Course Details for A.Y. 2016/2017
Name:
Analisi Numerica / Numerical Analysis
Basic information
Language:
Italian
![[info]](img/info16.png)
Course Objectives
provide thr mathematical instruments for the numerical solution of basic problems in applied sciences and create the ability to develope algorithms
by means of Matlab, a structured programming language. The course consists of 9 credits (90 hours).
Course Content
- Numerical problems and algorithms.
- Conditioning and stability.
- Numerical linear algebra.
- Methods to solve linear systems.
- Methods for eigenvalue problems.
- Approximation of functions.
- Interplolation.
- Least square approximation.
- Iterative methods for root finding.
- Quadrature formulas.
Learning Outcomes (Dublin Descriptors)
On successful completion of this course, the student should
- learn the following subjects:
-
Error analysis:
Representation of real numbers in a specific base. Floating point representation. Machine numbers. Rounding and truncation.
Machine operations. Error propagation. Conditioning and stability of algorithms. Elements of programming: the Matlab language.
-
Numerical linear algebra:
Vectors, matrices and their properties. Norms. Eigenvalues and spectral radius. Relationships between norms and spectral radius. Special
classes of matrices (Hermitian matrices, positive definite matrices, etc..). Direct methods for the solution of linear systems: triangular systems,
Gaussian elimination, pivoting. LU and L LH factorizations.
Cholesky factorization. Conditioning of a linear system. Condition numbers.
3. Elements of programming: the Matlab programming language.
- Eigenvalue and eigenvector computation. Localization of eigenvalues in the complex plane. Perturbation theormes for eigenvalues. Power method and
Wielandt variant for the computation of specific eigenvalues and eigenvectors.
5. Interpolation and approximation. Computation of an algebraic polynomial at a point. Lagrange form. Linear interpolation operator
Interpolation error. Chebyshev polynomials. Recursive formula, zeros, minimum norm property. Computation of the interpolating
polynomial. Newton devided difference representation.
6. Hints on convergence of interpolatory processes. Piecewise polynomial interpolation. Spline functions. Computation of the cubic spline.
7. Quadrature formulas. General form of a qf. Polynomial order. Interpolatory formulas. Convergence theorem. Newton-Cotes formulas.
Gauss formulas. Empiric error estimate. Adaptive quadrature (hints).
-
Iterative methods for the solution of large linear systems. Splitting methods; general convergence theorem; error control. Jacobi and Gauss-Seidel
methods. Convergence of Jacobi methods on diagonally dominant systems.
-
Numerical methods for approximating roots of a nonlinear equation (in one variable). Bisection method. Iterative methods: general theory.
Newton method and its variants.
Prerequisites and Learning Activities
Mathematical Analysis and Linear Algebra
Assessment Methods and Criteria
Written test, lab test (optional oral test)
Textbooks
- E.Isaacson, H.Keller, Analysis of numerical methods , J.Wiley & sons, New York. 1966.
- G.Monegato, Calcolo Numerico , Levrotto e Bella, Torino. 1985.
- J.Stoer, R.Bulirsch, Introduction to Numerical Analysis , Springer Verlag. 1993.
- W. J. Palm III, Matlab 6 per l'Ingegneria e le Scienze , Mc Graw Hill. 2003.
- D. Bini, M. Capovani e O. Menchi, Metodi numerici per l'algebra lineare , Zanichelli. 1988.
Course page updates
This course page is available (with possible updates) also for the following academic years:
To read the current information on this course, if it is still available, go to the university course catalogue .
Course information last updated on: 29 novembre 2016, 15:14
Italian version
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