The convergence of Euler-like method for the simultaneous inclusion of polynomial zeros

In this paper, we give the convergence analysis of the Euler-like iterative method for the simultaneous inclusion of all simple real or complex zeros of a polynomial. The established initial conditions provide the safe convergence of the considered method and the fourth order of convergence. These conditions are computationally verifiable, which is of practical importance. A procedure for the choice of initial inclusion disks is also given.     

On a second order method for the simultaneous inclusion of polynomial complex zeros in rectangular arithmetic

Starting from separated rectangles in the complex plane which contain polynomial complex zeros, an iterative method of second order for the simultaneous inclusion of these zeros is formulated in rectangular arithmetic. The convergence and a condition for convergence are considered. Applying Gauss-Seidel approach to the proposed method, two accelerated interval methods are formulated. TheR-order of convergence of these methods is determined. An analysis of the convergence order is given in the presence of rounding errors. The presented methods are illustrated numerically in examples of polynomial equations.     

On theR-order of some accelerated methods for the simultaneous finding of polynomial zeros

A Gauss-Seidel procedure is applied to increase the convergence of a basic fourth order method for finding polynomial complex zeros. Further acceleration of convergence is performed by using Newton's and Halley's corrections. It is proved that the lower bounds of theR-order of convergence for the proposed serial (single-step) methods lie between 4 and 7. Computational efficiency and numerical examples are also given.     

On initial conditions for the convergence of simultaneous root finding methods

Conditions for the convergence of iteration methods for the simultaneous approximation of polynomial complex roots, treated in the literature, are most frequently based on unattainable data. In this paper we give simple initial conditions involving only initial approximations to the roots of a polynomial and the polynomial degree. Convergence theorems are stated for the simultaneous methods in ordinary complex arithmetic and complex interval arithmetic.     

On the convergence condition of generalized root iterations for the inclusion of polynomial zeros

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 simultaneous determination of all zeros of a polynomial

A class of adaptive iterative methods of higher order for the simultaneous determination of all zeros of a polynomial is constructed. These methods preserve their order of convergence also in the case of multiple roots. Numerical examples are included.     

Higher-order simultaneous methods for the determination of polynomial multiple zeros

Starting from Laguerre's method and using Newton's and Halley's corrections for a multiple zero, new simultaneous methods of Laguerre's type for finding multiple (real or complex) zeros of polynomials are constructed. The convergence order of the proposed methods is five and six, respectively. By applying the Gauss–Seidel approach, these methods are further accelerated. The lower bounds of the R-order of convergence of the improved (single-step) methods are derived. Faster convergence of all proposed methods is attained with negligible number of additional operations, which provides a high computational efficiency of these methods. A detailed convergence analysis and numerical results are given.     

Convergence conditions of some methods for the simultaneous computation of polynomial zeros

The convergence of Newton's, Aberth's and Durand–Kerner's methods for polynomial root finding is analyzed; for each of them convergence theorems and error estimates are provided. Received: December 1995 / Revised version: May 1996     

A family of simultaneous methods for the determination of polynomial complex zeros

Using the development of a rational function by elementary fractions, a family of methods for the simultaneous determination of polynomial complex zeros is derived. All the methods of the family are cubically convergent for simple zeros. The known simultaneous procedures of the third order are included. The presented class of iteration functions is suitable for the parallel inclusion of polynomial complex zeros by circular regions. The family of methods, defined in complex circular arithmetic, gives a new interval method with cubic convergence. Numerical example is given.     

An improvement on two iteration methods for simultaneous determination of the zeros of a polynomial

The Durand-Kerner and the Ehrlich? Methods of respective quadratic and cubic convergence are two of the current methods for determining simultaneously all zeros of a polynomial. By respectively including a Durand-Kerner and a Newton correction term in the above formulae, we establish two new methods-the Improved Durand-Kerner and the Improved Ehrlich.

We show that the improvement is reflected by an increase of unity in the order of convergence of each of the two methods.     

A modified algorithm for the simultaneous extraction of polynomial roots

A modification of Viswanathan's algorithm for the simultaneous extraction of polynomial roots is proposed. The Fletcher-Powell adaptation of Davidson's algorithm is used instead of the method of steepest descent used by Viswanathan. The modified algorithm is illustrated using the example treated by Viswanathan.     

On the simultaneous calculation of two zeros

Differentiable functions are approximated by parabolas the zeros of which are approximations for two zeros of the function. Using the calculated zeros for the construction of new approximating parabolas, it arises an iteration method with quadratic convergence at single zeros.     

On the improved Newton-like methods for the inclusion of polynomial zeros

The aim of this paper is to present some modifications of Newton's type method for the simultaneous inclusion of all simple complex zeros of a polynomial. Using the concept of the R-order of convergence of mutually dependent sequences, the convergence analysis shows that the convergence rate of the basic method is increased from 3 to 6 using Jarratt's corrections. The proposed method possesses a great computational efficiency since the acceleration of convergence is attained with only few additional calculations. Numerical results are given to demonstrate convergence properties of the considered methods.     

A computational method for finding the zeros of a multivariable linear time invariant system

A simple computational method is presented in this paper for calculating the zeros of the transfer function which exists between an input and output of an arbitrary multivariable linear time invariant system. The method is simple to use; is computationally fast and is accurate. Some numerical examples for a 9th order system are included.     

On a generalisation of the root iterations for polynomial complex zeros in circular interval arithmetic

Consider a polynomialP (z) of degreen whose zeros are known to lie inn 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 thek-th root of a disc. The order of convergence of the proposed interval methods isk+2 (k≧1). Both procedures are suitable for simultaneous determination of interval approximations containing real or complex zeros of the considered polynomialP. A criterion for the choice of the appropriatek-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.     

A modified convergence theorem for a random optimization method

The purpose of this paper is to investigate the random optimization method which has been contrived for the minimization method. Concerning with the convergence of this method, Matyas gave interesting theorems. However, there is a questionable point in his proofs, and the assumptions used in them are too severe. In this paper, a modified theorem concerned with the convergence of the random optimization method is given.     

On the convergence of some methods for determining zeros of order-convex operators

LetF:X→Y be an order-convex operator, whereX, Y are partially ordered Banach spaces. Two related methods for the solution ofF(x)=0 are discussed, one of which has been studied by Pasquali (see [2]) and the other by Wolfe [12]. Existence-convergence theorems for the methods are given, and these are illustrated with the aid of example. Some remarks are also made on a method due to Traub [7] which has also been discussed by Wolfe [12].     

A higher order method for multiple zeros of nonlinear functions

A method of order three for finding multiple zeros of nonlinear functions is developed. The method requires two evaluations of the function and one evaluation of the derivative per step.     

An algorithm for locating all zeros of a real polynomial

We present a globally convergent algorithm for calculating all zeros of a polynomialp _n ,p _n (z) = ∑ _{v = 0} ⁿ a _v z ^v, with real coefficients. Splittingp _n (exp(it)) into its real and imaginary part we can decide via Euclidean division of Chebyshev expansions and Sturm sequence argumentations whetherp _n has some zeros on the unit circle and how many zeros lie on the boundary and in the interior of it. Hence, by a bisection strategy we get the moduli of all zeros to a prescribed accuracy, and additionally we find the arguments as real zeros of a low degree polynomial. In this way we generate starting approximations for all zeros which in a final step are refined by an iterative process of higher order of convergence (e.g. Newton's or Bairstow's method).