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Rather than a formal syllabus, this is a listing of topics I intend to
cover. This will be adjusted depending on the interest of the
enrollees. Every week or so, I will update this page with a
description of what was covered and what I plan to cover in the
immediate future.
Planned topics
Lec 1 (Sept 10): Representation of numbers by computers; roundoff
and truncation error.
Lecs 2 & 3 (Sept 15 & 17): Interpolation and extrapolation.
Polynomial representation, rational function representation, cubic
spline interpolation. Chebyshev representation. Chebyshev stuff
might spill into week 3.
Lec 4 (Sept 22): Wrap-up of Chebyshev representation.
Lecs 5 & 6 (Sept 24 & 29): Heaviside calculus and its
usefulness for deriving approximate formulas for derivatives and
integrals of functions. Introduction to quadrature methods:
Trapezoidal rule, Simpson's rule; construction of Simpson's rule from
trapezoidal rule; Richardson extrapolation and Romberg integration.
Concepts behind Gaussian quadrature.
Lecs 7, 8 & 9 (Oct 1, 6 & 8): Integration of ordinary
differential equations. "Mathematical" methods (Euler, Runge-Kutta,
Bulirsch-Stoer; methods for stiff equations) and "physical" methods
(especially symplectic integration techniques).
Lecs 10, 11 & 12 (Oct 15, 20 & 22): Monte Carlo methods.
Generation of uniform deviates (general principles behind bad methods,
some of the concepts behind good methods); generation of deviates with
a specified PDF. Application to Monte Carlo integration and Markov
Chain Monte Carlo.
Lecs 13 (Oct 27): Root finders: Overview of bisection, secant
and Newton's method; generalization of Newton's method to
multidimensional problems. (Note, no class was held on Oct 29th due
to a departmental event.)
Topics to be loosely planned to be covered
Solutions of systems of equations
Eigensystems
Fast fourier transforms
Solution of partial differential equations
This list is likely to expand ... and topics which are not covered
here are grist for student projects.
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