 [Search][Subject Index][MathMap][Tour][Help!] # Analysis of Numerical Topics

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 mathematical economics. [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.