Improvement of convergence condition of the square-root interval method for multiple zeros (original) (raw)
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Reliab. Comput., 2012
A new iterative method of Ostrowski’s type for the inclusion of one isolated simple or multiple complex zero of a polynomial is established in circular complex arithmetic. Cubic convergence is proved and computationally veriable initial condition that guarantees the convergence of this inclusion method is stated. In order to demonstrate convergence behavior of the proposed method, two numerical examples are given.
On the improved family of simultaneous methods for the inclusion of multiple zeros of polynomials
Starting from a family of iterative methods for the simultaneous inclusion of multiple complex zeros, we construct efficient iterative methods with accelerated convergence rate by the use of Gauss-Seidel procedure and the suitable corrections. The proposed methods are realized in the circular complex interval arithmetic and produce disks that contain the wanted zeros. The suggested algorithms possess a high computational efficiency since the increase of the convergence rate is attained without additional calculations. Using the concept of the R-order of convergence of mutually dependent sequences, the convergence analysis of the proposed methods is presented. Numerical results are given to demonstrate the convergence properties of the considered methods.
On an iterative method for simultaneous inclusion of polynomial complex zeros
Journal of Computational and Applied Mathematics, 1982
Starting from disjoint discs which contain polynomial complex zeros, the iterative interval method of the third order for the simultaneous finding inclusive discs for complex zeros is formulated. The Lagrangean interpolation formula and complex circular arithmetic are used. The convergence theorem and the conditions for convergence are considered. The proposed method has been applied for solving an algebraic equation.
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Mathematics and Computers in Simulation, 2008
Using a fixed point relation based on the logarithmic derivative of the k-th order of an algebraic polynomial and the definition of the k-th root of a disk, a family of interval methods for the simultaneous inclusion of complex zeros in circular complex arithmetic was established by Petković [M.S. Petković, On a generalization of the root iterations for polynomial complex zeros in circular interval arithmetic, Computing 27 (1981) 37-55]. In this paper we give computationally verifiable initial conditions that guarantee the convergence of this parallel family of inclusion methods. These conditions are significantly relaxed compared to the previously stated initial conditions presented in literature.
The improved square-root methods for the inclusion of multiple zeros of polynomials
2010
Starting from a fixed point relation, we construct very fast iterative methods of Ostrowski-root's type for the simultaneous inclusion of all multiple zeros of a polynomial. The proposed methods possess a great computational efficiency since the acceleration of the convergence is attained with only a few additional calculations. Using the concept of the R-order of convergence of mutually dependent sequences, we present the convergence analysis of the total-step method with Schröder's and Halley's corrections under computationally verifiable initial conditions. Further acceleration is attained by the Gauss-Seidel approach (single-step mode). Numerical examples are given to demonstrate properties of the proposed inclusion methods.
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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 an Interval Method for the Inclusion of One Polynomial Zero
2013
In this paper we construct a new interval method for the inclusion of one simple or multiple complex polynomial zero in circular complex arithmetic. We present the convergence analysis starting from the computationally verifiable initial condition that guarantees the convergence of this inclusion method. We also give two numerical examples in order to demonstrate convergence behavior of the proposed method.
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.