Numerical analysis: Difference between revisions

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{{Short description|StudyMethods of algorithms usingfor numerical approximationapproximations}}

{{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 + 51/60² + 10/60³ = 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 -20th century, computers calculate the required functions instead, but many of the same formulas continue to be used in software algorithms.<ref name="20c">{{cite book |last1=Brezinski |first1=C. |last2=Wuytack |first2=L. |title=Numerical analysis: Historical developments in the 20th century |publisher=Elsevier |date=2012 |isbn=978-0-444-59858-5 |url={{GBurl|dt3Z1yu2VxwC|pg=PP6}}}}</ref>

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]].

Numerical analysis continues this long tradition: rather than giving exact symbolic answers translated into digits and applicable only to real-world measurements, approximate solutions within specified error bounds are used.

Key aspects of numerical analysis include:

1. **Error Analysis**: Understanding and minimizing the errors that arise in numerical calculations, such as round-off errors, truncation errors, and approximation errors.

2. **Convergence**: Determining whether a numerical method will converge to the correct solution as more iterations or finer steps are taken.

3. **Stability**: Ensuring that small changes in the input or intermediate steps do not cause large changes in the output, which could lead to incorrect results.

4. **Efficiency**: Developing algorithms that solve problems in a reasonable amount of time and with manageable computational resources.

5. **Conditioning**: Analyzing how the solution to a problem is affected by small changes in the input data, which helps in assessing the reliability of the numerical solution.

Numerical analysis plays a crucial role in scientific computing, engineering simulations, financial modeling, and many other fields where mathematical modeling is essential.

==Applications==

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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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==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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*[[:Category:Numerical analysts]]

*[[Analysis of algorithms]]

*[[Approximation theory]]

*[[Computational science]]

*[[Computational physics]]

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*{{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===