Square-Root Families for the Simultaneous Approximation of Polynomial Multiple Zeros (original) (raw)
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Square-root families for the simultaneous approximation of
2005
One-parameter families of iterative methods for the simultaneous determination of multiple complex zeros of a polynomial are considered. Acceleration of convergence is performed by using Newton's and Halley's corrections for a multiple zero. It is shown that the convergence order of the constructed total-step methods is five and six, respectively. By applying the Gauss-Seidel approach, further improvements of these methods are obtained. The lower bounds of the R-order of convergence of the improved (single-step) methods are derived. Accelerated convergence of all proposed methods is attained with negligible number of additional operations, which provides a high computational efficiency of these methods. Convergence analysis and numerical results are given.
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.
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.
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Applied Mathematics and Computation, 2005
Using a suitable fixed point relation with a complex parameter, a new one parameter family of simultaneous methods of the fourth order for finding complex zeros of a polynomial is derived in ordinary complex arithmetic. Convergence analysis of the presented family is performed under computationally verifiable initial conditions which depend only on polynomial coefficients and initial approximations. Further improvements of the proposed family of methods are discussed and a modification for finding multiple zeros is presented. Some numerical results for various values of the parameter are given.
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Journal of Computational and Applied Mathematics, 1993
Carstensen, C. and M.S. PetkoviC, On iteration methods without derivatives for the simultaneous determination of polynomial zeros, Journal of Computational and Applied Mathematics 45 (1993) 251-266. Several algorithms for simultaneously approximating simple complex zeros of a polynomial are presented. These algorithms use Weierstrass' corrections and do not require any polynomial derivatives. It is shown that Nourein's method is, actually, regula falsi for Weierstrass' corrections. Convergence analysis and computational efficiency are given for the considered methods in complex and circular arithmetic. Special attention is paid to hybrid methods that combine the efficiency of floating-point arithmetic and the inclusion property of interval arithmetic.
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International Journal of Computer Mathematics, 2012
An improvement of the Farmer-Loizou method for the simultaneous determination of simple roots of algebraic polynomials is proposed. Using suitable corrections of Newton's type, the convergence of the basic method is increased from 4 to 5 without any additional calculations. In this manner, a higher computational efficiency of the improved method is achieved. We prove a local convergence of the presented method under initial conditions which depend on a geometry of zeros and their initial approximations. Numerical examples are given to demonstrate the convergence behaviour of the proposed method and related methods.
A highly efficient root-solver of very fast convergence
Applied Mathematics and Computation, 2008
The improved iterative method of Ehrlich-Aberth's type for the simultaneous determination of all simple complex zeros of a polynomial is proposed. The presented convergence analysis shows that the convergence rate of the basic third order method is increased from 3 to 6 using Ostrowski's corrections. The new iterative method is more efficient compared to all existing methods based on fixed point relations. Some computational aspects and numerical examples are given.
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.