Some improved inclusion methods for polynomial roots with Weierstrass' corrections (original) (raw)
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Applied Numerical Mathematics, 1994
The interval version of the complex third-order method of Maehly, Borsch-Supan, Ehrlich, and Aberth is the most efficient method for simultaneous inclusion of simple polynomial roots [14]. In this note, Gargantini's generalization of this third-order interval method for multiple roots is accelerated using Schroder's modification of Newton's corrections and modifying the required interval inversions. The underlying idea is that the iteration of the midpoints of the interval method should be similar to Nourein's acceleration of the above-mentioned complex third-order method improving the convergence of the midpoints. Since the convergence of the radii and the midpoints are coupled it can be proved that the R-orders of convergence of the radii of the newly presented Schroder-like interval methods are asymptotically greater than 3.5. Hence two of these methods are more efficient than the most efficient one known before. Numerical results and an analysis of computational efficiency are included.
Euler-Like Method for the Simultaneous Inclusion of Polynomial Zeros with Weierstrass’ Correction
Scientific Computing, Validated Numerics, Interval Methods, 2001
An improved iterative method of Euler's type for the simultaneous inclusion of polynomial zeros is considered. To accelerate the convergence of the basic method of fourth order, Carstensen-Petković's approach using Weierstrass' correction is applied. It is proved that the R-order of convergence of the improved Euler-like method is (asymptotically) 2 + √ 7 ≈ 4.646 or 5, depending of the type of applied inversion of a disk. The proposed algorithm possesses great computational efficiency since the increase of the convergence rate is obtained without additional calculations. Initial conditions which provide the guaranteed convergence of the considered method are also studied. These conditions are computationally verifiable, which is of practical importance.
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.
New higher-order methods for the simultaneous inclusion of polynomial zeros
Numerical Algorithms, 2011
Higher-order methods for the simultaneous inclusion of complex zeros of algebraic polynomials are presented in parallel (total-step) and serial (single-step) versions. If the multiplicities of each zeros are given in advance, the proposed methods can be extended for multiple zeros using appropriate corrections. These methods are constructed on the basis of the zero-relation of Gargantini's type, the inclusion isotonicity property and suitable corrections that appear in two-point methods of the fourth order for solving nonlinear equations. It is proved that the order of convergence of the proposed methods is at least six. The computational efficiency of the new methods is very high since the acceleration of convergence order from 3 (basic methods) to 6 (new methods) is attained using only n polynomial evaluations per iteration. Computational efficiency of the considered methods is studied in detail and two numerical examples are given to demonstrate the convergence behavior of the proposed methods.
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.
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.
Ostrowski-Like Method for the Inclusion of a Single Complex Polynomial Zero
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.
Derivative free method for the simultaneous inclusion of polynomial zeros
Filomat, 2003
A combined method for the simultaneous inclusion of complex zeros of a polynomial, composed of two circular arithmetic methods, is presented. This method does not use polynomial derivatives and has the order of convergence equals four. Computationally verifiable initial conditions that guarantee the convergence are also stated. Two numerical example are included to demonstrate the convergence speed of the presented method.
Inclusion Weierstrass-like root-finders with corrections
Filomat, 2003
In this paper we present iterative methods of Weierstrass's type for the simultaneous inclusion of all multiple zeros of a polynomial. The order of convergence of the proposed interval method is 1 + ?2 ? 2.414 or 3, depending on the type of the applied disk inversion. The criterion for the choice of a proper circular root-set is given. This criterion uses the already calculated entries which increases the computational efficiency of the presented algorithms. Numerical results are given to demonstrate the convergence behavior.
Weierstrass-like Methods with Corrections for the Inclusion of Polynomial Zeros
Computing, 2005
In this paper we present iteration methods of Weierstrass' type for the simultaneous inclusion of all (simple or multiple) zeros of a polynomial. The main advantage of the proposed methods consists of the increase of the convergence rate applying correction terms. Using the concept of the R-order of convergence of mutually dependent sequences, we present the convergence analysis for the total-step and the single-step methods. Numerical examples are given.