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{{Short description|
{{Use dmy dates|date=October 2020}}
[[Image:Ybc7289-bw.jpg|thumb|250px|right|Babylonian clay tablet [[YBC 7289]] (c. 1800–1600 BCE) with annotations. The approximation of the [[square root of 2]] is four [[sexagesimal]] figures, which is about six [[decimal]] figures. 1 + 24/60 + 51/60<sup>2</sup> + 10/60<sup>3</sup> = 1.41421296...<ref>{{Cite web |url=http://it.stlawu.edu/%7Edmelvill/mesomath/tablets/YBC7289.html |title=Photograph, illustration, and description of the ''root(2)'' tablet from the Yale Babylonian Collection |access-date=2 October 2006 |archive-date=13 August 2012 |archive-url=https://web.archive.org/web/20120813054036/http://it.stlawu.edu/%7Edmelvill/mesomath/tablets/YBC7289.html |url-status=dead }}</ref>]]
'''Numerical analysis''' is the study of [[algorithm]]s that use numerical [[approximation]] (as opposed to [[symbolic computation|symbolic manipulations]]) for the problems of [[mathematical analysis]] (as distinguished from [[discrete mathematics]]). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: [[ordinary differential equation]]s as found in [[celestial mechanics]] (predicting the motions of planets, stars and galaxies), [[numerical linear algebra]] in data analysis,<ref>{{cite book |first=J.W. |last=Demmel |title=Applied numerical linear algebra |publisher=[[Society for Industrial and Applied Mathematics|SIAM]] |date=1997 |isbn=978-1-61197-144-6 |doi=10.1137/1.9781611971446 |url=https://epubs.siam.org/doi/epdf/10.1137/1.9781611971446.fm}}</ref><ref>{{cite book |last1=Ciarlet |first1=P.G. |last2=Miara |first2=B. |last3=Thomas |first3=J.M. |title=Introduction to numerical linear algebra and optimization |publisher=Cambridge University Press |date=1989 |isbn=9780521327886 |oclc=877155729 }}</ref><ref>{{cite book |last1=Trefethen |first1=Lloyd |last2=Bau III |first2=David |title=Numerical Linear Algebra |publisher=SIAM |date=1997 |isbn=978-0-89871-361-9 |url={{GBurl|4Mou5YpRD_kC|pg=PR7}}}}</ref> and [[stochastic differential equation]]s and [[Markov chain]]s for simulating living cells in medicine and biology.
Before modern computers, [[numerical method]]s often relied on hand [[interpolation]] formulas, using data from large printed tables. Since the mid
The numerical point of view goes back to the earliest mathematical writings. A tablet from the [[Yale Babylonian Collection]] ([[YBC 7289]]), gives a [[sexagesimal]] numerical approximation of the [[square root of 2]], the length of the [[diagonal]] in a [[unit square]].
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* Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations.
* Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving [[partial differential equation]]s numerically.
*
Stephen Blyth.
[https://
2013.
page VII.
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Interpolation: Observing that the temperature varies from 20 degrees Celsius at 1:00 to 14 degrees at 3:00, a linear interpolation of this data would conclude that it was 17 degrees at 2:00 and 18.5 degrees at 1:30pm.
Extrapolation: If the [[gross domestic product]] of a country has been growing an average of 5% per year and was 100 billion last year, it might be extrapolated that it will be 105 billion this year.
[[Image:Linear-regression.svg|right|100px|A line through 20 points]]
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Optimization problems ask for the point at which a given function is maximized (or minimized). Often, the point also has to satisfy some [[Constraint (mathematics)|constraint]]s.
The field of optimization is further split in several subfields, depending on the form of the [[objective function]] and the constraint. For instance, [[linear programming]] deals with the case that both the objective function and the constraints are linear. A famous method in linear programming is the [[simplex algorithm|simplex method]].
The method of [[Lagrange multipliers]] can be used to reduce optimization problems with constraints to unconstrained optimization problems.
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{{Main|Numerical ordinary differential equations|Numerical partial differential equations}}
Numerical analysis is also concerned with computing (in an approximate way) the solution of [[differential equations]], both [[ordinary differential equations]] and [[partial differential equations]].<ref>{{cite book |first=A. |last=Iserles |title=A first course in the numerical analysis of differential equations |publisher=Cambridge University Press |edition=2nd |date=2009 |isbn=978-0-521-73490-5 |url={{GBurl|M0tkw4oUucoC|pg=PR5}} }}</ref>
Partial differential equations are solved by first discretizing the equation, bringing it into a finite-dimensional subspace.<ref>{{cite book |first=W.F. |last=Ames |title=Numerical methods for partial differential equations |publisher=Academic Press |edition=3rd |date=2014 |isbn=978-0-08-057130-0 |url={{GBurl|KmjiBQAAQBAJ|pg=PP7}} }}</ref> This can be done by a [[finite element method]],<ref>{{cite book |first=C. |last=Johnson |title=Numerical solution of partial differential equations by the finite element method |publisher=Courier Corporation |date=2012 |isbn=978-0-486-46900-3 |url={{GBurl|0IFCAwAAQBAJ|p=2}} }}</ref><ref>{{cite book |last1=Brenner |first1=S. |last2=Scott |first2=R. |title=The mathematical theory of finite element methods |publisher=Springer |edition=2nd |date=2013 |isbn=978-1-4757-3658-8 |url={{GBurl|ServBwAAQBAJ|pg=PR11}}}}</ref><ref>{{cite book |last1=Strang |first1=G. |last2=Fix |first2=G.J. |orig-year=1973 |title=An analysis of the finite element method |publisher=Wellesley-Cambridge Press |date=2018 |isbn=9780980232783 |url=https://archive.org/details/analysisoffinite0000stra |oclc=1145780513 |edition=2nd}}</ref> a [[finite difference]] method,<ref>{{cite book |last=Strikwerda |first=J.C. |title=Finite difference schemes and partial differential equations |publisher=SIAM |edition=2nd |date=2004 |isbn=978-0-89871-793-8 |url={{GBurl|mbdt5XT25AsC|pg=PP5}}}}</ref> or (particularly in engineering) a [[finite volume method]].<ref>{{cite book |first=Randall |last=LeVeque |title=Finite Volume Methods for Hyperbolic Problems |publisher=Cambridge University Press |date=2002 |isbn=978-1-139-43418-8 |url={{GBurl|mfAfAwAAQBAJ|pg=PT6}}}}</ref> The theoretical justification of these methods often involves theorems from [[functional analysis]]. This reduces the problem to the solution of an algebraic equation.
==Software==
{{Main|List of numerical-analysis software|Comparison of numerical-analysis software|List of programming languages by type#Numerical analysis|l3=Numerical analysis programming languages|List of numerical libraries}}
Since the late twentieth century, most algorithms are implemented in a variety of programming languages. The [[Netlib]] repository contains various collections of software routines for numerical problems, mostly in [[Fortran]] and [[C (programming language)|C]]. Commercial products implementing many different numerical algorithms include the [[IMSL Numerical Libraries|IMSL]] and [[Numerical Algorithms Group|NAG]] libraries; a [[free software|free-software]] alternative is the [[GNU Scientific Library]].
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The [[Naval Surface Warfare Center]] several times published its [https://apps.dtic.mil/sti/pdfs/ADA476840.pdf ''Library of Mathematics Subroutines''] (code [https://jblevins.org/mirror/amiller/#nswc here]).
There are several popular numerical computing applications such as [[MATLAB]],<ref>{{cite book |last1=Quarteroni |first1=A. |last2=Saleri |first2=F. |last3=Gervasio |first3=P. |title=Scientific computing with MATLAB and Octave |publisher=Springer |edition=4th |date=2014 |isbn=978-3-642-45367-0 |url={{GBurl|_0m9BAAAQBAJ|pg=PR11}}}}</ref><ref name="gh">{{cite book |editor1-last=Gander |editor1-first=W. |editor2-last=Hrebicek |editor2-first=J. |title=Solving problems in scientific computing using Maple and Matlab® |publisher=Springer |date=2011 |isbn=978-3-642-18873-2 |url={{GBurl|di2qCAAAQBAJ|pg=PR14}}}}</ref><ref name="bf">{{cite book |last1=Barnes |first1=B. |last2=Fulford |first2=G.R. |title=Mathematical modelling with case studies: a differential equations approach using Maple and MATLAB |publisher=CRC Press |edition=2nd |date=2011 |isbn=978-1-4200-8350-7 |oclc=1058138488 }}</ref> [[TK Solver]], [[S-PLUS]], and [[IDL (programming language)|IDL]]<ref>{{cite book |first=L.E. |last=Gumley |title=Practical IDL programming |publisher=Elsevier |date=2001 |isbn=978-0-08-051444-4 |url={{GBurl|1d-tNpm_x4gC|pg=PR9}}}}</ref> as well as free and open
Many [[computer algebra system]]s such as [[Mathematica]] also benefit from the availability of [[arbitrary-precision arithmetic]] which can provide more accurate results.<ref>{{cite book |first=R.E. |last=Maeder |title=Programming in mathematica |publisher=Addison-Wesley |edition=3rd |date=1997 |isbn=9780201854497 |oclc=1311056676 |url=https://archive.org/details/programminginmat0000maed_l2m6}}</ref><ref>{{cite book |first=Stephen |last=Wolfram |date=1999 |title=The MATHEMATICA® book, version 4 |publisher=[[Cambridge University Press]] |url={{GBurl|Xny77v_QPkEC|pg=PR19}} |isbn=9781579550042 }}</ref><ref>{{cite book |last1=Shaw |first1=W.T. |last2=Tigg |first2=J. |title=Applied Mathematica: getting started, getting it done |publisher=Addison-Wesley |date=1993 |isbn=978-0-201-54217-2 |oclc=28149048 |url=http://www.gbv.de/dms/bowker/toc/9780201542172.pdf}}</ref><ref>{{cite book |last1=Marasco |first1=A. |last2=Romano |first2=A. |title=Scientific Computing with Mathematica: Mathematical Problems for Ordinary Differential Equations |publisher=Springer |date=2001 |isbn=978-0-8176-4205-1 |url={{GBurl|iFRqemnmMqUC|pg=PR7}}}}</ref>
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*[[:Category:Numerical analysts]]
*[[Analysis of algorithms]]
*[[Approximation theory]]
*[[Computational science]]
*[[Computational physics]]
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{{refbegin}}
* {{cite book |author1 = Golub, Gene H. |author-link = Gene H. Golub |author2 = Charles F. Van Loan |author2-link = Charles F. Van Loan |title=Matrix Computations |edition=3rd |publisher=Johns Hopkins University Press |isbn=0-8018-5413-X |year = 1986 }}
* {{cite book |author1 = Ralston Anthony |author2 = Rabinowitz Philips | title=A First Course in Numerical Analysis | edition=2nd |publisher=Dover publications | isbn=978-0486414546 |year=2001 }}
*{{cite book |first=Nicholas J. |last=Higham |author-link=Nicholas Higham |title = Accuracy and Stability of Numerical Algorithms |url=https://archive.org/details/accuracystabilit0000high |url-access=registration |publisher=Society for Industrial and Applied Mathematics |isbn=0-89871-355-2 |orig-year=1996 |year=2002}}
* {{cite book |last=Hildebrand |first=F. B. |author-link=Francis B. Hildebrand | title=Introduction to Numerical Analysis |edition=2nd |year=1974 |publisher=McGraw-Hill |isbn= 0-07-028761-9 }}
* David Kincaid and Ward Cheney: ''Numerical Analysis : Mathematics of Scientific Computing'', 3rd Ed., AMS, ISBN 978-0-8218-4788-6 (2002).
* {{cite book |last=Leader |first=Jeffery J. |author-link=Jeffery J. Leader |title=Numerical Analysis and Scientific Computation |year=2004 |publisher=Addison Wesley |isbn= 0-201-73499-0 }}
* {{cite book|last= Wilkinson |first =J.H. |author-link=James H. Wilkinson |title=The Algebraic Eigenvalue Problem |url= https://archive.org/details/algebraiceigenva0000wilk |url-access= registration |publisher = Clarendon Press |orig-year=1965 |year=1988 |isbn=978-0-19-853418-1 }}
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*[https://dlmf.nist.gov/3 Numerical Methods], ch 3. in the ''[[Digital Library of Mathematical Functions]]''
*[https://personal.math.ubc.ca/~cbm/aands/page_875.htm Numerical Interpolation, Differentiation and Integration], ch 25. in the ''Handbook of Mathematical Functions'' ([[Abramowitz and Stegun]])
*[https://fncbook.com/ Tobin A. Driscoll and Richard J. Braun: ''Fundamentals of Numerical Computation'' (free online version)]
===Online course material===
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