Root clustering for convex combination of complex polynomials

In some important applications, such as the edge-theorem, it is required that a polynomial is α-stable along a parameter segment. Using the critical constraint, the necessary and sufficient conditions for root clustering (α-stability) of convex combinations of complex polynomials are presented. The approach is general and requires that a certain real polynomial has no zeros in the open interval (0,1)     

A root distribution criterion for interval polynomials

The problem of finding the conditions under which an interval polynomial has a given number of roots in the open left-half plane and the other roots in the open right-half plane, irrespective of the values of its coefficients, is considered. A simple criterion is provided to test interval polynomials for the root distribution invariance, viewed as an extension of Kharitonov's theorem. The goal is to provide an alternative theorem and then give an efficient means of checking the root distribution invariance     

Strong Kharitonov theorems for low-order polynomials

The root clustering property of low-order interval polynomials is examined, and it is shown that the number of polynomials required to check root clustering within the left sector may be reduced if the order of the interval polynomial is low. Specifically, the reduction is possible if nφ<3π, where n is the order of the polynomial and φ is the damping angle that defines the left sector     

On a root distribution criterion for interval polynomials

H. Kokame and T. Mori (1991) and C.B. Soh (1990) derived conditions under which an interval polynomial has a given number of roots in the open left-half plane and the other roots in the open right-half plane. However, the one-shot-test approach using Sylvester's resultant matrices and Bezoutian matrices implies that the implemented conditions are only sufficient (not necessary) for an interval polynomial to have at least one root in the open left-half plane and open right-half plane. Alternative necessary and sufficient conditions, which only require the root locations of four polynomials to check the root distribution of an interval polynomial, are presented     

The root-locus method of synthesis of stable polynomials by adjustment of all coefficients

The problem of synthesis of an asymptotically stable polynomial on the basis of the initial unstable polynomial is solved. For the purpose of its solution, the notion of the extended (complete) root locus of a polynomial is introduced, which enables one to observe the dynamics of all its coefficients simultaneously, to isolate the root-locus trajectories, along which values of each coefficient change, to establish their interrelation, which provides a way of using these trajectories as “conductors” for the movement of roots in the desired domains. Values of the coefficients that ensure the stability of a polynomial are chosen from the stability intervals found on the stated trajectories as the nearest values to the values of appropriate coefficients of the unstable polynomial or by any other criterion, for example, the criterion of provision of the required stability reserve. The sphere of application of the root locus, which is conventionally used for synthesis of characteristic polynomials through the variation of only one parameter (coefficient) of the polynomial, is extended for the synthesis of polynomials by way of changing all coefficients and with many changing coefficients. Examples of application of the developed algorithm are considered for the synthesis of stable polynomials with constant and interval coefficients.     

A note on sufficient conditions for positivity in x ε [− 1, 1]

In this note, sufficient conditions for a real polynomial to be positive in the interval [?1, 1] are presented. One such condition is obtained by using a nonlinear transformation which maps the interval [?1, 1] onto the periphery of the unit circle. The root distribution of the transformed polynomial determines the positivity condition.     

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.     

Optimal computation of the Bernstein algorithm for the bound of an interval polynomial

If a polynomial is expanded in terms of Bernstein polynomial over an interval then the coefficients of the expansion may be used to provide upper and lower bounds for the value of the polynomial over the interval. When applying this method to interval polynomials straightforwardly, the coefficients of the expansion are computed with an increase in width due to dependency intervals. In this paper we show that if the computations are rearranged suitably then the Bernstein coefficients can be computed with no increase in width due to dependency intervals.     

Linear matrix equations and root clustering

Given A ε ?^{n + n} and ? ? ?, we search for a criterion assuring that the spectrum of A is clustered in ?, σ(A)? ? One approach to root clustering is the linear matrix equation, whose half plane version dates back to Lyapunov. The existing literature deals with an algebraic region defined by a single polynomial. In this paper, we construct a novel linear matrix equation related to the intersection of algebraic regions. This considerably enlarges the family of regions with root clustering criteria.     

具有结构式不确定性线性系统特征根的鲁棒群聚

采用广义Ｌｙａｐｕｎｏｖ方程，讨论具有结构式不确定性线性系统的特征根在复平面上某些区域内的鲁棒群聚性，如果标称矩阵的所有特征根都在复平面上的特定区域内，本文所给出充分条件将保证当存在有结构式不确定性时矩阵的特征根也在同一区域内，所提出的判断比当前文献中的结论具有较少的保守性。     

On the computing time of the continued fractions method

The maximum computing time of the continued fractions method for polynomial real root isolation is at least quintic in the degree of the input polynomial. This computing time is realized for an infinite sequence of polynomials of increasing degrees, each having the same coefficients. The recursion trees for those polynomials do not depend on the use of root bounds in the continued fractions method. The trees are completely described. The height of each tree is more than half the degree. When the degree exceeds one hundred, more than one third of the nodes along the longest path are associated with primitive polynomials whose low-order and high-order coefficients are large negative integers. The length of the forty-five percent highest order coefficients and of the ten percent lowest order coefficients is at least linear in the degree of the input polynomial multiplied by the level of the node. Hence the time required to compute one node from the previous node using classical methods is at least proportional to the cube of the degree of the input polynomial multiplied by the level of the node. The intervals that the continued fractions method returns are characterized using a matrix factorization algorithm.     

Analysis of CZCS data for the U.K. South-western Approaches

The implementation of a non-iterative atmospheric correction algorithm is described in detail and the performance of the algorithm is illustrated for several CZCS images. Chlorophyll retrieval is attempted using linear, power and polynomial regression for ratios of corrected images and the best correlation coefficients are in the region of 0-9. The same images are analysed in three spectral bands using the ISOCLS clustering algorithm and ocean areas are stratified into subclass patterns which correlate well with ratios and sea-truth. The monocluster blocks approach is used to extract training statistics for maximum likelihood classification of ocean areas and the results compare favourably with corresponding ratio images.     

Stability analysis of polynomials with coefficients in disks

The aim of this note is to report results on the stability of a class of polynomials from the small gain theorem point of view. The authors consider families of polynomials whose coefficients lie in closed circular disks around their nominal values. Various measures of variation of polynomial coefficients around their nominal value are considered and in each case necessary and sufficient conditions are presented for stability of the resulting family of polynomials. The stability region could be any closed region of the complex plane. Based on similar ideas of small gain, the authors also provide sufficient conditions for testing the stability of systems with commensurate time delays, and for two-dimensional type systems. These conditions become both necessary and sufficient in some special cases. All tests are easy to implement and require checking the stability of a matrix (or equivalently checking the stability of the central polynomial) and evaluation of a norm     

On the thirty-two virtual polynomials to stabilize an intervalplant

This paper studies the conservatism of the 32 virtual polynomials to stabilize an interval plant. It is shown that working with the 32 virtual vertices is generally less conservative than with the Kharitonov polynomials of the smallest interval polynomial containing the characteristic polynomial polytope. By means of the former, it is possible to find all the controllers such that the value set of the polytope of characteristic polynomials is applied in two quadrants as a maximum for each ω; while using the latter, only some of them can be found. The cases in which both methods coincide are also analyzed, and the conditions on the numerator and denominator of the controller are developed. Thus, this coincidence can be known a priori from the characteristics of the coefficients of the numerator and denominator of the controller. It is shown that these conditions are satisfied by the first-order controllers     

散乱数据(2m-1,2n-1)次多项式自然样条插值

考虑对窄间散乱数据(2m-1,2n-1)次多项式自然样条插值,使得插值函数对x的m次偏导数和对y的n次偏导数平方积分极小(带自然边界条件).用希尔伯特空间样条方法,得出其解的结构,解的系数能够用线性方程组确定,方程组系数矩阵对称,可用改进的平方根法解.例子表明方法简单,效果良好.     

Necessary conditions for Hadamard factorizations of Hurwitz polynomials

A property of Hurwitz polynomials is related with the Hadamard product. Garloff and Wagner proved that Hadamard products of Hurwitz polynomials are Hurwitz polynomials, and Garloff and Shrinivasan shown that there are Hurwitz polynomials of degree 4 which do not have a Hadamard factorization into two Hurwitz polynomials of the same degree 4. In this paper, we give necessary conditions for an even-degree Hurwitz polynomial to have a Hadamard factorization into two even-degree Hurwitz polynomials; such conditions are given in terms of the coefficients of the given polynomial alone. Furthermore, we show that if an odd-degree Hurwitz polynomial has a Hadamard factorization then a system of nonlinear inequalities has at least one solution.     

The root and Bell’s disk iteration methods are of the same error propagation characteristics in the simultaneous determination of the zeros of a polynomial,Part II: Round-off error analysis by use of interval arithmetic

In Part I (Ikhile, 2008) [4], it was established that the root and Bell’s disk/point iteration methods with or without correction term are of the same asymptotic error propagation characteristics in the simultaneous determination of the zeros of a polynomial. This concluding part of the investigation is a study in round-offs, its propagation and its effects on convergence employing interval arithmetic means. The purpose is to consequently draw attention on the effects of round-off errors introduced from the point arithmetic part, on the rate of convergence of the generalized root and Bell’s simultaneous interval iteration algorithms and its enhanced modifications introduced in Part I for the numerical inclusion of all the zeros of a polynomial simultaneously. The motivation for studying the effects of round-off error propagation comes from the fact that the readily available computing devices at the moment are limited in precision, more so that accuracy expected from some programming or computing environments or from these numerical methods are or can be machine dependent. In fact, a part of the finding is that round-off propagation effects beyond a certain controllable order induces overwhelmingly delayed or even a severely retarded convergence speed which manifest glaringly as poor accuracy of these interval iteration methods in the computation of the zeros of a polynomial simultaneously. However, in this present consideration and even in the presence of overwhelming influence of round-offs, we give conditions under which convergence is still possible and derive the error/round-off relations along with the order/$R$-order of convergence of these methods with the results extended to similar interval iteration methods for computing the zeros of a polynomial simultaneously, especially to Bell’s interval methods for refinement of zeros that form a cluster. Our findings are instructive and quite revealing and supported by evidence from numerical experiments. The analysis is preferred in circular interval arithmetic.     

Parameterization of a Set Determined by the Generalized Discriminant of a Polynomial

A generalization of the classical discriminant of the real polynomial defined using the linear Hahn operator that decreases the degree of the polynomial by one is studied. The structure of the generalized discriminant set of the real polynomial, i.e., the set of values of the polynomial coefficients at which the polynomial and its Hahn operator image have a common root, is investigated. The structure of the generalized discriminant of the polynomial of degree n is described in terms of the partitions of n Algorithms for the construction of a polynomial parameterization of the generalized discriminant set in the space of the polynomial coefficients are proposed. The main steps of these algorithms are implemented in a Maple library. Examples of calculating the discriminant set are discussed.     

Robust stability region of fractional order PIλ controller for fractional order interval plant

In this article, by using the fractional order PI^λ controller, we propose a simple and effective method to compute the robust stability region for the fractional order linear time-invariant plant with interval type uncertainties in both fractional orders and relevant coefficients. The presented method is based on decomposing the fractional order interval plant into several vertex plants using the lower and upper bounds of the fractional orders and relevant coefficients and then constructing the characteristic quasi-polynomial of each vertex plant, in which the value set of vertex characteristic quasi-polynomial in the complex plane is a polygon. The D-decomposition method is used to characterise the stability boundaries of each vertex characteristic quasi-polynomial in the space of controller parameters, which can obtain the stability region by varying λ orders in the range (0,?2). These regions of each vertex plant are computed by using three stability boundaries: real root boundary (RRB), complex root boundary (CRB) and infinite root boundary (IRB). The method gives the explicit formulae corresponding to these boundaries in terms of fractional order PI^λ controller parameters. Thus, the robust stability region for fractional order interval plant can be obtained by intersecting stability region of each vertex plant. The robustness of stability region is tested by the value set approach and zero exclusion principle. Our presented technique does not require sweeping over the parameters and also does not need linear programming to solve a set of inequalities. It also offers several advantages over existing results obtained in this direction. The method in this article is useful for analysing and designing the fractional order PI^λ controller for the fractional order interval plant. An example is given to illustrate this method.     

Graph Characterization by Counting Sink Star Subgraphs

In this paper, we study the polynomial coefficients of the reduced Bartholdi zeta function for characterizing simple unweighted graphs and demonstrate how to use these coefficients for clustering graphs. The polynomial coefficients of the reduced Bartholdi zeta function are invariant to vertex order permutations and also carry information about counting the sink star subgraphs in a symmetric digraph of G. We also investigate the advantages of the reduced Bartholdi coefficients over other spectral methods such as the Ihara zeta function and Laplacian spectra. Experimental results indicate that the proposed method is more effective than the other spectral approaches, and compared to the Ihara zeta function, it has less sensitivity to structural noises such as omitting an edge.