Multiple Polynomial Zeros Research Articles

Construction and Convergence of H-S Combined Mean Method for Multiple Polynomial Zeros

Construction and Convergence of H-S Combined Mean Method for Multiple Polynomial Zeros

Construction and Convergence of the C-S Combined Mean Method for Multiple Polynomial Zeros

Construction and Convergence of the C-S Combined Mean Method for Multiple Polynomial Zeros

Unified Convergence Analysis of Chebyshev–Halley Methods for Multiple Polynomial Zeros

In this paper, we establish two local convergence theorems that provide initial conditions and error estimates to guarantee the Q-convergence of an extended version of Chebyshev–Halley family of iterative methods for multiple polynomial zeros due to Osada (J. Comput. Appl. Math. 2008, 216, 585–599). Our results unify and complement earlier local convergence results about Halley, Chebyshev and Super–Halley methods for multiple polynomial zeros. To the best of our knowledge, the results about the Osada’s method for multiple polynomial zeros are the first of their kind in the literature. Moreover, our unified approach allows us to compare the convergence domains and error estimates of the mentioned famous methods and several new randomly generated methods.

Local and Semilocal Convergence of Nourein’s Iterative Method for Finding All Zeros of a Polynomial Simultaneously

In 1977, Nourein (Intern. J. Comput. Math. 6:3, 1977) constructed a fourth-order iterative method for finding all zeros of a polynomial simultaneously. This method is also known as Ehrlich’s method with Newton’s correction because it is obtained by combining Ehrlich’s method (Commun. ACM 10:2, 1967) and the classical Newton’s method. The paper provides a detailed local convergence analysis of a well-known but not well-studied generalization of Nourein’s method for simultaneous finding of multiple polynomial zeros. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with verifiable initial condition and a posteriori error bound) for the classical Nourein’s method. Each of the new semilocal convergence results improves the result of Petković, Petković and Rančić (J. Comput. Appl. Math. 205:1, 2007) in several directions. The paper ends with several examples that show the applicability of our semilocal convergence theorems.

General Local Convergence Theorems about the Picard Iteration in Arbitrary Normed Fields with Applications to Super–Halley Method for Multiple Polynomial Zeros

In this paper, we prove two general convergence theorems with error estimates that give sufficient conditions to guarantee the local convergence of the Picard iteration in arbitrary normed fields. Thus, we provide a unified approach for investigating the local convergence of Picard-type iterative methods for simple and multiple roots of nonlinear equations. As an application, we prove two new convergence theorems with a priori and a posteriori error estimates about the Super-Halley method for multiple polynomial zeros.

Traub-Gander’s family for the simultaneous determination of multiple zeros of polynomials

By combining Traub-Gander’s family of third order for finding a multiple zero and suitable corrective approximations od Schröder’s and Halley’s type, a new family of iterative methods for the simultaneous approximation of multiple zeros of algebraic polynomials is proposed. Taking various forms of a function involved in the iterative formula, a number of different simultaneous methods can be obtained. It is proved that the order of convergence is 4, 5 or 6, depending of the type of employed corrective approximations. Two numerical examples are given to demonstrate the convergence properties of the proposed family of simultaneous methods. Displayed trajectories of the sequences of approximations point to global characteristics of the proposed family of iterative methods.

Convergence of Newton, Halley and Chebyshev iterative methods as methods for simultaneous determination of multiple polynomial zeros

In this paper, we provide a local convergence analysis of Newton, Halley and Chebyshev iterative methods considered as methods for simultaneous determination of all multiple zeros of a polynomial f over an arbitrary normed field K. Convergence theorems with a priori and a posteriori error estimates for each of the proposed methods are established. The obtained results for Newton and Chebyshev methods are new even in the case of simple zeros. Three numerical examples are given to compare the convergence properties of the considered methods and to confirm the theoretical results.

On the Convergence of Chebyshev’s Method for Multiple Polynomial Zeros

In this paper we investigate the local convergence of Chebyshev’s iterative method for the computation of a multiple polynomial zero. We establish two convergence theorems for polynomials over an arbitrary normed field. A priori and a posteriori error estimates are also provided. All of the results are new even in the case of simple zero.

Improved methods for the simultaneous inclusion of multiple polynomial zeros

Using a new fixed point relation, the interval methods for the simultaneous inclusion of complex multiple zeros in circular complex arithmetic are constructed. 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 with Schröder’s and Halley-like corrections under computationally verifiable initial conditions. The suggested algorithms possess a great computational efficiency since the increase of the convergence rate is attained without additional calculations. Two numerical examples are given to demonstrate convergence characteristics of the proposed method.

On the Convergence of Halley’s Method for Multiple Polynomial Zeros

In this paper, we investigate the local convergence of Halley’s method for the computation of a multiple polynomial zero with known multiplicity. We establish two local convergence theorems for Halley’s method for multiple polynomial zeros under different initial conditions. The convergence of these results is cubic right from the first iteration. Also we find an initial condition which guarantees that an initial guess is an approximate zero of the second kind for Halley’s method. All of the results are new even in the case of simple zeros.

Efficient Halley-like Methods for the Inclusion of Multiple Zeros of Polynomials

AbstractStarting from suitable zero-relation, we derive higher-order iterative methods for the simultaneous inclusion of polynomial multiple zeros in circular complex interval arithmetic. The convergence rate is increased using a family of two-point methods of the fourth order for solving nonlinear equations as a predictor. The methods are more efficient compared to existing inclusion methods for multiple zeros, based on fixed point relations. Using the concept of the R-order of convergence of mutually dependent sequences, we present the convergence analysis of the total-step and the single-step methods. The proposed self-validated methods possess a great computational efficiency since the acceleration of the convergence rate from four to seven is achieved only by a few additional calculations. To demonstrate convergence behavior of the presented methods, two numerical examples are given.

General local convergence theory for a class of iterative processes and its applications to Newton’s method

General local convergence theorems with order of convergence r ≥ 1 are provided for iterative processes of the type x n + 1 = T x n , where T : D ⊂ X → X is an iteration function in a metric space X . The new local convergence theory is applied to Newton iteration for simple zeros of nonlinear operators in Banach spaces as well as to Schröder iteration for multiple zeros of polynomials and analytic functions. The theory is also applied to establish a general theorem for the uniqueness ball of nonlinear equations in Banach spaces. The new results extend and improve some results of [K. Dočev, Über Newtonsche Iterationen, C. R. Acad. Bulg. Sci. 36 (1962) 695–701; J.F. Traub, H. Woźniakowski, Convergence and complexity of Newton iteration for operator equations, J. Assoc. Comput. Mach. 26 (1979) 250–258; S. Smale, Newton’s method estimates from data at one point, in: R.E. Ewing, K.E. Gross, C.F. Martin (Eds.), The Merging of Disciplines: New Direction in Pure, Applied, and Computational Mathematics, Springer, New York, 1986, pp. 185–196; P. Tilli, Convergence conditions of some methods for the simultaneous computation of polynomial zeros, Calcolo 35 (1998) 3–15; X.H. Wang, Convergence of Newton’s method and uniqueness of the solution of equations in Banach space, IMA J. Numer. Anal. 20 (2000) 123–134; I.K. Argyros, J.M. Gutiérrez, A unified approach for enlarging the radius of convergence for Newton’s method and applications, Nonlinear Funct. Anal. Appl. 10 (2005) 555–563; M. Giusti, G. Lecerf, B. Salvy, J.-C. Yakoubsohn, Location and approximation of clusters of zeros of analytic functions, Found. Comput. Math. 5 (3) (2005) 257–311], and others.

An efficient higher order family of root finders

A one parameter family of iterative methods for the simultaneous approximation of simple complex zeros of a polynomial, based on a cubically convergent Hansen–Patrick's family, is studied. We show that the convergence of the basic family of the fourth order can be increased to five and six using Newton's and Halley's corrections, respectively. Since these corrections use the already calculated values, the computational efficiency of the accelerated methods is significantly increased. Further acceleration is achieved by applying the Gauss–Seidel approach (single-step mode). One of the most important problems in solving nonlinear equations, the construction of initial conditions which provide both the guaranteed and fast convergence, is considered for the proposed accelerated family. These conditions are computationally verifiable; they depend only on the polynomial coefficients, its degree and initial approximations, which is of practical importance. Some modifications of the considered family, providing the computation of multiple zeros of polynomials and simple zeros of a wide class of analytic functions, are also studied. Numerical examples demonstrate the convergence properties of the presented family of root-finding methods.

A higher order family for the simultaneous inclusion of multiple zeros of polynomials

Starting from a suitable fixed point relation, a new family of iterative methods for the simultaneous inclusion of multiple complex zeros in circular complex arithmetic is constructed. The order of convergence of the basic family is four. Using Newton’s and Halley’s corrections, we obtain families with improved convergence. Faster convergence of accelerated methods is attained with only few additional numerical operations, which provides a high computational efficiency of these methods. Convergence analysis of the presented methods and numerical results are given.

Relationships between order and efficiency of a class of methods for multiple zeros of polynomials

The behavior of a class of high order methods for solving polynomial equations is examined. It is shown that the number of iterations for local convergence to a multiple zero, to the limits of attainable accuracy, is bounded independent of the multiplicity of the zero, and decreases as the order of the method increases. For the higher order methods, the number of iterations decreases as the multiplicity increases. Computational efficiency as a function of degree, order and multiplicity is investigated, and an effective choice of order is recommended. Numerical examples are provided.

Some higher-order methods for the simultaneous approximation of multiple polynomial zeros

Applying Newton's and Halley's corrections, some modified methods of higher order for the simultaneous approximation of multiple zeros of a polynomial are derived. Further acceleration of convergence of these methods is performed by approximating all zeros in a serial fashion using the new approximations as they become available. Faster convergence is attained without additional calculations. Lower bounds of the R-order of convergence for the serial (single-step) methods are given.

A note on the computation of multiple zeros of polynomials by Newton's method

A note on the computation of multiple zeros of polynomials by Newton's method