Scientific Disciplinary Sector (SSD)
MAT/08 - NUMERICAL ANALYSIS
Primo semestre dal Oct 3, 2022 al Jan 27, 2023.
The course will discuss, from both the analytic and computational points of view, the numerical solution of Mathematical problems such as: non linear systems, linear systems, matrix eigenvalues, interpolation and approximation, Gaussian quadrature. The objective therefore is to expand on the material introduced in Calcolo Numerico I and to introduce new and more sophisticated solution algorithms. In particular, we will present techniques that are fundamental for important modern problems of Applied Mathematics such as that of high dimensional datasets (SVD and PCoA) and optimization (conjugate gradient method). The course has a Laboratory component where the methods studied will be implemented using the MATLAB programming platform (using either the official Matlab from Mathworks or else the open source version GNU OCTAVE). At the end of the course the student will be expected to demonstrate that s/he has attained a level of competence in the computational and computer aspects of the course subject, as well as the ability to recognize which algorithms are appropriate for basic and advanced problems of numerical analysis.
Prerequisites and basic notions
Linear algebra, differential calculus in one and more variables, integral calculus, basic methods of numerical analysis.
The course will discuss the following topics:
* Methods for finding zeros of (systems of) functions (fixed point iterations)
* Methods for linear systems (classical iterative methods, conjugate gradient, QR factorization, SVD factorization, overdetermined systems)
* Methods for finding eigenvalues and eigenvectors (the Power method, the QR iteration)
* Spline and Bezier curve interpolation
* Gaussian quadrature
* Introduction to preconditioning and iterative methods for non symmetric systems (GMRES)
* Introduction to numerical optimization
It is expected that there will be a tutor to help with the correction of assigned exercises and with the Laboratory sessions.
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The teaching will be delivered in 52 classroom hours, of which about 12 in the computer lab.
Learning assessment procedures
The purpose of the exam is to see if the student is able to recall and produce the theory of the Numerical Analysis presented during the lectures and Laboratory and knows how to use Computer resources for possible further investigation. Moreover, the student must show that s/he knows how to program in the specific software introduced during the course. The exam will consist of two parts. The first part will be held in a Laboratory where the student will be given two hours to individually implement the numerical methods necessary for the solution of the assigned questions. The questions will be based on the entire course material. A pass will be given for a mark of 18/30 or higher. To be admitted to the second part of the exam, the oral, it is required to have first passed the written part. Marks for the written part will remain valid until, and not after, the beginning of the following semester. The oral exam will be based on the topics discussed during the classroom lectures. The final course mark will be the weighted average of the marks for the two parts of the exam, with weight 1/4 for the written part and 3/4 for oral.
To pass the exam, students must demonstrate: * knowing and understanding the fundamental iterative methods of the numerical resolution of linear systems * knowing and understanding the fundamental methods of numerical resolution of non-linear systems * knowing and understanding the fundamental methods of 'piecewise numerical approximation * knowing and understanding the fundamental methods of the Gaussian quadrature * having an adequate capacity for analysis and synthesis and abstraction * knowing how to apply this knowledge to solve problems and exercises, knowing how to argue one's reasoning with mathematical rigor.
Criteria for the composition of the final grade
The final mark will be the weighted average: 3/4 times the Oral mark plus 1/4 times the Laboratory mark.