A new formula for the approximation of root of polynomials with complex coe#cients is presented. For each simple root there exists a neighborhood such that given any input within this neighborhood, the formula generates a convergent sequence, computed via elementary operations on the input and the corresponding derivative values. Each element of the sequence is defined in terms of the quotient of two determinants, computable via a recursive formula. Convergence is proved by deriving an explicit error estimate. For special polynomials explicit neighborhoods and error estimates are derived that depend only on the initial error. In particular, the latter applies to the approximation of root of numbers. The proof of convergence utilizes a family of iteration functions, called the Basic Family; a nontrivial determinantal generalization of Taylor's theorem; a lower bound on determinants; Gerschgorin's theorem and Hadamard's inequality; as well as several new key results. The convergence resu...

We present an algorithm which is able to compute all roots of a given univariate polynomial within a given interval. In each step, we use degree reduction to generate a strip bounded by two quadratic polynomials which encloses the graph of the polynomial within the interval of interest. The new interval(s) containing the root(s) is (are) obtained by intersecting this strip with the abscissa axis. In the case of single roots, the sequence of the lengths of the intervals converging towards the root has the convergence rate 3. For double roots, the convergence rate is still superlinear ( 3 2). We show that the new technique compares favorably with the classical technique of Bézier clipping. Key words: root finding, polynomial, Bézier clipping

For each natural number m greater than one, and each natural number k less than or equal to m, there exists a rootfinding iteration function, B (k) m defined as the ratio of two determinants that depend on the first m - k derivatives of the given function. This infinite family is derived in [4] and its order of convergence is analyzed in [5]. In this paper we give a computational study of the first nine rootfinding methods. These include Newton, secant, and Halley methods. Our computational results with polynomials of degree up to 30 reveal that for small degree polynomials B (k-1) m is more e#cient than B (k) m , but as the degree increases, B (k) m becomes more e#cient than B (k-1) m . The most e#cient of the nine methods is B (4) 4 , having theoretical order of convergence equal to 1.927. Newton's method which is often viewed as the method of choice is in fact the least e#cient method. Keywords: Polynomial Zeros, Order of Convergence. AMS Subject Classification. 65H05, ...

INTRODUCTION Deconvolution in its most basic form can be described as the task of calculating the input to a discrete-time system from its output. It is usually assumed that the system is linear and that its input-output mapping is known with good accuracy. In acoustics and audio, single-channel deconvolution is particularly useful since it can compensate for the response of imperfect transducers such as headphones, loudspeakers, and amplifiers [1]. Multi-channel deconvolution is necessary in the design of cross-talk cancellation systems and virtual source imaging systems [2], [3]. Regularisation is a method that is commonly used when one is faced with an ill-conditioned problem [4, Section 18.4]. The basic idea is to prevent the solution from having some undesirable feature by adding a "smoothness" term to the cost function that we wish to minimise. A suitable choice of the smoothness term can improve the conditioning of the problem substantially but it will inevitably be at

. Let p(z) be a complex polynomial of degree n. To each complex number a we associate a sequence called the Basic Sequence {Bm(a) = a - p(a)Dm-2 (a)/Dm-1 (a)}, where Dm (a) is defined via a homogeneous linear recurrence relation and depends only on the normalized derivatives p (i) (a)/i!. Each Dm (a) is also representable as a Toeplitz determinant. Except possibly for the locus of points equidistant to two distinct roots, given any input a, the Basic Sequence converges to a root of p. The roots of p partition the Euclidean plane into Voronoi regions. Under some regularity assumption (e.g. simplicity of the roots), for almost all inputs within the Voronoi polygon of a root, the corresponding Basic Sequence converges to that root. The discovery of the Basic Sequence, its error estimates, and several of its properties are consequences of our previous analysis of a fundamental family of iteration functions {Bm(z)}, called the Basic Family. Given any fixed m # 2 and an appropriat...

Polynomiography is defined to be “the art and science of visualization in approximation of the zeros of complex polynomials, via fractal and non-fractal images created using the mathematical convergence properties of iteration functions. ” An individual image is called a “polynomiograph.” The word polynomiography is a combination of the word “polynomial ” and the suffix “-graphy. ” It is meant to convey the idea that it represents a certain graph of polynomials, but not in the usual sense of graphing, say a parabola for a quadratic polynomial. Polynomiographs are obtained using algorithms requiring the manipulation of thousands of pixels on a computer monitor. Depending upon the degree of the underlying polynomial, it is possible to obtain beautiful images on a laptop computer in less time than a TV commercial. Polynomials form a fundamental class of mathematical objects with diverse applications; they arise in devising algorithms for such mundane task as multiplying two numbers, much faster than the ordinary way we have all learned to do this task (FFT). According to the Fundamental Theorem of Algebra, a polynomial of degree n, with real or complex coefficients, has n zeros (roots) which may or may not be distinct. The task of approximation of the zeros of polynomials is a problem that was known to Sumerians (third millennium B.C.). This problem

this article, the coeOEcients are most often rational, but could also be complex. The fundamental computational problem of algebra is to compute all values of x for which the polynomial evaluates to zero. These values are called zeros, roots, or solutions

We introduce an exact subdivision algorithm CEVAL for isolating complex roots of a square-free polynomial. The subdivision predicates are based on evaluating the original polynomial or its derivatives, and hence is easy to implement. It can be seen as a generalization of a previous real root isolation algorithm called EVAL. Under suitable conditions, the algorithm is applicable for general analytic functions. We provide a complexity analysis of our algorithm on the benchmark problem of isolating all complex roots of a square-free polynomial with Gaussian integer coefficients. The analysis is based on a novel technique called δ-clusters. This analysis shows, somewhat surprisingly, that the simple EVAL algorithm matches (up to logarithmic factors) the bit complexity bounds of current practical exact algorithms such as those based on Descartes, Continued Fraction or Sturm methods. Furthermore, the more general CEVAL also achieves the same complexity.

. Algebraic eigenvalues with associated left and right eigenvectors (nearly) orthogonal, and polynomial roots that are multiple, have been known to be sensitive to perturbations in numerical computation and thereby ill-conditioned. By constructing an extended polynomial system, it is proved that, under certain conditions, traditionally defined ill-conditioned eigenvalues and polynomial roots can be remarkably stable and insensitive to perturbations, if they are computed as numerical multiple eigenvalues/roots. Furthermore, using the extended polynomial system which is easy to solve, those numerical multiple eigenvalues can be computed with high forward accuracy, while backward errors are estimated as by-products through auxiliary variables of the extended system. Key words. eigenvalue, ill-condition, polynomial, multiple root AMS subject classifications. 65F15, 65F35, 65H05, 65H17 1. Introduction. It is known in classical numerical linear algebra theories that an algebraic eigenvalue...

This article, along with [Elkadi and Mourrain 1996], explain the correlation between residue theory and the Dixon matrix, which yields an alternative method for studying and approximating all common solutions. In 1916, Macaulay [1916] constructed a matrix whose determinant is a multiple of the classical resultant for n homogeneous polynomials in n variables. The Macaulay matrix si16 multaneously generalizes the Sylvester matrix and the coefficient matrix of a system of linear equations [Kapur and Lakshman Y. N. 1992]. As the Dixon formulation, the Macaulay determinant is a multiple of the resultant. Macaulay, however, proved that a certain minor of his matrix divides the matrix determinant so as to yield the exact resultant in the case of generic homogeneous polynomials. Canny [1990] has invented a general method that perturbs any polynomial system and extracts a non-trivial projection operator. Using recent results pertaining to sparse polynomial systems [Gelfand et al. 1994, Sturmfels 1991], a matrix formula for computing the sparse resultant of n + 1 polynomials in n variables was given by Canny and Emiris [1993] and consequently improved in [Canny and Pedersen 1993, Emiris and Canny 1995]. The determinant of the sparse resultant matrix, like the Macaulay and Dixon matrices, only yields a projection operation, not the exact resultant. Here, sparsity means that only certain monomials in each of the n + 1 polynomials have non-zero coefficients. Sparsity is measured in geometric terms, namely, by the Newton polytope