# TOPICS IN NUMERICAL ANALYSIS

Degree course:
Corso di First cycle degree in MATHEMATICS
Academic year when starting the degree:
2017/2018
Year:
3
Academic year in which the course will be held:
2019/2020
Course type:
Compulsory subjects, characteristic of the class
Seat of the course:
Como - Università degli Studi dell'Insubria
Credits:
8
Period:
Second semester
Standard lectures hours:
80
Detail of lecture’s hours:
Lesson (80 hours)
Requirements:

The course is designed not only for Math students, but also for students of Physics or Chemistry, with interests in scientific computing. The basic notions of analysis and geometry we will use are: Taylor's expansions, the notion of linear combinations and basis in a vector space. From numerical analysis, notions on the solution of linear algebraic systems of equations and the concept of condition number will be useful. If these concepts are not familiar, they will be recalled.

Final Examination:
Orale

The exam is oral, and consists of two parts, which usually take place on the same day.
In the first part, the student discusses a brief report on a computational project (previously agreed with the teacher) that leverages on a topic taught during the course.
The second part of the exam is an oral examination of the material covered by the course.

Assessment:
Voto Finale

The purpose of the course is to complete the preparation of Numerical Analysis by introducing the main ideas and algorithms of classical numerical computing, designed to solve problems in approximation of functions, definite integrals and ordinary differential equations.

1. Interpolation of data and functions
2. Approximation of functions and data
4. Integration of Ordinary Differential Equations

1. Interpolation of data and functions: Polynomial interpolation (Lagrange basis, basis of monomials and divided differences), Hermite interpolation (divided differences with coincident nodes), piecewise polynomial interpolation (splines).
2. Approximation of functions and data: families of orthogonal polynomials; least squares approximation.

3. Numerical quadrature: integration on an interval, Newton-Cotes formulas and composite quadrature, adaptive methods with uniform and non-uniform refinement, gaussian formulas.

4. Integration of Ordinary Differential Equations: analysis of the Euler method; analysis of one step methods; stability; Runge-Kutta Methods; automatic step control; linear multistep methods.

For Chapters 1, 2 and 3, the main reference will be the chapters 5,6,7 of “Metodi Numerici” by Bevilacqua, Bini, Capovani, Meini (Zanichelli). Alternatively, “Numerical mathematics” by Quarteroni, Sacco, Saleri (Springer) is suggested.
For Chapter 4, in addition to Quarteroni's book, we will also use Randy Leveque's "Finite Differential Methods for Ordinary and Partial Differential Equations".

Additional material, expecially for the lab sessions, will be made available by the teacher through the e-learning web site.

Lectures are at the blackboard, and in the computing lab

Office hours are booked on demand, by email or at lecture time.