Instructor: Paolo Piersanti Time: Tue, Thu 3:00 PM — 4:15 PM
Prerequisites: M301 or M303, M311 or S311, M343 or S343
Textbook: R. Burden, J. Faires and A. Burden, Numerical Analysis – Tenth edition (2016).
Tentative Course Outline: In principle, mathematical analysis can be used to study questions in fields
from biology to quantum mechanics. However, the answers it gives are not always as conclusive as one might
hope. For example, a solution may contain expressions such as sin(1). Though we know precisely what is
meant by this expression, in a certain sense it is incomplete. What is missing are instructions on how to
convert it into a numerical value. This final step is nontrivial- in the case of sin(x) it can only be done
approximately- and, from a practical standpoint, is the most important part of the analysis. The situation
can be worse; for example, in fluid dynamics, the mathematical description of a fluid flow is sometimes only
that it is a solution of a certain partial differential equation. Numerical analysis is the branch of mathematics
which concerns this problem of extracting numerical values from “pure math” solutions, usually with the
aid of high-speed computers.
This course will introduce the basic theory and application of modern numerical approximation techniques,
including: standard computational methods, design of algorithms, and error analysis, in particular
of the stability and convergence of various schemes.
The course contents are divided into two parts.
- Part 1: Root-finding Algorithms and Polynomial Interpolation. The first part of the course
will begin with the presentation of some of the classical methods for computing the roots of an algebraic
equation. Afterwards, we will study how to approximate a smooth functions by means of polynomial
splines.
The contents for this part will be drawn from Chapters 1–3 of the main textbook.
- Part 2: Numerical Differentiation and Integration. The second part of the course will be
devoted to the analysis of techniques aiming to approximate the values of the derivatives and the
integrals of certain smooth functions. We will also survey how to apply these techniques to approximate
the solutions of ordinary differential equations.
The contents for this part will be drawn from Chapters 4–6 of the main textbook.
Filed under: Academic Help