TOPICS IN NUMERICAL ANALYSIS
- Overview
- Assessment methods
- Learning objectives
- Contents
- Full programme
- Bibliography
- Teaching methods
- Contacts/Info
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.
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.
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
3. Numerical quadrature
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.