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 are shown in the MathMap in green.
We have bundled together these areas into a stop on the tour since
each is concerned with issues which, from a theoretical perspective, are
trivial (e.g., addition or ordering of a set of real numbers!) but without
which a naive appeal to machine computation, say, would be impossible.
It must be admitted, however, that the links among these fields are not
remarkably strong than their links with other parts of mathematics.
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 figuratively
described as the study of optimal resource allocation. Depending on
the options and constraints in the setting, this may involve linear
programming, or quadratic-, convex-, integer-, or boolean-programming.
This category also includes game theory, which is actually not about
games at all but rather about optimization; which combination of
strategies leads to an optimal outcome. This area also includes
[Starting in the year 2000, a new section
91: Game theory, economics, social and behavioral sciences
For the more abstract theory of algorithms or information flow,
jump to the Computer Sciences part of the tour.
If you've clicked on topics "in order", you've now visited all the general
areas of analysis. If you missed any, or if you'd like to continue the tour
of other areas besides analysis, you can do so by returning now to the analysis page.
You can reach this page through http://www.math-atlas.org/welcome.html
Last modified 1999/05/12 by Dave Rusin. Mail: email@example.com