Complements of mathematics and calculus (F6X)

A.Y. 2019/2020
Overall hours
MAT/01 MAT/02 MAT/03 MAT/04 MAT/05 MAT/06 MAT/07 MAT/08 MAT/09
Learning objectives
The course aims at: completing the students' knowledge in Mathematics, by studying some of the problems frequently encountered in Applied Sciences; providing the basic tools regarding the numerical simulation of mathematical problems of applicative interest, and the basic tools for an appropriate usage of Scientific Computing software.
Expected learning outcomes
The student will acquire a good knowledge of the mathematical foundations of linear algebra, of descriptive statistics and of numerical calculation; he/she will be able to frame some mathematical problems of applicative interest, and to correctly use the Scientific Calculation software to process data and simulate simple problems in the chemical field.
Course syllabus and organization

Single session

Course syllabus
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.
Prerequisites for admission
To properly face the couse, the Student should have a basic knowledge of Elementary Mathematics and of standard Calculus.
Teaching methods
The course will be given by means of traditional lessons, using the blackboard. Furthermore, for the lab lessons the software MATLAB will be used to implement the studied methods.
Teaching Resources
- 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 methods and Criteria
The final examination consists of two parts: a written exam and a lab exam.

- During the written exam, the student must solve some exercises in the format of open-ended and/or short answer questions, with the aim of assessing the student's ability to solve simple problems in Linear Algebra and Scientific Computing. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed two hours).

-The lab exam consists in solving simple exercises using the software MATLAB. The lab portion of the final examination serves to assess the capability of the student to use a classical Scientific Computing software.

The complete final examination is passed if both parts (written, lab) are successfully passed. Final marks are given using the numerical range 0-30, and will be will be available in the SIFA service through the UNIMIA portal.
MAT/01 - MATHEMATICAL LOGIC - University credits: 0
MAT/02 - ALGEBRA - University credits: 0
MAT/03 - GEOMETRY - University credits: 0
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0
MAT/06 - PROBABILITY AND STATISTICS - University credits: 0
MAT/07 - MATHEMATICAL PHYSICS - University credits: 0
MAT/08 - NUMERICAL ANALYSIS - University credits: 0
MAT/09 - OPERATIONS RESEARCH - University credits: 0
Practicals: 24 hours
Lessons: 36 hours