TOPICS IN NUMERICAL ANALYSIS
- Overview
- Assessment methods
- Learning objectives
- Contents
- 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 can take place on the same day.
In the first part, the student presents a brief report, which can also be carried out by groups of 2 or 3 people, on a project assigned during the course. Usually the project is computational, and the student must demonstrate a certain degree of independence.
The second part of the exam is an oral the topics discussed during 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 and differential calculus.
1. Approximation of functions: Polynomial interpolation, Hermite interpolation, least squares, piecewise polynomial interpolation, extension to the two dimensional case
2. Numerical Integration: integration on an interval, Newton-Cotes formulas and composite quadrature, adaptive methods, integration in multiple dimension, Monte Carlo quadrature and dimensional curse
3. Approximation and integration with orthogonal polynomials: families of orthogonal polynomials, Gaussian integration, integration on infinite intervals with Hermite polynomials, Trigonometric interpolation, and discrete Fourier transform
4. Integration of Ordinary Differential Equations: a few results from Analysis, elementary methods and their implementation, analysis of one step methods, stability, Runge-Kutta Methods, linear multistep methods, adaptivity, stiff problems and implicit methods, Runge-Kutta IMEX Methods
For Chapters 1, 2 and 3, the main text will be Quarteroni, Sacco, Saleri: "Numerical Mathematics," Springer. 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".
Additionally, slides and notes written by the teacher will be available for the lab sessions.
Lectures are at the blackboard, and in the computing lab
Office hours are booked on demand, by email or at lecture time.
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