Complex Zeros Of Polynomials Research Articles

Zeros of complex polynomials and Kaplan classes

We carry out complete membership to Kaplan classes of polynomials with all zeros on unit circle. In this way we extend Sheil-Small’s and Jahangiri’s theorems.

Ehrlich-type Methods with King's Correction for the Simultaneous Approximation of Polynomial Complex Zeros

There are many simultaneous iterative methods for approximating complex polynomial zeros, from more traditional numerical algorithms, such as the well-known third order Ehrlich–Aberth method, to the more recent ones. In this paper, we present a new family of combined iterative methods for the simultaneous determination of simple complex zeros of a polynomial, which uses the Ehrlich iteration and a correction based on King's family of iterative methods for nonlinear equations. The use of King's correction allows increasing the convergence order of the basic method from three to six. Some numerical examples are given to illustrate the convergence behaviour and effectiveness of the proposed sixth order Ehrlich-like family of combined iterative methods for the simultaneous approximation of simple complex polynomial zeros.

Vibration of Mechanical System with Higher Degrees of Freedom: Solution of the Frequency Equations

The solution of the motion equation of the rigid body systems with higher degrees of freedom 2 ≤ p° ≤ 10 is difficult. The presented method allows solving the motion equations of such systems by its transformation to higher degrees’ algebraic characteristic equations. The vibration of the system is then described by frequencies obtained from solution of characteristic equations. The proposed method follows Bezout's factor theorem, Bairstow-Hitchcock's method, method of synthetic division and other presuppositions given in the article. The solution is based on the determination of the complex zeros of polynomials.

Polynomials with coefficients from a finite set

Abstract This paper focuses on the problem concerning the location and the number of zeros of those polynomials when their coefficients are restricted with special conditions. The problem of the number of the zeros of reciprocal Littlewood polynomials on the unit circle $\mathbb{T}$ is discussed, the interest on bounds for the number of the zeros of reciprocal polynomials on the unit circle arose after 1950 when Erdös began introducing problems on zeros of various types of polynomials. Our main result is the problem of finding the number of zeros of complex polynomials in an open disk.

The Quality of Zero Bounds for Complex Polynomials

In this paper, we evaluate the quality of zero bounds on the moduli of univariate complex polynomials. We select classical and recently developed bounds and evaluate their quality by using several sets of complex polynomials. As the quality of priori bounds has not been investigated thoroughly, our results can be useful to find optimal bounds to locate the zeros of complex polynomials.

Location of Zeros of Complex Polynomials

Location of Zeros of Complex Polynomials

Preface

Following the fashion of the conference Applied Linear Algebra—in honor of Richard Varga, which took place in Palić in October 2005, a celebration honoring another influential person in applied linear algebra community, the international conference Applied Linear Algebra—in honor of Ivo Marek was held in Novi Sad, Serbia, on 28–30 April 2008. The aim of ALA 2008 was to review numerous contributions of Ivo Marek and to report and discuss recent progress through the participation of international leaders in the field, who gathered in his honor. The conference was attended by 67 participants from 15 different countries around the world. The program included invited and contributed talks, as well as speeches and social activities. The talks presented at the conference covered a wide range of topics in numerical analysis and scientific computation, with an emphasis on matrix problems and their numerical solution. Eleven papers published in this special issue represent a selection of those presented at the conference. Four of them are dealing with eigenvalues: from a generalization of the Perron–Frobenius theory and an investigation of minimal eigenvalues of totally non-negative matrices, via generalized eigenvalues and matrix pencils, to the nonlinear eigenvalue problem and variational eigenproblem. Four papers deal with systems of equations: nonsymmetric linear systems connected with preconditioning, bilinear systems, and some properties of the Schur complement, as well. Iterative methods are the main subject of two papers related to matrix sector function, and simultaneous determination of complex polynomial zeros. Well-conditioned matrices are investigated in one paper, in which Fredholm integration equation is considered. The papers appear in the order described above, and they are all a fitting homage to Ivo Marek, a very distinguished person in mathematics in the Czech Republic and abroad. Ivo Marek was born in Prague in 1933. He received his BSc in mathematics in 1956, and his PhD in 1962, both from Charles University Prague. He became a Doctor of Sciences at Academy of Sciences, Prague in 1968, and Professor of Mathematics in 1977. According to his own words, his work in Nuclear Research Institute in Rez, a city not far from Prague, from 1956 to 1963, made a significant influence upon his further research and work. Since 1963 he is working as a Professor of Mathematics at Charles University of Prague, and later at Czech Technical University in Prague, where he is still an active developer of modern numerical analysis. He is an active member of New York Academy since 1995, and an active member of Academia Scientiarum et Artium Europaea, Salzburg. Besides his regular appointments, Ivo Marek has held visiting positions at several universities and research institutions in U.S.A., Russia, Spain, Germany, Sweden, and the Netherlands. His list of publications consists of more than 120 research papers. His research contributions span several areas of numerical analysis and operator theory: numerical solution of operator equations, numerical solution of partial differential equations, large-scale computations, multigrid and aggregation methods, K-positive operators with applications to problems of mathematical physics, homogenization, data processing. He has always been very active and encouraging his students and colleagues. He also served (and still serves) in several very important and highly rating journals (including this one). Among his honors, let us mention the Bolzano Silver Medal for Merits in Mathematics (1983), the Medal of Peace of the Charles University (1988), and especially the Gold Felber Medal of Czech Technical University for 2008. Ivo Marek, however, is not only a great scientist, but also a living proof that mathematics and music fit together, for he is a big connoisseur of music and of world composers. He used to play flute. He was also an excellent tennis player. He and his wife became champions of Czech republic in tennis—mixed pairs, some ‘few’ years ago. We dedicate this issue of Numerical Linear Algebra with Applications to our friend, Professor Ivo Marek, a creator, who has immensely contributed to the treasury of science as well as to opening the roads that lead to enrichment of knowledge and its application.

Nodal lines, ergodicity and complex numbers

This article reviews two rigorous results about the complex zeros of eigenfunctions of the Laplacian, that is, the zeros of the analytic continuation of the eigenfunctions to the complexification of the underlying space. Such a complexification of the problem is analogous to studying the complex zeros of polynomials with real coefficients. The first result determines the limit distribution of complex zeros of `ergodic eigenfunctions' such as eigenfunctions of classically chaotic systems. The second result determines the expected distribution of complex zeros for complexifications of Gaussian random waves adapted to the Riemannian manifold. The resulting distribution is the same in both cases. It is singular along the set of real points.

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.

On efficient computation and asymptotic sharpness of Kalantari’s bounds for zeros of polynomials

We study an infinite family of lower and upper bounds on the modulus of zeros of complex polynomials derived by Kalantari. We first give a simple characterization of these bounds which leads to an efficient algorithm for their computation. For a polynomial of degree n n our algorithm computes the first m m bounds in Kalantari’s family in O ( m n ) O(mn) operations. We further prove that for every complex polynomial these lower and upper bounds converge to the tightest annulus containing the roots, and thus settle a problem raised in Kalantari’s paper.

Polynomiography and applications in art, education, and science

Polynomiography is the art and science of visualizing approximation of the zeros of complex polynomials. Informally speaking, polynomiography allows one to create colorful images of polynomials. These images can subsequently be re-colored in many ways, using one's own creativity and artistry. Polynomiography has tremendous applications in the visual arts, education, and science. The paper describes some of those applications. Artistically, polynomiography can be used to create quite a diverse set of images reminiscent of the intricate patterning of carpets and elegant fabrics, abstract expressionist and minimalist art, and even images that resemble cartoon characters. Educationally, polynomiography can be used to teach mathematical concepts, theorems, and algorithms, e.g., the algebra and geometry of complex numbers, the notions of convergence and continuity, geometric constructs such as Voronoi regions, and modern notions such as fractals. Scientifically, polynomiography provides not only a tool for viewing polynomials, present in virtually every branch of science, but also a tool to discover new theorems.

On the guaranteed convergence of the square-root iteration method

The construction of initial conditions which guarantee the convergence of the applied iterative method, is one of the most important problems in solving nonlinear equations which attracted the great attention for many years. In this paper, we give a precise convergent analysis of the Ostrowski-like method of the fourth order for the simultaneous determination of polynomial zeros. Using a procedure based on Smale's point estimation theory and some recent results related to the localization of complex polynomial zeros, we state initial conditions which enable both the guaranteed and fast convergence of this method. These conditions are computationally verifiable since they depend only on polynomial coefficients, its degree and initial approximations, which is of practical importance.

The guaranteed convergence of Laguerre-like method

The construction of initial conditions that provide a guaranteed convergence of zero-finding methods has attracted a great deal of attention for many years. In this paper, we consider convergent properties of the Laguerre-like method of the fourth order for the simultaneous approximation of polynomial zeros. Using a procedure based on Smale's point estimation theory and some recent results concerned with localization of complex polynomial zeros, we state initial conditions which enable both the guaranteed and fast convergence of this method. These conditions are computationally verifiable since they depend only on initial approximations, polynomial coefficients, and polynomial degree, which is of practical importance.

A new simultaneous method of fourth order for finding complex zeros in circular interval arithmetic

Starting from disjoint disks which contain polynomial complex zeros, the new iterative interval method for simultaneous finding of inclusive disks for complex zeros is formulated. The convergence theorem and the conditions for convergence are considered, and the convergence is shown to be fourth. Numerical examples are included.

On the perturbation of the zeros of complex polynomials

We investigate the deviation of the zeros of a polynomial. The deviation grows with the nth root of the perturbation of the polynomial, where n is the degree of the polynomial, and the task is to determine the factor in front of this root. We improve upon earlier results by Ostrowski and Schonhage and give sharp estimates. Applications are given in estimating the differences of root-radii, which is important for the computation of the zeros of polynomials.

Approximating Complex Polynomial Zeros: Modified Weyl's Quadtree Construction and Improved Newton's Iteration

We propose a new algorithm for the classical and still practically important problem of approximating zeros zj of an nth degree polynomial p(x) within error bound 2−bmaxj|zj|. The algorithm uses O((n2logn)log(bn)) arithmetic operations and comparisons for approximating all the n zeros and O((knlogn)log(bn)) for approximating the k zeros lying in a fixed domain (disc or square) and isolated from the other zeros. Unlike the previous fast algorithms of this kind, the new algorithm has its simple elementary description, is convenient for practical implementation, and allows the users to adapt the computational precision to the current level of approximation achieved in the process of computing and ultimately to the requirements to the output precision for each zero of p(x). The algorithm relies on our novel versions of Weyl's quadtree construction and Newton's iteration.

An algebraic algorithm to isolate complex polynomial zeros using Sturm sequences

In this paper, algorithms to enumerate and isolate complex polynomial roots are developed, analyzed, and implemented. We modified an algorithm due to Wilf, in which Sturm sequences and the principle of argument are used, by employing algebraic methods, aiming to enumerate zeros inside a rectangle in an exact way. Several improvements are introduced, such as dealing with zeros on the boundary of the rectangle. The performance of this new algorithm is evaluated in a theoretical as well as from a practical point of view, by means of experimental tests. The robustness of the algorithm is verified by using tests with ill-conditioned polynomials. We also compare the performance of this algorithm with the results of a recent paper, using different polynomial classes.

Graeffe's, Chebyshev-like, and Cardinal's Processes for Splitting a Polynomial into Factors

Numerical splitting of a real or complex univariate polynomial into factors is the basic step of the divide-and-conquer algorithms for approximating complex polynomial zeros. Such algorithms are optimal (up to polylogarithmic factors) and are quite promising for practical computations. In this paper, we develop some new techniques, which enable us to improve numerical analysis, performance, and computational cost bounds of the known splitting algorithms. In particular, we study a Chebyshev-like modification of Graeffe's lifting iteration (which is a basic block of the splitting algorithms, as well as of several other known algorithms for approximating polynomial zeros), analyze its numerical performance, compare it with Graeffe's, prove some results on numerical stability of both lifting processes (that is, Graeffe's and Chebyshev-like), study their incorporation into polynomial root-finding algorithms, and propose some improvements of Cardinal's recent effective technique for numerical splitting of a polynomial into factors. Our improvement relies, in particular, on a modification of the matrix sign iteration, based on the analysis of some conformal mappings of the complex plane and of techniques of recursive lifting/recursive descending. The latter analysis reveals some otherwise hidden correlations among Graeffe's, Chebyshev-like, and Cardinal's iterative processes, and we exploit these correlations in order to arrive at our improvement of Cardinal's algorithm. Our work may also be of some independent interest for the study of applications of conformal maps of the complex plane to polynomial root-finding and of numerical properties of the fundamental techniques for polynomial root-finding such as Graeffe's and Chebyshev-like iterations.

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.

A hybrid method for a polynomial complex zero

Using classical Weierstrass' result concerning polynomial zeros in a suitable interval form and some iterative method in floating-point arithmetic, a hybrid method for finding an isolated simple complex zero of a polynomial is established. This method possesses a high computational efficiency and the capability of automatic determination of error bounds of the approximate solution. A computationally verifiable test for the existence of a polynomial complex zeros in a given disk is also presented.