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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>
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The Chebyshev nodes are a common choice for the initial approximation because of their role in the theory of polynomial interpolation. For the initialization of the optimization problem for function ''f'' by the Lagrange interpolant ''L''<sub>n</sub>(''f''), it can be shown that this initial approximation is bounded by
:<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 |
* 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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*[https://www.boost.org/doc/libs/1_47_0/libs/math/doc/sf_and_dist/html/math_toolkit/toolkit/internals2/minimax.html Minimax Approximations and the Remez Algorithm], background chapter in the [[Boost (C++ libraries)|Boost]] Math Tools documentation, with link to an implementation in C++
*[http://www.bores.com/courses/intro/filters/4_equi.htm Intro to DSP]
*{{MathWorld|urlname=RemezAlgorithm|title=Remez Algorithm|author1-link=Ronald Aarts|author=Aarts, Ronald M.|author2=Bond, Charles|author3=Mendelsohn, Phil|author4= Weisstein, Eric W.|name-list-style=amp}}
[[Category:Polynomials]]
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