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# 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 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.

If you've been clicking 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. Otherwise, you may wish to proceed to the next portion of the tour: Probability and Statistics