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{{Short description|Algorithm to approximate functions}}
The '''Remez algorithm'''
A typical example of a Chebyshev space is the subspace of [[Chebyshev polynomials]] of order ''n'' in the [[Vector space|space]] of real [[continuous function]]s on an [[interval (mathematics)|interval]], ''C''[''a'', ''b'']. The polynomial of best approximation within a given subspace is defined to be the one that minimizes the maximum [[absolute difference]] between the polynomial and the function. In this case, the form of the solution is precised by the [[equioscillation theorem]].
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The result is called the polynomial of best approximation or the [[minimax approximation algorithm]].
A review of technicalities in implementing the Remez algorithm is given by W. Fraser.<ref>{{cite journal |doi=10.1145/321281.321282 |first=W. |last=Fraser |title=A Survey of Methods of Computing Minimax and Near-Minimax Polynomial Approximations for Functions of a Single Independent Variable |journal=J. ACM |volume=12 |pages=295–314 |year=1965 |issue=3 |s2cid=2736060 |doi-access=free }}</ref>
===Choice of initialization===
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:<math>\lVert f - L_n(f)\rVert_\infty \le (1 + \lVert L_n\rVert_\infty) \inf_{p \in P_n} \lVert f - p\rVert</math>
with the norm or [[
:<math>\lVert L_n\rVert_\infty = \overline{\Lambda}_n(T) = \max_{-1 \le x \le 1} \lambda_n(T; x),</math>
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:<math>\lambda_n(T; x) = \sum_{j = 1}^{n + 1} \left| l_j(x) \right|, \quad l_j(x) = \prod_{\stackrel{i = 1}{i \ne j}}^{n + 1} \frac{(x - t_i)}{(t_j - t_i)}.</math>
Theodore A. Kilgore,<ref>{{cite journal |doi=10.1016/0021-9045(78)90013-8 |first=T. A. |last=Kilgore |title=A characterization of the Lagrange interpolating projection with minimal Tchebycheff norm |journal=J. Approx. Theory |volume=24 |pages=273–288 |year=1978 |issue=4 |doi-access=
For Chebyshev nodes, which provides a suboptimal, but analytically explicit choice, the asymptotic behavior is known as<ref>{{cite journal |first1=F. W. |last1=Luttmann |first2=T. J. |last2=Rivlin |title=Some numerical experiments in the theory of polynomial interpolation |journal=IBM J. Res. Dev. |volume=9 |pages=187–191 |year=1965 |issue=3 |doi= 10.1147/rd.93.0187}}</ref>
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:<math>0 < \alpha_n < \frac{\pi}{72 n^2}</math> for <math>n \ge 1,</math>
and upper bound<ref>{{cite book |first=T.J. |last=Rivlin
:<math>\overline{\Lambda}_n(T) \le \frac{2}{\pi} \log(n + 1) + 1</math>
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<math>p_2(x)</math> to the ordinates <math>(-1)^i</math>
:<math>p_1(x_i) = f(x_i), p_2(x_i) = (-1)^i, i = 0, ..., n.</math>
To this end, use each time [[Newton polynomial|Newton's interpolation formula]] with the [[divided differences]] of order <math>0, ...,n</math> and <math>O(n^2)</math> arithmetic operations.
The polynomial <math>p_2(x)</math> has its ''i''-th zero between <math>x_{i-1}</math> and <math>x_i,\ i=1, ...,n</math>, and thus no further zeroes between <math>x_n</math> and <math>x_{n+1}</math>: <math>p_2(x_n)</math> and <math>p_2(x_{n+1})</math> have the same sign <math>(-1)^n</math>.
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:<math>p(x_i) - f(x_i) \ = \ -(-1)^i E,\ \ i = 0, ... , n\!+\!1. </math>
The
<math>p(x)-\tilde p(x) = (p(x) - f(x)) - (\tilde p(x) - f(x))</math> would still be positive/negative at the ''n''+2 nodes <math>x_i</math> and therefore have at least ''n''+1 zeros which is impossible for a polynomial of degree ''n''.
Thus, this ''E'' is a lower bound for the minimum error which can be achieved with polynomials of degree ''n''.
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In each P-region, the current node <math>x_i</math> is replaced with the local maximizer <math>\bar{x}_i</math> and in each N-region <math>x_i</math> is replaced with the local minimizer. (Expect <math>\bar{x}_0</math> at ''A'', the <math>\bar {x}_i</math> near <math>x_i</math>, and <math>\bar{x}_{n+1}</math> at ''B''.) No high precision is required here,
the standard ''line search'' with a couple of ''quadratic fits'' should suffice. (See <ref>
Let <math>z_i := p(\bar{x}_i) - f(\bar{x}_i)</math>. Each amplitude <math>|z_i|</math> is greater than or equal to ''E''. The Theorem of ''de La Vallée Poussin'' and its proof also
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==Variants==
Some modifications of the algorithm are present on the literature.<ref>{{Citation |last1=Egidi |first1=Nadaniela |title=A New Remez-Type Algorithm for Best Polynomial Approximation |date=2020 |url=http://link.springer.com/10.1007/978-3-030-39081-5_7 |work=Numerical Computations: Theory and Algorithms |volume=11973 |pages=56–69 |editor-last=Sergeyev |editor-first=Yaroslav D. |place=Cham |publisher=Springer
* Replacing more than one sample point with the locations of nearby maximum absolute differences.{{Citation needed|date=March 2022}}
* Replacing all of the sample points with in a single iteration with the locations of all, alternating sign, maximum differences.<ref name="toobs">{{cite journal |last1=Temes
* Using the relative error to measure the difference between the approximation and the function, especially if the approximation will be used to compute the function on a computer which uses [[floating point]] arithmetic;
* Including zero-error point constraints.<ref name="toobs" />
* The Fraser-Hart variant, used to determine the best rational Chebyshev approximation.<ref>{{Cite journal |last=Dunham |first=Charles B. |date=1975 |title=Convergence of the Fraser-Hart algorithm for rational Chebyshev approximation |url=https://www.ams.org/mcom/1975-29-132/S0025-5718-1975-0388732-9/ |journal=Mathematics of Computation |language=en |volume=29 |issue=132 |pages=1078–1082 |doi=10.1090/S0025-5718-1975-0388732-9 |issn=0025-5718|doi-access=free |url-access=subscription }}</ref>
== See also ==
{{Portal|Mathematics}}
* [[Approximation theory]]▼
* {{annotated link|Hadamard's lemma}}
* {{annotated link|Laurent series}}
* {{annotated link|Padé approximant}}
* {{annotated link|Newton series}}
* {{annotated link|Function approximation}}
==References==
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