الرئيسية
Numerical analysis
Introduction
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.
History
Applications and related fields
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.
This image slightly hand-edited for clarity.
Subfields
65A05: Tables
65B: Acceleration of convergence
65C: Probabilistic methods, simulation and stochastic differential equations.
65D: Numerical approximation, Primarily algorithms; for theory, see 41-XX
65E05: Numerical methods in complex analysis (potential theory, etc.), For numerical methods in conformal mapping, See 30C30
65F: Numerical linear algebra
65G: Error analysis and interval analysis
65H: Nonlinear algebraic or transcendental equations
65J: Numerical analysis in abstract spaces
65K: Mathematical programming, optimization and variational techniques
65L: Ordinary differential equations
65M: Partial differential equations, initial value and time-dependent initial-boundary value problems
65N: Partial differential equations, boundary value problems
65P: Numerical problems in dynamical systems [See also 37Mxx] [new in 2000]
65Q05: Difference and functional equations, recurrence relations
65R: Integral equations, integral transforms, see also 45LXX
65S05: Graphical methods
65T: Numerical methods in Fourier analysis
65Y: Computer aspects of numerical algorithms
65Z05: Applications to physics [new in 2000]
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.
Textbooks, reference works, and tutorials
"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:
http://www.indra.com/~sullivan/q10.html Mathcom
This FAQ is usually available from MIT's rtfm and its mirrors: ftp://rtfm.mit.edu/pub/usenet/news.answers/num-analysis/faq/part1 MIT's rtfm
If not, a compressed (with gzip) version is at: ftp://ftp.mathcom.com/mathcom/nafaq.txt.gz Mathcom ftp
There is a separate newsgroup for Matlab (comp.soft-sys.matlab)
There is a directory of newsletters related to scientific computation at the site of the . See in addition the Numerical Analysis Digest. There is also a mailing list on Reliable Computing; send email to majordomo@interval.usl.edu 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.
Software and tables
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 68W30: Symbolic computation; the largest of those have numerical capabilities as well.
Pointer to Indiana University's "Knowledge Base" (Computer FAQs)
Home page for Matlab: http://www.mathworks.com/.
Home page for Mideva and Matcom (Matlab interpreter/compiler): http://www.mathtools.com
Pointer to Minitab (statistical software)
Pointer to Octave, a noncommercial alternative to Matlab
Pointer to SciLab (free numerical software).
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.
Illustration of search of GAMS numerical software library.
For example, the GAMS software tree has a node for numerical differentiation.
Description of Netlib from D-Lib Magazine [Digital Libraries]
[Offsite] Finite Element Method site
Pointer to sample Finite Element code.
Pointer to Numerical Recipes site, and citation to alternative text.
Pointer to shareware versions of Numerical Recipes codes.
Other web sites with this focus
Highly recommended: Numerical methods (or this European mirror)
Numerical Analysis at Concordia. (Nice overview, including online lessons on basic topics)
NA-Net
NAFEMS (Professional organization)
Internet Finite Element Resources
German Scientific Computing Home Page
Numerical analysis for physicists
Numerical Recipes in C
Numerical Recipes in C
GAMS : Guide to Available Mathematical Software
Netlib Repository at UTK/ORNL (Software)
Here are the AMS and Goettingen resource pages for area 65.
Selected topics at this site
Pointers for General numerical analysis software
Pointer to an interval arithmetic tutorial.
Performing arithmetic on real numbers using continued fractions.
Best way to compute numerical solutions to a quadratic polynomial.
Calculating the roots of polynomials. (general guide).
Laguerre's method for determining zeroes of complex analytic functions (in a region).
Bairstow's method for finding the roots of a polynomial.
Dekker's algorithm of finding zeros of functions.
Software to compute all zeros of an analytic function in a rectangle.
Example of a bad Newton's method problem.
Example of oscillation in Newton's method.
Humdrum instance of Newton's method.
Use Newton's method for sets of functions of several variables? (Yes)
Citation: variations of Newton's method (better around multiple roots).
Improvements to Newton's method
How might Newton's method fail?
Computing inverses without division, using Newton's method.
Numerical calculation of "special functions" (trig/exp/log/sqrt...)
Numerical evaluation of the error function.
CORDIC algorithms for evaluating elementary (trig. etc.) functions -- citations, summary, pointers to code.
How do calculators compute sin(x) (etc.)?
Basic algorithm for computing trigonometric functions with CORDIC algorithms.
[OFFSITE] CORDIC bibliography and related informations.
Citation: computation of elementary functions.
Efficient iterative computation of sqrt(x)
Calculating logarithms of the gamma function (and factorials).
Applications of 3D (discrete) Fourier transforms to data compression.
Discussion of FFT procedures for sizes not a power of two; pointers to implementations
Pointers to FFT code and descriptions
Interpolating a function on R^2 from values at discrete points.
Interpolating a function on R^2 from values at discrete points.
Overview of options and pitfalls of (1-dimensional) interpolation
Basic pointers: netlib, gams
Pointer to HOMPACK (numerically solve systems of polynomial equations).
Description of AXIOM (numerically solve systems of polynomial equations by continuation).
Citations, cautions to numerical fitting of polynomial to data
How to fit a curve y(x)=a+b*sin(c*x+d) to data: ODRPACK
Source code for Hough transform
Pointer for C++ package Range -- variable precision 'range arithmetic'
Runge-Kutta methods of integration.
How to select points and weights for Gauss quadrature when computing numerical integrals?
Use of orthogonal polynomials for quadrature (numerical integration a la Gauss-Legendre).
Pointer to code for computing elliptic integrals
Iterative construction of a conformal mapping between two domains.
Pointers to basic Finite Element resources.
Finite-volume method for solving partial differential equations.
Pointers to techniques for mesh generation for PDEs.
Mesh generation with mgnet for solving Laplace's equation.
Summary of Runge-Kutta methods for solving ODEs.
Adams method for solving ODEs (a predictor-corrector method).
Gear's method for solving ODEs.
Solving the Ricatti equation.
Solving delay differential equations.
Summary of basic methods for integrating PDEs.
Basic comparison of root-finding methods
Challenge problems for root-finding algorithms
Various methods to find (all) the zeros of a complex function
Hirano's method root finding
Numerical root-finding methods appropriate with multiple roots
Examples of failure of Newton's method
Multidimensional secant methods (quasi-Newton methods) for finding zeros
Continuous Newton's Method
Intersection points of two cubic parametric curves
Outline of Runge-Kutta method (ODE) and Jenkins-Traub method (polynomial root-finding)
First appearance of Gaussian quadrature
Basic description of Gauss quadrature
General comparison of numerical integration methods
Comparison of errors from various numerical integration algorithms
Pointer to Clenshaw-Curtis method of integration
Pointer to Runge Kutta 8th order methods of solving differential equations
Semi-infinite Gauss integration rule
Integrals over infinite range
Filon's numerical integration formula
Evaluating numerical integrals of oscillating functions
Computing a double integral
Numerical integration methods over the cube
Gaussian integration formula on a triangle
Numerically integrating monomials on 2- and 3-dimensional cells
Integration in R^n: Monte Carlo or grid methods better?
Integration and interpolation over a sphere
Karatsuba multiplication of large integers
Using Newton's method to divide by using multiplication
Computing sqrt or quotients with very many digits of accuracy
Computing derivatives numerically, using piecewise-polynomial fit, Savitsky-Golay filters or complex integration
Interpolating from scattered data in R^3
Thin plate splines
Akima's interpolating spline
Differences between Fast Fourier Transform and Discrete Fourier Transform
Pointers to Fast Fourier Transform codes
Does FFT require code length a power of 2?
What is a Monte Carlo method?
Finding inflection points from numerical data
Numerical computation of surface minimizing a functional
Computing best approximation of a positive function by positive functions from appropriate families
Quick integer square root algorithm
Methods of describing roots of high-order polynomials; decision procedure to decide whether distinct
Reference: algorithms for solving ordinary differential equations numerically
Tutorials on Finite Element Methods
Different approaches in algebraic and geometric multigrid methods
Using multigrid solvers for hyperbolic CFD problems
Question on numerical solution of the diffusion equation
Numerical solution of Maxwell's equations
A collection of examples of calculator and computer errors (hardware constraints and symbolic-algebra bloopers)
» رمضان مبارك
» اقتراح للادارة !!
» سلام خاص الى استاذي الغالي
» نظائر الكلور
» الصداقة الحقيقية
» الابتسامة وفوائدها
» العمليات الكيميائية لاستخلاص غاز الكلور
» هل تعلم