Remez algorithm

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The Remez algorithm (sometimes also called Remes algorithm, Remez/Remes exchange algorithm), published by Evgeny Yakovlevich Remez in 1934^[1] is an iterative algorithm for best approximation in the uniform norm L_∞ in the Chebyshev space. A typical example of Chebyshev space is the subspace of polynomial of order n in the space of real continuous function on an interval, C[a, b].

The polynomial of best approximation of a given degree is defined to be the one that minimizes the maximum absolute difference between the polynomial and the function.

1 Procedure
- 1.1 On the choice of initialization
2 Variants
3 References
4 See also
5 External links

[edit] Procedure

The Remez algorithm starts with a set of sample points in the approximation interval, usually the Chebyshev nodes linearly mapped to the interval.

A polynomial approximation of the function at the sample points is obtained through Lagrange interpolation.
The difference between the approximation and the function is measured across the interval. The point where the difference has the largest absolute value is found.
The sample point nearest the location of the maximum absolute difference is replaced with the location of the maximum absolute difference.
The process is repeated until all sample points converge to the points of maximum absolute difference, and the sign of the difference between the function and the polynomial alternates.

The result is called the polynomial of best approximation, the Chebyshev approximation, or the minimax approximation.

A review of technicalities in implementing the Remez algorithm is given by W. Fraser^[2].

[edit] On the choice of initialization

The reason for the Chebyshev nodes being a common choice for the initial approximation is in its behavior in the theory of polynomial interpolation.

For the initialization of the optimization problem for function f by the Lagrange interpolant L_n(f), it can be shown that this initial approximation is bounded by

$\lVert f - L_n(f)\rVert_\infty \le (1 + \lVert L_n\rVert_\infty) \inf_{p \in P_n} \lVert f - p\rVert$

with the norm or Lebesgue constant of the Lagrange interpolation operator L_n of the nodes (t₁, ..., t_n + 1) being

$\lVert L_n\rVert_\infty = \overline{\Lambda}_n(T) = \max_{-1 \le x \le 1} \lambda_n(T; x),$

T being the zeros of the Chebyshev polynomials, and the Lebesgue functions being

$\lambda_n(T; x) = \sum_{j = 1}^{n + 1} \left| l_j(x) \right|, \quad l_j(x) = \prod_{\stackrel{i = 1}{i \ne j}}^{n + 1} \frac{(x - t_i)}{(t_j - t_i)}$

Theodore A. Kilgore^[3], Carl de Boor, and Allan Pinkus^[4] proved that there exists an unique t_i for each L_n, although not known explicitly for (ordinary) polynomials. Similarly, $\underline{\Lambda}_n(T) = \min_{-1 \le x \le 1} \lambda_n(T; x)$ , and the optimality of a choice of nodes can be expressed as $\overline{\Lambda}_n - \underline{\Lambda}_n \ge 0.$

For Chebyshev nodes, which provides a suboptimal, but analytically explicit choice, the asymptotic behavior is known as^[5]

$\overline{\Lambda}_n(T) = \frac{2}{\pi} \log(n + 1) + \frac{2}{\pi}\left(\gamma + \log\frac{8}{\pi}\right) + \alpha_{n + 1}$

(γ being the Euler-Mascheroni constant) with

$0 < \alpha_n < \frac{\pi}{72 n^2}$ for $n \ge 1,$

and upper bound^[6]

$\overline{\Lambda}_n(T) \le \frac{2}{\pi} \log(n + 1) + 1$

Lev Brutman^[7] obtained the bound for $n \ge 3$ , and $\hat{T}$ being the zeros of the expanded Chebyshev polynomials:

$\overline{\Lambda}_n(\hat{T}) - \underline{\Lambda}_n(\hat{T}) < \overline{\Lambda}_3 - \frac{1}{6} \cot \frac{\pi}{8} + \frac{\pi}{64} \frac{1}{\sin^2(3\pi/16)} - \frac{2}{\pi}(\gamma - \log\pi)\approx 0.201.$

Rüdiger Günttner^[8] obtained from a sharper estimate for $n \ge 40$

$\overline{\Lambda}_n(\hat{T}) - \underline{\Lambda}_n(\hat{T}) < 0.0196.$

[edit] Variants

Sometimes more than one sample point is replaced at the same time with the locations of nearby maximum absolute differences.

Sometimes relative error is used 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.