A remez algorithm with simultaneous exchanges is described for minimax approximation with Lagrange-type interpolation by varisolvent families, in particular, families of Meinardus and Schwedt.
A remez algorithm with simultaneous exchanges is described for minimax approximation with Lagrange-type interpolation by varisolvent families, in particular, families of Meinardus and Schwedt.
In recent decades, several generalizations of the real remez algorithm to the complex domain have been proposed. For example, a recent paper by Tseng, [SIAM J. Numer. Anal., 33 (1996), pp. 2017-2049] presents a genera...
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In recent decades, several generalizations of the real remez algorithm to the complex domain have been proposed. For example, a recent paper by Tseng, [SIAM J. Numer. Anal., 33 (1996), pp. 2017-2049] presents a generalized multiple exchange method for solving Chebyshev approximation problems by polynomials on the unit circle. His method is particularly efficient when the number of extremal points characterizing the optimal solution is close to its lower bound n + 1. (n - 1 is the degree of the polynomial approximant.) Under the same assumptions, the aim of this paper is to show that the complex problem can be solved by considering a real polynomial Chebyshev approximation. Hence, we apply the real remez algorithm and we illustrate the efficiency of this approach by various numerical experiments, e.g., in digital filter design.
Here a more accurate piecewise approximation (PWA) scheme for non-linear activation function is proposed. It utilizes a precision-controlled recursive algorithm to predict a sub-range;after that, the remez algorithm i...
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Here a more accurate piecewise approximation (PWA) scheme for non-linear activation function is proposed. It utilizes a precision-controlled recursive algorithm to predict a sub-range;after that, the remez algorithm is used to find the corresponding approximation function. The PWA realized in three ways: using first-order functions, that is, piecewise linear model, second-order functions (piecewise non-linear model), and hybrid-order model (a mixture of first-order and second-order functions). The hybrid-order approximation employs the second-order derivative of non-linear activation function to decide the linear and non-linear sub-regions, correspondingly the first-order and second-order functions are predicted, respectively. The accuracy is compared to the present state-of-the-art approximation schemes. The multi-layer perceptron model is designed to implement XOR-gate, and it uses an approximate activation function. The hardware utilization is measured using the TSMC 0.18-mu m library with the Synopsys Design Compiler. Result reveals that the proposed approximation scheme efficiently approximates the non-linear activation functions.
Uniform approximation on an interval [αβ] by an alternating family when positive deviations (errors) are magnified by a bias factor is considered. This problem is related to one-sided uniform approximation from abov...
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Uniform approximation on an interval [αβ] by an alternating family when positive deviations (errors) are magnified by a bias factor is considered. This problem is related to one-sided uniform approximation from above for large bias factors. Best approximations are characterized by alternation, suggesting use of the first author's variant of the remez 2nd algorithm for generalized weight functions. Previously written subprograms of the authors are combined with minor modifications to produce a general subprogram for families satisfying the hypotheses of Meinardus and Schwedt.
Best one-sided minimax approximations from above on an interval by alternating families have an alternating characterization and may be computed by a simple modification of the classical remez algorithm for ordinary C...
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Best one-sided minimax approximations from above on an interval by alternating families have an alternating characterization and may be computed by a simple modification of the classical remez algorithm for ordinary Chebyshev approximation.
The classical remez algorithm was developed for constructing the best polynomial approximations for continuous and discrete functions in an interval [a, b]. In this paper, the classical remez algorithm is generalized ...
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The classical remez algorithm was developed for constructing the best polynomial approximations for continuous and discrete functions in an interval [a, b]. In this paper, the classical remez algorithm is generalized to the problem of linear spline approximation with certain conditions on the spline parameters. Namely, the spline parameters have to be nonnegative and the values of the splines at one of the borders (or both borders) of the approximation intervals may be fixed. This type of constraint occurs in some practical applications, e.g. the problem of taxation tables restoration. The results of the numerical experiments with a remez-like algorithm developed for this class of conditional optimization problems, are presented.
In this paper, a general algorithm for finding of function extrema in the remez algorithm is presented. The proposed algorithm utilizes neural network, gradient method and least square method which allows one to achie...
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ISBN:
(纸本)9788388309472
In this paper, a general algorithm for finding of function extrema in the remez algorithm is presented. The proposed algorithm utilizes neural network, gradient method and least square method which allows one to achieve precise solution of the problem. Moreover since neural network allows a parallel solving for several points, it means the proposed algorithm is time efficient. The gradient method and the least square method make the algorithm ease to implement.
The classical remez algorithm was developed for constructing the best polynomial approximations for continuous and discrete functions in an interval [a, b]. In this paper, the classical remez algorithm is generalized ...
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The classical remez algorithm was developed for constructing the best polynomial approximations for continuous and discrete functions in an interval [a, b]. In this paper, the classical remez algorithm is generalized to the problem of linear spline approximation with certain conditions on the spline parameters. Namely, the spline parameters have to be nonnegative and the values of the splines at one of the borders (or both borders) of the approximation intervals may be fixed. This type of constraint occurs in some practical applications, e.g. the problem of taxation tables restoration. The results of the numerical experiments with a remez-like algorithm developed for this class of conditional optimization problems, are presented.
The best approximation problem is a classical topic of the approximation theory and the remez algorithm is one of the most famous methods for computing minimax polynomial approximations. We present a slight modificati...
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ISBN:
(纸本)9783030390815
The best approximation problem is a classical topic of the approximation theory and the remez algorithm is one of the most famous methods for computing minimax polynomial approximations. We present a slight modification of the (second) remez algorithm where a new approach to update the trial reference is considered. In particular at each step, given the local extrema of the error function of the trial polynomial, the proposed algorithm replaces all the points of the trial reference considering some "ad hoc" oscillating local extrema and the global extremum (with its adjacent) of the error function. Moreover at each step the new trial reference is chosen trying to preserve a sort of equidistribution of the nodes at the ends of the approximation interval. Experimentally we have that this method is particularly appropriate when the number of the local extrema of the error function is very large. Several numerical experiments are performed to assess the real performance of the proposed method in the approximation of continuous and Lipschitz continuous functions. In particular, we compare the performance of the proposed method for the computation of the best approximant with the algorithm proposed in [17] where an update of the remez ideas for best polynomial approximation in the context of the chebfun software system is studied.
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