TOPICS IN NUMERICAL ANALYSIS
The course is designed not only for Math students, but also for other students with interests in scientific computing. The basic notions of analysis and geometry we will use are: Taylor's expansions, linear combinations and basis in a vector space. It is useful, but not mandatory, to be familiar with the numerical solution linear algebraic systems of equations and the concept of condition number (see the Numerical Analysis course).
The final score will be determined: for 1/3 by the evaluation of a computational project agreed with the teacher and for 2/3 by the oral examination.
The computational project must be agreed with the teacher and submitted together with the computer code developed. The project shall be a case-study that leverages on a topic taught during the course. The subject of the evaluation will be the appropriateness and quality of the software produced and the ability to present and discuss critically the results.
The oral examination will be on the material covered by the course. The evaluation will be based on knowledge of the course contents, the mathematical accuracy of the discourse, the ability to master the technical jargon of the subject, critical reasoning and ability to link the various topics.
Numerical Analysis develops and analyse methods to compute approximate solutions of mathematical problems, controlling the computational effort required and the error of the computed approximation; for this reason it’s one of the pillars of Scientific Computing and it is a fundamental discipline in the training of a modern mathematician.
The aim of this course is to complement the knowledge in this field already acquired in the courses “Matematica Computazionale” and “Analisi Numerica”.
At the end of the course the student will know and will be able to apply the main classical algorithms of numerical computing for problems involving (1) interpolation of functions and data, (2) approximations of functions and data, (3) computation of definite integrales and (4) solution of ordinary differential equations.
1. Interpolation of functions and data: Lagrange polynomials, interpolation error estimate, Newton form of the interpolation polynomial, divided differences with distinct and coincident nodes, Hermite interpolation, introduction to poice-wise polynomial interpolation.
2. Approssimation of functions and data: the linear approximation problem, continous and discrete least-squares approximation, orthogonal polynomials.
3. Numerical quadrature: simple and composite Newton-Cotes rules, Richardson extrapolation and error estimate, automatic quadrature with uniform and non-uniform refinement, gaussian quadrature rules, weighted gaussian rules, Radau and Lobatto rules.
4. Ordinary differential equations. Numerical differentiation formulae. Initial value problems: Euler’s method, a-priori error estimates, Taylor methods, Runge-Kutta and linear multistep methods, automatic step control. Stiff equations. Parameter dependent differential equations and sensitivity equations.
Lectures (2/3 of the hours) are conducted mainly at the blackboard. Exercises to help the individual study of each pupil will be made available and discussed during the following lectures upon request.
One third of the hours will be in the computing lab to teach the students how to implement (in MatLab), test and employ numerical analysis algorithms (some of the tools explained in the lectures will be used for this).
Office hours are booked on demand, by email or at lecture time.