Complements of Mathematics and Calculus (F6X)

Goals
The course aims at:
1) completing the Students' knowledge in Mathematics, by studying some of the problems frequently encountered in Applied Sciences;
2) providing the basic tools regarding the numerical simulation of mathematical problems of applicative interest;
3) providing the basic tools for an appropriate usage of Scientific Computing software.

Acquired skills
Expertise in dealing with some of the mathematical problems arising from Applied Sciences; basic knowledge of Scientific Computing software.


Course content

Linear Algebra. Vector and matrices. Linear maps. Matrix determinant. Eigenvalues and eigenvectors of a matrix. Inverse matrix. Some relevant classes of matrices: symmetric matrices, definite matrices, triangular matrices, etc.
Numerical methods for solving linear systems. Direct methods: LU decomposition end Gauss method; Cholesky decomposition. Iterative methods: Jacobi and Gauss-Seidel methods; stopping criteria.
Polynomial approximation of functions and data. Polynomila interpolation: Lagrange representation and error analysis; spline functions; least squares method and linear regression.
Non-linear equations. Bisection method; Newton method and its convergence properties; stopping criteria.
Numerical quadrature. Open and closed Newton- Côtes quadrature formulae; error analysis and composite quadrature.
Ordinary differential equations. One step methods (forward Euler, backward Euler, Cranck-Nicolson, Heun methods); consistency and local truncation error, convergence order; A-stability.

Suggested prerequisites
The course "Istituzioni di Matematica"


Reference material
- A. Quarteroni, F. Saleri, P. Gervasio, Calcolo scientifico. Springer, 2012.
- G. Naldi, L. Pareschi: MATLAB Concetti e progetti. Milano, Apogeo 2002.
- G. Naldi, L. Pareschi, G. Russo, Introduzione al calcolo scientifico. Metodi e applicazioni con Matlab, McGraw-Hill Education.

Assessment method
Written exam split into two parts: the first part will require the development and the solution of theoretical exercises; the second one will require the development of exercises using the software MATLAB. The course exam is passed if the Student is successful in both the parts.

Language of instruction
Italian

Attendance Policy
Strongly recommended


Mode of teaching
Traditional frontal lectures, at the blackboard, as far as the theoretical part is concerned. Lab lectures for what concerns the numerical experiments using the software MATLAB.