29227 - Foundations of Informatics T

Learning outcomes

Floating point operations, linear algebra problems, non linear equations, data and function approximation, numerical integration, ordinary differential equations.

Course contents

I. Theory lectures:

1) Floating point operations and errors in numerical analysis. Physical problem, mathematical problem, numerical problems and idealization errors; treatment of real numbers in the computer; floating point approximation and machine numbers; rounding errors and floating point calculations; absolute and relative errors; exact decimals and significative digits; numerical cancellation; numerical problems and alogorithms; input data errors and conditioning of a numerical problems; truncation errors in an iterative process; stability of an algorithm. 2) Matrices and their properties. Permutation and linear combination matrices; sparse and dense matrices; diagonal dominant matrices; non singular matrices; vector and matrix norms; compatible norms. 3) Numerical solution of linear systems of equations. Conditioning of the numerical problem; direct and iterative methods; Vandermonde matrices; direct methods: Gauss method with partial pivoting;PLU decomposition of a matrix; permutaion matrices P; linear combination matrices M; doublesweep method for tridiagonal matrices;iterative method of Jacobi; iterative method of Gauss-Seidel; Sor Method; convergence criteria. 4) Numerical solution of non linear equations.Conditioning of the numerical problem; bisection method; regula falsi method; secants method; Newton method; convergence order for an iterative method; convergence criteria based on solution value and function; fixed point method; systems of non-linear equations; Newton method and related approximate methods. 5) Data and function approximations. Polynomial interpolation; Lagrange fundamental polynomials and related errors; Vandermonde matrices; least squares method and its normal equations. 6) Numerical methods for the solution of definite integrals. Interpolatory quadrature formulas; weights and nodes of the quadrature formulas; weighted quadrature formulas; Newton-Cotes quadrature formulas; trapezi rule; Simpson rule; repeated trapezi and Simpson rules; Gaussian quadrature formulas

II: Laboratory lectures with MATLAB on all the arguments treated during theory lectures.

Readings/Bibliography

MONEGATO G. 100 PAGINE DI ELEMENTI DI CALCOLO NUMERICO, LIBRERIA UNIVERSITARIA LEVROTTO & BELLA TORINO 1995

Teaching methods

Lectures at the blackboard and laboratory lectures with Personal Computers

Tutor assistance

Assessment methods

Examination procedues: written test with personal computer.

The examination test is structured with a series of questions with multiple answers. Some answers must be motivated with the use of a computational tool named MATLAB. Simple algorithms must be written in MATLAB.

Teaching tools

Lectures with overhead projector, slide, blackboard

Office hours

See the website of Vittorio Colombo