65: Numerical analysis
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.
Numerical linear algebra topics: solutions of linear systems AX = B, eigenvalues and eigenvectors, matrix factorizations. Calculus topics: numerical differentiation and integration, interpolation, solutions of nonlinear equations f(x) = 0. Statistical topics: polynomial approximation, curve fitting.
For papers involving machine computations and programs in a specific mathematical area, See Section --04 in that area. This includes computational issues in group theory, number theory, geometry, statistics, and so on; for each of these fields there are software packages or libraries of code which are discussed on those index pages. (On the other hand, most results of numerical integration, say, are in this section rather than Measure and Integration; topics in optimization are in section 65K rather than Operations Research.)
For calculations of a combinatorial nature and for graph-theoretic questions such as the traveling salesman problem or scheduling algorithms, see Combinatorics. (These are distinguished by the discrete nature of the solution sought.) Portions of that material -- particularly investigations into the complexity of the algorithms -- is also treated in Computer Science.
General issues of computer use, such as system organization and methodology, or artificial intelligence, are certainly in computer science. Topics in computer algebra or symbolic calculation are treated separately.
Issues concerning limitations of specific hardware or software are not strictly speaking part of mathematics at all but often illustrate some of the issues addressed in numerical analysis. Some of these can be seen in examples seen below.
Applications of numerical analysis occur throughout the fields of applied (numerical) mathematics, in particular in the fields of physics (sections 70-86). Many of these areas including subheading e.g. for finite element methods (which are primarily treated here in 65L - 65P).
There are also applications to areas typically considered part of pure mathematics; for example, there is substantial work done on the roots of 26C:Polynomials and rational functions.
This area is undergirded by the areas of analysis. See for example Real analysis or Complex analysis for general topics of convergence.
The study of whole numbers and their properties (e.g. solving equations in integers) is not numerical analysis at all but Number Theory.
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This is one of the largest areas of the Math Reviews database, with many of the subfields being also large. (65N30, Finite Element Methods, is among the largest of the 5-digit areas, too). This field has more subfields than almost any other!
Browse all (old) classifications for this area at the AMS.
"The state of the art in numerical analysis", Proceedings of the conference held at the University of York, York, April 1996. Edited by I. S. Duff and G. A. Watson. The Institute of Mathematics and its Applications Conference Series. New Series, 63. The Clarendon Press, Oxford University Press, New York, 1997. 562 pp. ISBN 0-19-850014-9 MR99a:65008 (There are also older proceedings with the same title.)
Shampine, L. F.: "What everyone solving differential equations numerically should know", Computational techniques for ordinary differential equations (Proc. Conf. Univ. Manchester, Dec. 18--20, 1978), pp. 1--17, Academic Press, London-New York-Toronto, Ont., 1980. 84e:65089
Nievergelt, Yves: "Total least squares: state-of-the-art regression in numerical analysis", SIAM Rev. 36 (1994), no. 2, 258--264. MR95a:65077
Sobol, Ilya M.: "A primer for the Monte Carlo method", CRC Press, Boca Raton, FL, 1994. 107 pp. ISBN 0-8493-8673-X MR95e:65001
"State-of-the-art surveys on finite element technology", Edited by Ahmed K. Noor and Walter D. Pilkey. American Society of Mechanical Engineers (ASME), New York, 1983. 530 pp. MR85h:65003
Babuska, I.: "The p and h-p versions of the finite element method: the state of the art", Finite elements (Hampton, VA, 1986), 199--239, ICASE/NASA LaRC Ser.; Springer, New York-Berlin, 1988. MR90b:65197
PostScript/PDF versions of Numerical Recipes (well-known and oft-debated general introduction)
"Reviews in Numerical Analysis 1980-1986", AMS
There is a USENET newsgroup sci.math.num-analysis. Newsgroup FAQ:
Numerical Analysis Digest
There is a mailing list on Reliable Computing; send email to email@example.com with the message "subscribe reliable_computing"
Online introductory book (U.S. Department of Energy)
Old-timers may remember that optimal formulae were often precomputed; see e.g. Abramowitz, M. and Stegun, C.A. (Ed.). "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables", 9th printing, 1972, New York: Dover.
Finite Element Analysis programs.
FElt Finite Elements software.
Following is a (still rather haphazard) collection of links to some popular numerical mathematical computation software. For other links to major all-in-one mathematical computing environments see the corresponding list in 68Q40: Symbolic computation; the largest of those have numerical capabilities as well.
Finally, some pointers to classes of algorithms. Note that mathematical algorithms constitute one of the most web-accessible bodies of human knowledge available. A search at one of these sites can easily save considerable programming headaches! Highly recommended are NETLIB and GAMS. See also TOMS.