Roots Of Polynomial Research Articles

The Weierstrass–Durand–Kerner root finder is not generally convergent

Finding roots of univariate polynomials is one of the fundamental tasks of numerics, and there is still a wide gap between root finders that are well understood in theory and those that perform well in practice. We investigate the root-finding method of Weierstrass, also known as the Durand–Kerner-method: this is a root finder that tries to approximate all roots of a given polynomial in parallel. This method has been introduced 130 years ago and has since then a good reputation for finding all roots in practice except in obvious cases of symmetry. Nonetheless, very little is known about its global dynamics and convergence properties. We show that the Weierstrass method, like the well-known Newton method, is not generally convergent: there are open sets of polynomials p p of every degree d ≥ 3 d \ge 3 such that the dynamics of the Weierstrass method applied to p p exhibits attracting periodic orbits. Specifically, all polynomials sufficiently close to Z 3 + Z + 175 Z^3 + Z + 175 have attracting cycles of period 4 4 . Here, period 4 4 is minimal: we show that for cubic polynomials, there are no periodic orbits of length 2 2 or 3 3 that attract open sets of starting points. We also establish another convergence problem for the Weierstrass method: for almost every polynomial of degree d ≥ 3 d\ge 3 there are orbits that are defined for all iterates but converge to ∞ \infty ; this is a problem that does not occur for Newton’s method. Our results are obtained by first interpreting the original problem coming from numerical mathematics in terms of higher-dimensional complex dynamics, then phrasing the question in algebraic terms in such a way that we could finally answer it by applying methods from computer algebra. The main innovation here is the translation into an algebraic question, which is amenable to (exact) computational methods close to the limits of current computer algebra systems.

DOA Estimation for Massive MIMO Systems with Unknown Mutual Coupling Based on Block Sparse Bayesian Learning.

Obtaining accurate angle parameters using direction-of-arrival (DOA) estimation algorithms is crucial for acquiring channel state information (CSI) in massive multiple-input multiple-output (MIMO) systems. However, the performance of the existing algorithms deteriorates severely due to mutual coupling between antenna elements in practical engineering. Therefore, for solving the array mutual coupling, the array output signal vector is modeled by mutual coupling coefficients and the DOA estimation problem is transformed into block sparse signal reconstruction and parameter optimization in this paper. Then, a novel sparse Bayesian learning (SBL)-based algorithm is proposed, in which the expectation-maximum (EM) algorithm is used to estimate the unknown parameters iteratively, and the convergence speed of the algorithm is enhanced by utilizing the approximate approximation. Moreover, considering the off-grid error caused by discretization processes, the grid refinement is carried out using the polynomial roots to realize the dynamic update of the grid points, so as to improve the DOA estimation accuracy. Simulation results show that compared with the existing algorithms, the proposed algorithm is more robust to mutual coupling and off-grid error and can obtain better estimation performance.

On computing root polynomials and minimal bases of matrix pencils

We revisit the notion of root polynomials, thoroughly studied in (Dopico and Noferini, 2020 [9]) for general polynomial matrices, and show how they can efficiently be computed in the case of a matrix pencil λE+A. The method we propose makes extensive use of the staircase algorithm, which is known to compute the left and right minimal indices of the Kronecker structure of the pencil. In addition, we show here that the staircase algorithm, applied to the expansion (λ−λ0)E+(A−λ0E), constructs a block triangular pencil from which a minimal basis and a maximal set of root polynomials at the eigenvalue λ0, can be computed in an efficient manner.

Common index divisor of the number fields defined by

AbstractLet $K={\mathbf {Q}}(\theta )$ be an algebraic number field with $\theta$ a root of an irreducible polynomial $x^5+ax+b\in {\mathbf {Z}}[x]$. In this paper, for every rational prime $p$, we provide necessary and sufficient conditions on $a,\,~b$ so that $p$ is a common index divisor of $K$. In particular, we give sufficient conditions on $a,\,~b$ for which $K$ is non-monogenic. We illustrate our results through examples.

Continuity of roots for polynomials over valued fields

We study connections between polynomials which are close to each other, i.e., whose respective coefficients are close in the topology induced by a valuation. This paper consists of both an overview on known root continuity theorems and new results on the subject. We present theorems which have been published over the years, correcting and improving some of their formulations and proofs. We also give a sketch of several approaches to root continuity, such as employing the Newton Polygon and induction on the degree of the polynomial. We study the behavior of the irreducible factors of a given polynomial, the extensions generated by its roots and invariants connected to that polynomial under transition to a second polynomial which is sufficiently close. Further, we present applications of root continuity to the study of valued field extensions and ramification theory, including the theory of the defect.

AN APPROACH TO THE STABILITY OF THE GAUSE-LOTKA-VOLTERRA COMPETITION ECOLOGICAL MODEL USING THE ROUTH-HURWITZ CRITERION

In the qualitative study of systems of ordinary differential equations, we can analyze the dynamics of the solutions by observing whether small variations or modifications in the initial conditions produce small changes in the future, this intuitive idea was formalized and studied by Lyapunov and reflects the concept of stability. For both linear and nonlinear systems of differential equations, we can analyze stability using criteria to obtain Hurwitz type polynomials. In this paper, we study the stability of a prey-predator model where three species compete considering the Gause effect generated by the Gause-Lotka-Volterra system, using an analytical stability technique to obtain Hurwitz-type polynomials called Routh-Hurwitz criterion. This criterion provides the necessary and/or sufficient conditions to analyze the dynamics of the system by studying the location of the roots of the characteristic polynomial associated with the system. This article describes and presents an analytical method to analyze the stability of the Gause-Lotka-Volterra model, based on the study of Hurwitz type polynomials. This technique allows establishing the asymptotic stability of the system by analyzing the location of the eigenvalues of the matrix associated with said system. The research results allow establishing a region of asymptotic stability of the Gause-Lotka-Volterra model that allows analyzing the interaction dynamics between the species.

Univariate rational sums of squares

Given rational univariate polynomials f and g such that gcd(f, g) and f / gcd(f, g) are relatively prime, we show that g is non-negative on all the real roots of f if and only if g is a sum of squares of rational polynomials modulo f. We complete our study by exhibiting an algorithm that produces a certificate that a polynomial g is non-negative on the real roots of a non-zero polynomial f , when the above assumption is satisfied.

Roots and critical points of polynomials over Cayley–Dickson algebras

The roots of polynomials over Cayley–Dickson algebras over an arbitrary field and of arbitrary dimension are studied. It is shown that the spherical roots of a polynomial f(x) are also roots of its companion polynomial . We generalize the classical theorems for complex and real polynomials by Gauss–Lucas and Jensen to locally-complex Cayley–Dickson algebras: it is proved that the spherical roots of belong to the convex hull of the roots of , and we also show that all roots of are contained in the snail of f(x), as defined by Ghiloni and Perotti.

Root vectors of polynomial and rational matrices: Theory and computation

The notion of root polynomials of a polynomial matrix P(λ) was thoroughly studied in Dopico and Noferini (2020) [6]. In this paper, we extend such a systematic approach to general rational matrices R(λ), possibly singular and possibly with coalescent pole/zero pairs. We discuss the related theory for rational matrices with coefficients in an arbitrary field. As a byproduct, we obtain sensible definitions of eigenvalues and eigenvectors of a rational matrix R(λ), without any need to assume that R(λ) has full column rank or that the eigenvalue is not also a pole. Then, we specialize to the complex field and provide a practical algorithm to compute them, based on the construction of a minimal state space realization of the rational matrix R(λ) and then using the staircase algorithm on the linearized pencil to compute the null space as well as the root polynomials in a given point λ0. If λ0 is also a pole, then it is necessary to apply a preprocessing step that removes the pole while making it possible to recover the root vectors of the original matrix: in this case, we study both the relevant theory (over a general field) and an algorithmic implementation (over the complex field), still based on minimal state space realizations.

An Equivalent Condition for Stability Analysis of LTI Systems with Bounded Time-invariant Delay

This paper deals with the stability analysis of Linear Time-Invariant (LTI) systems in the presence of a bounded and time-invariant delay parameter. The paper extends the theory of characteristic quasi-polynomial to derive equivalent stability conditions. Based on the quasi-polynomial, a two-variate polynomial is defined and it is proved that system stability is equivalently guaranteed if all roots of the new polynomial are not allocated in a certain region of the right half plane (RHP). Moreover, the algorithm uses the proposed equivalent stability conditions to develop a set of implementable stability conditions based on the exposed edges theorem. The algorithm is applied to three simulation examples that admit some novel results for these systems. The first example shows the step-by-step procedure for the proposed algorithm. The second and the third ones compare various aspects of the proposed algorithm with some existing methods. Two hundred and fifty delay systems are randomly generated and the proposed algorithm is compared with them in terms of conservativeness and computational burden. The results reveal the superiority of the proposed method over the existing ones.

Об интервальном оценивании с высоким порядком минимума функции

The article addresses the problem of two-sided estimation of the minimum of a function of one real variable using interval methods. These methods are widely used for finding the ranges of functions, but their order of accuracy often does not exceed 3 for the functions of one variable and 2 for the case of many variables. The paper proposes a method for finding an interval estimate for the minimum of a function of one variable with any odd order greater than 2. The proposed technique uses so-called functional intervals, the boundaries of which are described by polynomials of even degrees obtained from the Taylor formula for the function under consideration. The article provides a derivation of interval estimates for the minimum, and also proves an estimate for the order of convergence of the width of the resulting interval with the decrease in the size of the domain of the function. An example is given in which estimating the minimum of a function reaches the theoretical order of convergence. The approach developed in this paper allows, in principle, to achieve any odd accuracy order in the interval estimation of the minimum of a function. In practice, this method is convenient to use for the third order of accuracy, since, on the one hand, we need to find an interval estimate for derivatives of high degrees, and, on the other hand, we need to find roots of polynomials of even degrees. The new approach is shown to be useful and practical, as it uses the local Taylor expansion and therefore behaves stably as the width of the function domain decreases. The new approach can be used to improve the performance of interval global optimization algorithms based on adaptive domain partitioning. An increase in the order of estimation accuracy entails the acceleration of these algorithms, as well as a decrease in the number of subintervals generated in the process of partitioning the domain.

Estimation of time series models using residuals dependence measures

We propose new estimation methods for time series models, possibly noncausal and/or noninvertible, using serial dependence information from the characteristic function of model residuals. This allows to impose the i.i.d. or martingale difference assumptions on the model errors to identify the unknown location of the roots of the lag polynomials for ARMA models without resorting to higher order moments or distributional assumptions. We consider generalized spectral density and cumulative distribution functions to measure residuals dependence at an increasing number of lags under both assumptions and discuss robust inference to higher order dependence when only mean independence is imposed on model errors. We study the consistency and asymptotic distribution of parameter estimates and discuss efficiency when different restrictions on error dependence are used simultaneously, including serial uncorrelation. Optimal weighting of continuous moment conditions yields maximum likelihood efficiency under independence for unknown error distribution. We investigate numerical implementation and finite sample properties of the new classes of estimates.

Arbitrage-free smile construction on FX option markets using Garman-Kohlhagen deltas and implied volatilities

This paper addresses arbitrage-free FX smile construction from near-term implied volatility dynamics proposed by Carr (J Financ Econ, 120(1), 1–20, 2016). The approach is directly applicable to FX option market conventions. Prices of market benchmark contracts (risk reversals and butterflies) are identified as the roots of a cubic polynomial and ATM-volatility can be matched by construction. Implied volatilities are computed with respect to (non-premium adjusted) option deltas. The approach is compared to the Vanna Volga Approach, which does not guarantee arbitrage-free prices. An empirical application to a normal and a stress scenario demonstrates that arbitrage-free implied volatilities coincide with those from the Vanna Volga Approach when prices are interpolated between the Delta25-call and Delta25-put options. Differences are observed when implied volatilities are extrapolated to the wings. Empirically, these differences are particularly relevant in a stress scenario during the Coronavirus crises (2020).

Enhancement of stability delay margins by virtual inertia control for microgrids with time delay

Large-scale deployment of renewable energy sources (RESs) contributes to fluctuations in the system frequency due to their inherent reduced inertia feature. Time delays have emerged as a major source of concern in microgrids (MGs) as a result of the broad adoption of open communication networks since significant delays inevitably reduce the controller?s performance and even cause instabilities. In this article, a frequency-domain direct method is used to evaluate the impact of the virtual inertia (VI) control on the stability delay margins of MG with communication delays. By avoiding approximation, the approach first removes transcendental terms from characteristic equations and turns the transcendental characteristic equations into regular polynomials. With this method, roots of the original characteristic equation on the imaginary axis correspond to exactly the positive real roots of the new regular polynomial not including any exponential term. This new polynomial can be used to find out whether the system stability is delay-dependent or not and enables us to compute stability delay margin for the delay-dependent stability case. The proposed analytical method is utilized for evaluating stability delay margins with regard to system parameters for various values of proportional-integral (PI) gains where the MG is marginally stable. Moreover, quantitative effect of virtual inertia and damping gains is comprehensively investigated. Based on the results, it is concluded that incorporating VI control enhances stability delay margins and enhances the MG's stability performance. Theoretical delay margin results are verified using time-domain simulations and quasipolynomial mapping-based root finder (QPmR) algorithm

СПОСТЕРІГАЧ СТАНУ ЕЛЕКТРОПРИВОДІВ НАВЕДЕННЯ БОЙОВИХ МАШИН

The paper substantiates the use of adaptive state observers of combat vehicles guiding electric drives instead of imperfect rotation speed sensors of executive motors, as which tachogenerators are traditionally used. The synthesis is carried out, the structure and mathematical description of the Luenberger observer of the direct current executive motor with independent excitation by the modal method are obtained. The performance of the synthesized observer is confirmed by checking the fulfillment of the observability condition. An expression for determining the geometric mean root, which makes it possible to simplify the structure of the observer and to evaluate not only the rotation speed, but also the load torque on the motor shaft is proposed. For definite structure of the observer, expressions for the coefficients of the Luenberger matrix are obtained. The modernized Luenberger observer was studied by simulation in the Matlab/Simulink environment. The parameters of the DC executive motor D-135 of the vertical guidance system of the 9K35 anti-aircraft missile system were used as the initial data of the model. The simulation results with the accepted binomial distribution of the roots of the characteristic polynomial proved the efficiency of the proposed structure of the observer. The estimation of the electric drive coordinates occurred aperiodically, without fluctuations, however, it had a relatively large error due to the underestimated speed of the observer. To eliminate this shortcoming, the distribution of the roots of the characteristic polynomial of the observer according to the Chebyshev linear form was proposed. With this distribution of roots, the performance of the Luenberger observer increased significantly, which limited the error in tracking the coordinates of the electric drive to 4%. The prospects for further research of the proposed method for the synthesis of state observers of the executive motors of the combat vehicles guidance systems are outlined. Keywords: Luenberger observer, executive motor, guidance system, combat vehicle.

Exchange Rate, Interest Rate and Agricultural Export Earnings: An analysis Using Panel Data Vector Autoregressive Model

The study modeled the dynamic interaction between exchange rate, interest rate and agricultural export earnings using panel VAR Model. The specific objectives of the study include to; interdependencies in the dynamic interaction between exchange rate, interest rate and agricultural export earnings, parameters of panel VAR model using PVAR Stata code developed by Abrigo and love, determine the shocks associated with their dynamic interactions between these variables, investigate direction of causality between interest rate, exchange rate and agricultural export earnings from six African countries and make appropriate recommendations. The data used for the study was secondary data extracted from index mundi website and world data indicators for the period of 40 years (1980-2020). The data was on exchange rate, interest rate and agricultural export earnings. Geographically, the six African countries include; Algeria, Angola, Egypt, Libya, Gabon and Nigeria. The study uses vector Autoregressive model estimation results with PVAR Stata code developed by Abrigo and love. The post estimation test on the Vector Autoregressive (VAR) model shows a contemporary Co-efficient of Correlation analysis. It was found that lending interest rate and exchange rate are negatively associated with Co-efficient of Correlation of (-0.0873). Also, it was found that there exist a positive association between exchange rate and agricultural export earnings. Also, there is a positive association between lending interest rate and agricultural export earnings. The inverse roots of a characteristic polynomial of the estimated Panel VAR model satisfied the stability condition (of the diagnostic test) since no root lied outside the unit root circle. Therefore, the estimated VAR is stable. However, it was confirmed that there is no directional relationship that exist between the variables. Also, the results show that exchange rate and lending rate have positive on agricultural export earnings, whereas exchange rate is likely to reduce the level of lending interest rate slightly. Therefore, it is recommended that in estimating the dynamic interaction between variables in a panel data system, there is need for the inclusion of the lags of the response variable among the determinants to measures the dynamic interaction as well capture heterogeneities in the series and also, policies should be formulated to stabilized exchange and lending rates in order to improve and strengthen the countries’ agricultural economy amongst others

Modeling the Effect of Monetary Policy on Commercial Bank’s Lending Operation in Nigeria using Vector Error Correction Model (1991- 2020)

The study examines the effect of monetary policy on commercial bank’s lending operation in Nigeria between February, 1991 to October, 2020 using the macroeconomic time series variables of exchange rate, interest rate, maximum lending rate and prime lending rate. The vector error correction model was employed to analyze these interactions as well as the effect and pattern of causality among the variables under investigation. Monthly data spanning from February, 1991 to October, 2020 which covered a period of 29 years and 7 months (357 observations) were sourced from the Statistical Bulletin of the Central Bank of Nigeria (CBN). Preliminary statistical approach such as descriptive statistics and time plot were carried out to test if the data set obey the normality assumption and to verify if there is trend on the series. Pre-estimation diagnosis such as unit root test, lag order selection criteria and co-integration test were carried out and the results shows that at level, all variables had unit root, then, at first difference all variables were stationary. The lag order selection criteria chose lag 3 (14.7844*) of Akaike information criteria, but the vector error correction model (VECM) was done at lag 2 indicating losing a lag. The co-integration test shows the presence of long run relationship among the variables. The vector error correction model (VECM) estimated from the results obtained shows that all the variables had positive effect on commercial bank’s lending operation. The post estimation test on vector error correction model such as normality of the residuals, serial correlation and heteroscedasticity shows that the VEC model was multivariate normal, no serial correlation and homoscedastic. The inverse root of AR characteristic polynomial shows that the estimated VECM satisfy the stability condition of the diagnostic test. The variance decomposition test shows that the variables have a very weak influence in predicting one another. Recommendations were made based on the result of findings in the study

Identification of the differencing operator of a non-stationary time series via testing for zeroes in the spectral density

A nonparametric procedure for identifying the differencing operator of a non-stationary time series is presented and tested. Any proposed differencing operator is first applied to the time series, and the spectral density is tested for zeroes corresponding to the polynomial roots of the operator. A nonparametric tapered spectral density estimator is used, and the subsampling methodology is applied to obtain critical values. Simulations explore the effectiveness of the procedure under a variety of scenarios involving non-stationary processes.

The Flow of Polynomial Roots Under Differentiation

The question about behavior of gaps between zeros of polynomials under differentiation is classical and goes back to Marcel Riesz. Recently, Stefan Steinerberger [42] formally derived a nonlocal nonlinear partial differential equation which models dynamics of roots of polynomials under differentiation. In this paper, we connect rigorously solutions of Steinerberger’s PDE and evolution of roots under differentiation for a class of trigonometric polynomials. Namely, we prove that the distribution of the zeros of the derivatives of a polynomial and the corresponding solutions of the PDE remain close for all times. The global in time control follows from the analysis of the propagation of errors equation, which turns out to be a nonlinear fractional heat equation with the main term similar to the modulated discretized fractional Laplacian $$(-\Delta )^{1/2}$$ .

High-Performance Polynomial Root Finding for Graphics

We present a computationally-efficient and numerically-robust algorithm for finding real roots of polynomials. It begins with determining the intervals where the given polynomial is monotonic. Then, it performs a robust variant of Newton iterations to find the real root within each interval, providing fast and guaranteed convergence and satisfying the given error bound, as permitted by the numerical precision used. For cubic polynomials, the algorithm is more accurate and faster than both the analytical solution and directly applying Newton iterations. It trivially extends to polynomials with arbitrary degrees, but it is limited to finding the real roots only and has quadratic worst-case complexity in terms of the polynomial's degree. We show that our method outperforms alternative polynomial solutions we tested up to degree 20. We also present an example rendering application with a known efficient numerical solution and show that our method provides faster, more accurate, and more robust solutions by solving polynomials of degree 10.