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| In mathematics, '''Neville's algorithm''' is an algorithm used for [[polynomial interpolation]] that was derived by the mathematician [[Eric Harold Neville]]. Given ''n'' + 1 points, there is a unique polynomial of degree ''≤ n'' which goes through the given points. Neville's algorithm evaluates this polynomial.
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| Neville's algorithm is based on the [[Newton polynomial|Newton form]] of the interpolating polynomial and the recursion relation for the [[divided differences]]. It is similar to Aitken's algorithm (named after [[Alexander Aitken]]), which is nowadays not used.
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| ==The algorithm==
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| Given a set of ''n''+1 data points (''x''<sub>''i''</sub>, ''y''<sub>''i''</sub>) where no two ''x''<sub>''i''</sub> are the same, the interpolating polynomial is the polynomial ''p'' of degree at most ''n'' with the property
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| <!--:<math>p(x_i) = y_i \mbox{ , } i=0,\ldots,n.</math>-->
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| :''p''(''x''<sub>''i''</sub>) = ''y''<sub>''i''</sub> for all ''i'' = 0,…,''n''
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| This polynomial exists and it is unique. Neville's algorithm evaluates the polynomial at some point ''x''.
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| Let ''p''<sub>''i'',''j''</sub> denote the polynomial of degree ''j'' − ''i'' which goes through the points (''x''<sub>''k''</sub>, ''y''<sub>''k''</sub>) for ''k'' = ''i'', ''i'' + 1, …, ''j''. The
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| ''p''<sub>''i'',''j''</sub> satisfy the recurrence relation
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| :{|
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| | <math> p_{i,i}(x) = y_i, \, </math> || <math> 0 \le i \le n, \, </math>
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| |-
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| | <math> p_{i,j}(x) = \frac{(x_j-x)p_{i,j-1}(x) + (x-x_i)p_{i+1,j}(x)}{x_j-x_i}, \, </math> || <math> 0\le i < j \le n. \, </math>
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| |}
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| This recurrence can calculate
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| <!--<math>p_{0,n}(x)</math>,-->
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| ''p''<sub>0,''n''</sub>(''x''),
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| which is the value being sought. This is Neville's algorithm.
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| For instance, for ''n'' = 4, one can use the recurrence to fill the triangular tableau below from the left to the right.
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| :{|
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| | <math> p_{0,0}(x) = y_0 \, </math>
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| |-
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| | || <math> p_{0,1}(x) \, </math>
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| |-
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| | <math> p_{1,1}(x) = y_1 \, </math> || || <math> p_{0,2}(x) \, </math>
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| |-
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| | || <math> p_{1,2}(x) \, </math> || || <math> p_{0,3}(x) \, </math>
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| |-
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| | <math> p_{2,2}(x) = y_2 \, </math> || || <math> p_{1,3}(x) \, </math> || || style="border: 1px solid;" | <math> p_{0,4}(x) \, </math>
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| |-
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| | || <math> p_{2,3}(x) \, </math> || || <math> p_{1,4}(x) \, </math>
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| |-
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| | <math> p_{3,3}(x) = y_3 \, </math> || || <math> p_{2,4}(x) \, </math>
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| |-
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| | || <math> p_{3,4}(x) \, </math>
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| |-
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| | <math> p_{4,4}(x) = y_4 \, </math>
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| |}
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| This process yields
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| <!--<math>p_{0,4}(x)</math>,-->
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| ''p''<sub>0,4</sub>(''x''),
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| the value of the polynomial going through the ''n'' + 1 data points (''x''<sub>''i''</sub>, ''y''<sub>''i''</sub>) at the point ''x''.
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| This algorithm needs [[big O notation|O]](''n''<sup>2</sup>) floating point operations.
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| ==Application to numerical differentiation==
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| Lyness and Moler showed in 1966 that using undetermined coefficients for the polynomials in Neville's algorithm, one can compute the Maclaurin expansion of the final interpolating polynomial, which yields numerical approximations for the derivatives of the function at the origin. While "this process requires more arithmetic operations than is required in finite difference methods", "the choice of points for function evaluation is not restricted in any way". They also show that their method can be applied directly to the solution of linear systems of the Vandermonde type.
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| ==References==
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| *{{cite book | last = Press | first = William | coauthors = Saul Teukolsky, William Vetterling and Brian Flannery | title = [[Numerical Recipes|Numerical Recipes in C. The Art of Scientific Computing]] | edition = 2nd edition | year = 1992 | publisher = Cambridge University Press | isbn = 978-0-521-43108-8 | doi=10.2277/0521431085 | chapter = §3.1 Polynomial Interpolation and Extrapolation (encrypted) | chapterurl = http://www.nrbook.com/ub30001/nr3-3-2.pdf }} (link is bad)
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| * J. N. Lyness and C.B. Moler, Van Der Monde Systems and Numerical Differentiation, Numerishe Mathematik 8 (1966) 458-464
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| ==External links==
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| *{{MathWorld|title=Neville's Algorithm|urlname=NevillesAlgorithm}}
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| *[http://math.fullerton.edu/mathews/n2003/NevilleAlgorithmMod.html Module for Neville Interpolation by John H. Mathews]
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| *[https://s3.amazonaws.com/torkian/torkian/Site/Research/Entries/2008/2/29_Nevilles_algorithm_Java_Code.html Java Code by behzad torkian]
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| [[Category:Polynomials]]
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| [[Category:Interpolation]]
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| [[de:Polynominterpolation#Algorithmus_von_Neville-Aitken]]
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Ed is what people call me and my spouse doesn't like it at all. Kentucky is exactly where I've always been living. Doing ballet is something she would by no means give up. Distributing manufacturing has been his occupation for some time.
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