methods for triangular matrices; Gauss elimination algorithm; LU factorization; Cholesky factorization; iterative methods; Jacobi method; Gauss-Seidel method.
Approximation of data and functions: simple Lagrangian interpolation; Newton’s form of the interpolation polynomial; Lagrange composite interpolation; least squares method.
Numerical integration: interpolatory quadrature formulas; rectangle formula; trapezoid formula; Cavalieri-Simpson formula; composite formulas.
Bibliography
- "Numerical analysis". L.W. Johnson, R.D. Riess. Addison-Wesley (1982).
Teaching methods
After an initial introduction to the Matlab programming language, the course contents will be analyzed highlighting the problems related to the introduced numerical techniques. The course will also provide a part of re-elaboration in cooperative learning, supervised by the professor, consisting in the application of numerical techniques, through
programming in Matlab. This activity will allow the student to acquire the ability to face "numerical" difficulties and to evaluate the reliability and consistency of the obtained results.
Assessment methods and criteria
The exam includes a laboratory test regarding knowledge and skills acquired during the course. The threshold of sufficiency is fixed to the knowledge of the algorithms proposed during the course and to their implementation
in the Matlab language. Moreover there will be questions on theoretical issues. The exam is in presence (or online if requested by the guidelines of the University).
Other information
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2030 agenda goals for sustainable development
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