Analysis of Numerical Topics
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Here we consider fields of mathematics which address the issues of how
to carry out --- numerically or even in principle --- those computations
and algorithms which are treated formally or abstractly in other branches
of analysis. These fields have shown enormous growth in recent decades
in response to demands for effective, robust solutions to demanding
problems from science, engineering, and other quantitative applications.
65: Numerical analysis involves the study
of methods of computing numerical data. In many problems this implies
producing a sequence of approximations; thus the questions involve the
rate of convergence, the accuracy (or even validity) of the answer,
and the completeness of the response. (With many problems it is
difficult to decide from a program's termination whether other
solutions exist.) Since many problems across mathematics can be
reduced to linear algebra, this too is studied numerically; here there
are significant problems with the amount of time necessary to process
the initial data. Numerical solutions to differential equations
require the determination not of a few numbers but of an entire
function; in particular, convergence must be judged by some global
criterion. Other topics include numerical simulation, optimization,
and graphical analysis, and the development of robust working code.
41: Approximations and expansions primarily
concern the approximation of classes of real functions by functions of
special types. This includes approximations by linear functions,
polynomials (not just the Taylor polynomials), rational functions, and
so on; approximations by trigonometric polynomials is separated into the separate field of
Fourier analysis. Topics include criteria for goodness of fit,
error bounds, stability upon change of approximating family, and
preservation of functional characteristics (e.g. differentiability)
under approximation. Effective techniques for specific kinds of
approximation are also prized. This is also the area covering
interpolation and splines.
90: Operations research may be loosely
described as the study of optimal resource allocation.
Mathematically, this is the study of optimization. Depending on the
options and constraints in the setting, this may involve linear
programming, or quadratic-, convex-, integer-, or boolean-programming.
For the more abstract theory of algorithms or information flow,
jump to the Computer Sciences part of the tour.
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Last modified 2000/01/26 by Dave Rusin. Mail: