This is an overview course to be offered to 3rd and 4th year undergraduate, and postgraduate students. The course is structured to systematically build on and provide an overview of several ideas and topics that comprise the basics of discretizations of continuous mathematics. In this setup, we will concern ourselves with computational as well as stability analyses of both methods and algorithms.
We will begin with an introduction to scientific computing. Then we will analyze and study methods in numerical linear algebra: matrix factorizations, direct solution of linear systems, solutions of linear least square problems, and solutions to eigenvalue problems.
This will be followed by solutions of nonlinear equations in 1d and then more generally. We will apply this learning to unconstrained optimization in 1d and again more generally. We will also delve into some constrained optimization and nonlinear least squares problems.
The next part of the course will discuss polynomial interpolation (including using splines) of discrete data in 1d. This will be utilized in methods for numerical differentiation of sampled data, and for numerically carrying out integration in 1d (also known as quadrature).
The last part of the course will deal with numerical solutions of initial and boundary value ordinary differential equations in 1d. This will be followed by a foray into numerical solution of model partial differential equations.
A student registering for the MTH573 version of the course will be required to work on an additional Pass/No Pass project for getting the course credits. Graduate students, in particular, will not be allowed to register for the 373 listing of this course.
Students understand notions of accuracy, conditioning and stability. They can apply this to analyzing linear, least squares and eigenvalue problems. They can solve these using a variety of methods as appropriate.
Students know how to find roots of equations of the form f(x) = 0, apply this to solving unconstrained optimization problems, and understand elements of constrained optimization.
Students can interpolate a point data set using polynomials and piecewise polynomials. They learn to numerically integrate and differentiate functions.
Students learn about ordinary differential equations, their numerical solution schemes, and stability analysis for such methods. They will also learn about representing and solving sparse linear systems.
Students learn to derive and analyze finite difference methods for solving parabolic and hyperbolic partial differential equations. They will understand analysis of their accuracy and stability. Finally, they will learn about elliptic partial differential equations, and their discretization via finite difference and finite element methods.