BASIC NUMERICAL METHODS

Knowledge and understanding

Acquisition of the main techniques of numerical approximation with the execution of many exercises in order to clear the main focuses of the course. During a tutoring in the afternoon the student will learn the algorithms with the Python language (Canopy version) related tto the main problems of Numerical Analysis. In particular, the student will get acquainted with matrices and the resolution of linear systems with direct and iterative methods, minquad theory, eigenvalues and eigenvectors, quadrature formulas and roots of nonlinear functions approximation.

Applying knowledge and understanding

During the course the student will be encouraged to solve many exercises which will be solved and commented in class. If the time will not be enough, they will be given as homeworks and after commented in class. Also practical lectures will be given to solve, through the use of the computer, the numerical methods.

Making judgements

If the time will be enough,there will be laboratory exercises in computer labs, otherwise students can work on their own with a free version of Matlab (Octave).

Lo studente sarà invitato, sia in classe che a casa, a svolgere degli esercizi e implementare dei codici in modo da rendersi conto del livello raggiunto nella comprensione della materia.

The student will be invited, either in class and at home, to solve by himself exercises and writing codes to realize the learning level of the matter.

Talkative skills

The homeworks will be analyzed and discussed together in class and students will be encouraged to work in small groups either in class and at home.

Learning skills

It will be possible to interacte with the teacher to discuss personal problems to enhance the knowledge skills.

The lectures will be given with slides, exercises on the blackboard and Python codes to be written on personal computers or in laboratory.

Error analysis theory: relative and absolut error , chopping and rounding, machine epsilon, convergence order, conditioning

Linear systems: matrix conditioning numbers. Gauss Method naif and with pivot. Matricial reformulation and LU factorization. Itherative methods: Jacobi and Gauss-Seidel matricial reformulation and convergence. SOR method and convergence.

Interpolation. Vandermond matrix, theorem of existence and unicity, Lagrange polynomials. Divided difference method. Error of lagrangian interpolation. Hermitian interpolation. Linear and cubic splines. Trigonometric interpolation.

Least square method approximation theory and solution of overdetermined systems. Linear regressione. Orthogonal polynomials.Chebichev polynomials.

Numerical integration .Newton-Cotes formulae, trapezes and Simpson rules. Composed formulae. Polynomial order. Gaussian integration: Mid-point rule.

Eigenvalues and eigenvectors. Conditioning. Power method, Gram-Schmidt othogonalization procedure. QR methods. Similarity transformations, Householder and Givens methods.

Non linear equations. Bisection,Newton and secants, gradient methods.

1. K.E. Atkinsons, An Introduction to Numerical Analysis, J.Wiley and sons, 1988

2. L.W. Johnson, R.D. Riess, Numerical Analysis, Addison-Wesley Publ, Co. ,1982

3. R. Sacco, A. Quarteroni, F. Saleri, Matematica Numerica, Springer, 2001.

4. G.Naldi, L.Pareschi Matlab: concetti e progetti, Apogeo 2002.