Nowaday, the Scientific Calculus is a very important tool in order to simulate the behavior of new devices before the construction of a prototype and to optimize the performances of such devices.
The aim of the course is the introduction of basic methods of the numerical analysis. In particular the following problems will be considered: solution of linear and nonlinear systems, function interpolation, numerical integration and differentiation, ordinary differential equations. The convergence and the properties of the numerical methods are discussed. An important feature of the course is the use of MATLAB® in order to develop some programs to solve simple problems and verify the theoretical results.
Programming with Matlab
Computer arithmetic
Nonlinear systems of equations
Approximation of functions and data
Numerical differentiation and integration
Ordinary differential equations
Systems of linear equations
Eigenvalue problems
Programming with Matlab
* Scalar variables and arrays
* Arithmetic operations and operations with arrays
* Functions and graphical elements
* Branching and loops
Computer arithmetic
* Number representation
* Floating point number systems
* Rounding error
* Arithmetic in floating point number systems and error propagation
Approximation of functions and data
* Interpolation by polynomials
* The interpolation error
* Spline functions
* Least square methods for data approximation
Numerical differentiation and integration
* Numerical integration
* Quadrature by interpolation formulas
* Adaptive Simpson formula
* Finite differences
Ordinary differential equations
* The Cauchy problem
* Euler and Cranc-Nicolson methods
* Theory for one-step methods
* Absolute stability region
* High order methods
* Systems of ordinary differential equations
Systems of linear equations
* LU factorization and Gauss method
* Error analysis
* Iteration methods and their convergence
* Jacobi, Gauss-Seidel and Richardson methods
Eigenvalue problems
* Introductory remarks
* Localization of eigenvalues
* Eigenvalues and eigenvectors of symmetric positive definte matrices
* Power methods
* Rayleigh quotient
* QR algorithm
Alfio Quarteroni, Fausto Saleri, Paola Gervasio, Calcolo Scientifico,
Springer-Verlag Italia, 2012 e precedenti edizioni
Lectures and exercises in computer laboratory using Matlab
The exam is composed of an oral exam at the end of the course and by
practical tests in computer laboratory