On iteration methods without derivatives for the simultaneous determination of polynomial zeros (original) (raw)
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On the efficiency of some combined methods for polynomial complex zeros
Journal of Computational and Applied Mathematics, 1990
Interval methods for the simultaneous inclusion of polynomial zeros produce the approximations that contain the exact zeros providing not only error bounds automatically but also take into account rounding errors without altering the fundamental structure of the interval formula. However, at present, the computational costs of most interval methods are still great, in general. In this paper several effective algorithms which preserve the inclusion property concerning the complex zeros and which have a high computational efficiency are constructed. These algorithms combine the efficiency of ordinary floating-point iterations with the accuracy control that may be obtained by the iterations in interval arithmetic. Several examples are included to illustrate the efficiency and some advantages of the proposed combined methods.
On Euler-like methods for the simultaneous approximation of polynomial zeros
Japan Journal of Industrial and Applied Mathematics, 1998
In this paper we consider some iterative methods of higher order for the simultaneous determination of polynomial zeros. The proposed methods are based on Euler's third order method for finding a zero of a given function and involve Weierstrass' correction in the case of simple zeros. We prove that the presented methods have the order of convergence equal to four or more. Based on a fixed -point relation of Euler's type, two inclusion methods are derived. Combining the proposed methods in floating-point arithmetic and complex interval arithmetic, an efficient hybrid method with automatic error bounds is constructed. Computational aspect and the implementation of the presented algorithms on parallel computers are given.
On some iteration functions for the simultaneous computation of multiple complex polynomial zeros
BIT, 1987
Second order methods for simultaneous approximation of multiple complex zeros of a polynomial are presented. Convergence analysis of new iteration formulas and an efficient criterion for the choice of the appropriate value of a root are discussed. A numerical example is given which demonstrates the effectiveness of the presented methods Subject Classifications: AMS(MOS): 65H05; CR: 5.15.
Computing, 1981
Consider a polynomial P (z) of degree n whose zeros are known to lie in n closed disjoint discs, each disc containing one and only one zero. Starting from the known simultaneous interval processes of the third and fourth order, based on Laguerre iterations, two generalised iterative methods in terms of circular regions are derived in this paper. These interval methods make use of the definition of the k-th root of a disc. The order of convergence of the proposed interval methods is k+2 (k >_-1). Both procedures are suitable for simultaneous determination of interval approximations containing real or complex zeros of the considered polynomial P. A criterion for the choice of the appropriate k-th root set is also given. For one of the suggested methods a procedure for accelerating the convergence is proposed. Starting from the expression for interval center, the generalised iterative method of the (k + 2)-th order in standard arithmetic is derived.
Symmetry
This paper describes the extended method of solving real polynomial zeros problems using the single-step method, namely, the interval trio midpoint symmetric single-step (ITMSS) method, which updates the midpoint at each forward-backward-forward step. The proposed algorithm will constantly update the value of the midpoint of each interval of the previous roots before entering the preceding steps; hence, it always generate intervals that decrease toward the polynomial zeros. Theoretically, the proposed method possesses a superior rate of convergence at 16, while the existing methods are known to have, at most, 9. To validate its efficiency, we perform numerical experiments on 52 polynomials, and the results are presented, using performance profiles. The numerical results indicate that the proposed method surpasses the other three methods by fine-tuning the midpoint, which reduces the final interval width upon convergence with fewer iterations.
ACM Transactions on Mathematical Software, 1987
A measure of efficiency of simultaneous methods for determination of polynomial zeros, defined by the coefficient of efficiency, is considered. This coefficient takes into consideration (1) the R-order of convergence in the sense of the definition introduced by Ortega and Rheinboldt (Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York, 1970) and (2) the number of basic arithmetic operations per iteration, taken with certain weights depending on a processor time. The introduced definition of computational efficiency was used for comparison of the simultaneous methods with various structures.
Symmetry
In 2016, Nedzhibov constructed a modification of the Weierstrass method for simultaneous computation of polynomial zeros. In this work, we obtain local and semilocal convergence theorems that improve and complement the previous results about this method. The semilocal result is of significant practical importance because of its computationally verifiable initial condition and error estimate. Numerical experiments to show the applicability of our semilocal theorem are also presented. We finish this study with a theoretical and numerical comparison between the modified Weierstrass method and the classical Weierstrass method.
Laguerre-like Methods for the Simultaneous Approximation of Polynomial Zeros
2001
Two new methods of the fourth order for the simultaneous determination of multiple zeros of a polynomial are proposed. The presented methods are based on the fixed point relation of Laguerre's type and realized in ordinary complex arithmetic as well as circular complex interval arithmetic. The derived iterative formulas are suitable for the construction of modified methods with improved convergence rate with negligible additional operations. Very fast convergence of the considered methods is illustrated by two numerical examples.