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| [[Image:Ybc7289-bw.jpg|thumb|250px|right|Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) 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>[http://it.stlawu.edu/%7Edmelvill/mesomath/tablets/YBC7289.html Photograph, illustration, and description of the ''root(2)'' tablet from the Yale Babylonian Collection]</ref>]]'''Numerical analysis''' studies different [[algorithm]]s to get [[approximation]]s for problems of [[mathematical analysis]]. Approximations are used for the following reasons: | | In [[mathematics]], a [[theory]] is called '''consistent''', if does not contain any [[contradiction]]s. In the context of [[predicate logic]] it is impossible to deduce both <math> \varphi</math> as well as <math>\neg \varphi</math> from <math>\Phi</math> |
| #There are no known ways to solve a problem using calculus. Examples for such problems are the [[Navier–Stokes equations]] or the [[Three-body problem]]
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| #There is a way to solve a problem using calculus. Getting an exact solution is impractical though, because it requires a long time, or many resources. An example for this is calculating [[Power series]].
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| One of the earliest known uses of numerical analysis is a [[Babylonian]] [[clay]] [[tablet]], which approximates the [[square root]] of 2. In a [[unit square]], the diagonal has this length. Being able to compute the sides of a triangle is extremely important, for instance, in [[carpentry]] and [[construction]].<ref>The New Zealand Qualification authority specifically mentions this skill in document 13004 version 2, dated 17 October 2003 titled [http://www.nzqa.govt.nz/nqfdocs/units/pdf/13004.pdf CARPENTRY THEORY: Demonstrate knowledge of setting out a building]</ref>
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| Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation of <math>\sqrt{2}</math>, modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors.
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| Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21st century, the life sciences and even the arts have adopted elements of scientific computations. Ordinary [[differential equation]]s appear in [[Proper motion|star movement]]; [[Mathematical optimization|optimization]] occurs in portfolio management; [[numerical linear algebra]] is important for data analysis; [[Probability theory|stochastic]] [[differential equation]]s and [[Markov chain]]s are essential in simulating living cells for [[medicine]] and [[biology]].
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| [[Computer]]s greatly helped this task. Before there were computers, numerical methods often depended on hand [[interpolation]] in large printed tables. Since the mid 20th century, computers calculate the required functions instead. These same interpolation formulas nevertheless continue to be used as part of the software [[algorithms]] for solving [[differential equations]].
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| ==References==
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| {{reflist}}
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| {{math-stub}} | | {{math-stub}} |
| [[Category:Mathematics]] | | [[Category:Logic]] |
Latest revision as of 16:40, 13 November 2015