Roots Of Polynomial Research Articles

Iterative schemes for finding all roots simultaneously of nonlinear equations

In this paper, we propose a procedure that can be added to any iterative scheme in order to turn it into an iterative method for approximating all roots simultaneously of any nonlinear equations. By applying this procedure to any iterative method of order p, we obtain a new scheme of order of convergence 2p. Some numerical tests allow us to confirm the theoretical results and to compare the proposed schemes with other known methods for simultaneous roots of polynomial and non-polynomial functions.

A Computationally Efficient and Virtualization-Free Two-Dimensional DOA Estimation Method for Nested Planar Array: RD-Root-MUSIC Algorithm.

To address the problem of expensive computation in traditional two-dimensional (2D) direction of arrival (DOA) estimation, in this paper, we propose a 2D DOA estimation method based on a reduced dimension and root-finding MUSIC algorithm for nested planar arrays (NPAs). Specifically, the algorithm proposed in this paper transforms the problem based on 2D spectral peak search into two one-dimensional estimation problems by reducing the dimension, and then transforms the one-dimensional estimation problem into a problem of polynomial root finding. Finally the parameters are paired to realize the 2D DOA estimation. The proposed algorithm not only performs two root finding operations directly according to the 2D spectral function transformation, avoiding the performance degradation caused by intermediate operations, but can also fully exploit the enlarged array aperture offered by NPAs with reduced computational complexity and no need for virtualization. The superiorities of the proposed algorithm in terms of estimation accuracy and complexity are verified by simulations.

A Uniform Bound for the Distance to a Root of Complex Polynomials Under Newton’s Method

A Uniform Bound for the Distance to a Root of Complex Polynomials Under Newton’s Method

Isoparametric singularity extraction technique for 3D potential problems in BEM

To solve boundary integral equations for potential problems using collocation Boundary Element Method (BEM) on smooth curved 3D geometries, an analytical singularity extraction technique is employed. By adopting the isoparametric approach, curved geometries that are represented by mapped rectangles or triangles from the parametric domain are considered. The singularity extraction on the governing singular integrals can be performed either as an operation of subtraction or division, each having some advantage.A particular series expansion of a singular kernel about a source point is investigated. The series in the parametric coordinates consists of functions of a type Rpxqyr, where R is a square root of a quadratic bivariate homogeneous polynomial with coefficients corresponding to the first fundamental form of a smooth surface, and p,q,r are integers, satisfying p≤−1 and q,r≥0. By extracting more terms from the series expansion of the singular kernel, the smoothness of the regularized kernel at the source point can be increased. Analytical formulae for integrals of such terms are obtained from antiderivatives of Rpxqyr, using recurrence formulae, and by evaluating them at the edges of rectangular or triangular parametric domains.Numerical tests demonstrate that the singularity extraction technique can be a useful prerequisite for a numerical quadrature scheme to obtain high order of convergence for the governing singular integrals in 3D collocation BEM with respect to the number of quadrature nodes or with respect to size of the discretization mesh cells.

On the distribution of real roots of almost-periodical polynomials

On the distribution of real roots of almost-periodical polynomials

On the Location of Roots of the Characteristic Polynomial of (p, q)-Distance Fibonacci Sequences

On the Location of Roots of the Characteristic Polynomial of (p, q)-Distance Fibonacci Sequences

Infinitely many roots of unity are zeros of some Jones polynomials

Let \(N=2n^2-1\) or \(N=n^2+n-1\), for any \(n\ge 2\). Let \(M=\frac{N-1}{2}\). We construct families of prime knots with Jones polynomials \((-1)^M\sum _{k=-M}^{M} (-1)^kt^k\). Such polynomials have Mahler measure equal to 1. If N is prime, these are cyclotomic polynomials \(\Phi _{2N}(t)\), up to some shift in the powers of t. Otherwise, they are products of such polynomials, including \(\Phi _{2N}(t)\). In particular, all roots of unity \(\zeta _{2N}\) occur as roots of Jones polynomials. We also show that some roots of unity cannot be zeros of Jones polynomials.

Co-Optimization of the Spooling and Cross-Current Trajectories of an Energy Harvesting Marine Hydrokinetic Kite

Abstract This paper examines the problem of simultaneously optimizing the spooling and cross-current flight trajectory of a tethered marine hydrokinetic kite using an analytic solution of its inverse dynamics. Tethered kites hold considerable promise for energy production, especially when undergoing cross-current motion. The novelty of this work lies in the use of an analytic solution of the inverse dynamics of the kite to solve the trajectory optimization problem. The term “inverse dynamics” refer to the process of obtaining an exact solution for the actuator inputs from the position, velocity, and acceleration of the kite. While the literature on tethered kites explores trajectory optimization in great detail, most of the work exploits the forward dynamics of the kite, and does not simultaneously optimize the spooling motion and cross-current trajectory. This paper formulates the co-optimization of the kite spooling and cross-current trajectory using a three degrees-of-freedom kite model, paired with an inelastic tether model. The analytic solution of the inverse dynamics is solved in terms of the roots of a fourth-order polynomial in terms of the angle of attack. A simulation study validates the optimization approach and shows that the kite is able to achieve significant energy production.

Passive Scalar Function Approximation Using SOS Polynomials

Signal and power integrity design in the time domain requires equivalent circuit models for interconnects and packages, whose descriptions may only be available as tabulated impedance or admittance parameters. Accurate models for these components should maintain their physical properties including causality, stability, and passivity. Even though a heuristic approach, pole flipping (i.e., changing the sign of the real part of an unstable pole) has proven to be sufficiently accurate for many applications, resulting in models with ensured causality and stability. In this article, we address the problem of generating passive scalar models, such as driving point impedances or admittances, based on an existing causal, stable, but nonpassive model. Our approach is based on using sum-of-squares (SOS) polynomials and results in a convex optimization problem, hence a global optimum is obtained. We obtain approximations through two distinct SOS algorithms: a coefficient and a sampling-based method. We also present a simple passivity test for such functions based on calculating the roots of numerator and denominator polynomials, which provides an intuitive link among existing approaches based on solving an eigenvalue problem.

Legendre polynomials roots and the F-pure threshold of bivariate forms

We provide a direct computation of the $F$-pure threshold of degree four homogeneous polynomial in two variables and, more generally, of certain homogeneous polynomials with four distinct roots. The computation depends on whether the cross ratio of the roots satisfies a specific M\"{o}bius transformation of a Legendre polynomial. We then make a connection between a long lasting open question, involving the relationship between the $F$-pure and the log canonical threshold, and roots of Legendre polynomials over $\mathbb{F}_p$.

Properties of q-Differential Equations of Higher Order and Visualization of Fractal Using q-Bernoulli Polynomials

We introduce several q-differential equations of higher order which are related to q-Bernoulli polynomials and obtain a symmetric property of q-differential equations of higher order in this paper. By giving q-varying variations, we identify the shape of the approximate roots of q-Bernoulli polynomials, a solution of q-differential equations of higher order, and find several conjectures associated with them. Furthermore, based on q-Bernoulli polynomials, we create a Mandelbrot set and a Julia set to find a variety of related figures.

Identification and Estimation of Structural VARMA Models Using Higher Order Dynamics

We use information from higher order moments to achieve identification of non-Gaussian structural vector autoregressive moving average (SVARMA) models, possibly nonfundamental or noncausal, through a frequency domain criterion based on higher order spectral densities. This allows us to identify the location of the roots of the determinantal lag matrix polynomials and to identify the rotation of the model errors leading to the structural shocks up to sign and permutation. We describe sufficient conditions for global and local parameter identification that rely on simple rank assumptions on the linear dynamics and on finite order serial and component independence conditions for the non-Gaussian structural innovations. We generalize previous univariate analysis to develop asymptotically normal and efficient estimates exploiting second and higher order cumulant dynamics given a particular structural shocks ordering without assumptions on causality or invertibility. Finite sample properties of estimates are explored with real and simulated data.

Locating eigenvalues of quadratic matrix polynomials

The location of the roots of a quadratic scalar polynomial may be identified from its coefficients. This paper shows that when the coefficients of the polynomial are square matrices, then appropriate generalizations of some of these statements hold for the eigenvalues of the resulting quadratic matrix polynomial. The locations of the eigenvalues are described with respect to the imaginary axis, the unit circle or the real line. The results lead to upper bounds on some important distances associated with quadratic matrix polynomials. The principal tool used is an eigenvalue localization technique using block Geršgorin sets applied to certain linearizations of these polynomials that come from well known vector spaces. New bounds on the eigenvalues of the matrix polynomial arising from these localizations are also presented.

On the Computation of Gaussian Quadrature Rules for Chebyshev Sets of Linearly Independent Functions

We consider the computation of quadrature rules that are exact for a Chebyshev set of linearly independent functions on an interval $[a,b]$. A general theory of Chebyshev sets guarantees the existence of rules with a Gaussian property, in the sense that $2l$ basis functions can be integrated exactly with just $l$ points and weights. Moreover, all weights are positive and the points lie inside the interval $[a,b]$. However, the points are not the roots of an orthogonal polynomial or any other known special function as in the case of regular Gaussian quadrature. The rules are characterized by a nonlinear system of equations, and earlier numerical methods have mostly focused on finding suitable starting values for a Newton iteration to solve this system. In this paper we describe an alternative scheme that is robust and generally applicable for so-called complete Chebyshev sets. These are ordered Chebyshev sets where the first $k$ elements also form a Chebyshev set for each $k$. The points of the quadrature rule are computed one by one, increasing exactness of the rule in each step. Each step reduces to finding the unique root of a univariate and monotonic function. As such, the scheme of this paper is guaranteed to succeed. The quadrature rules are of interest for integrals with non-smooth integrands that are not well approximated by polynomials.

Global Regularity for a Nonlocal PDE Describing Evolution of Polynomial Roots Under Differentiation

In this paper, we analyze a nonlocal nonlinear partial differential equation formally derived by Steinerberger [Proc. Amer. Math. Soc., 147 (2019), pp. 4733--4744] to model dynamics of roots of polynomials under differentiation. This partial differential equation is critical and bears striking resemblance to hydrodynamic models used to describe collective behavior of agents (such as birds, fish, or robots) in mathematical biology. We consider a periodic setting and show global regularity and exponential in time convergence to uniform density for solutions corresponding to strictly positive smooth initial data.

Salem–Zygmund inequality for locally sub-Gaussian random variables, random trigonometric polynomials, and random circulant matrices

In this manuscript we give an extension of the classic Salem–Zygmund inequality for locally sub-Gaussian random variables. As an application, the concentration of the roots of a Kac polynomial is studied, which is the main contribution of this manuscript. More precisely, we assume the existence of the moment generating function for the iid random coefficients for the Kac polynomial and prove that there exists an annulus of width O(n-2(logn)-1/2-γ),γ>1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \ ext {O}( n^{-2}(\\log n)^{-1/2-\\gamma }), \\quad \\gamma >1/2\\end{aligned}$$\\end{document}around the unit circle that does not contain roots with high probability. As an another application, we show that the smallest singular value of a random circulant matrix is at least n^{-rho }, rho in (0,1/4) with probability 1-text {O}( n^{-2rho }).

A Method for Locating the Real Roots of the Symbolic Quintic Equation Using Quadratic Equations

AbstractA method is proposed with which the locations of the roots of the monic symbolic quintic polynomial can be determined using the roots of two resolvent quadratic polynomials: and , whose coefficients are exactly those of the quintic polynomial. The different cases depend on the coefficients of and and on some specific relationships between them. The method is illustrated with the full analysis of one of the possible cases. Some of the roots of the symbolic quintic equation for this case have their isolation intervals determined and, as this cannot be done for all roots with the help of quadratic equations only, finite intervals containing 1 or 3 roots, or 0 or 2 roots, or, rarely, 0, or 2, or 4 roots of the quintic are identified. Knowledge of the stationary points of the quintic lifts this indeterminacy and allows finding the isolation interval of each root. Separately, using the complete root classification of the quintic, one can also lift this indeterminacy. The method also allows to see how variation of the individual coefficients of the quintic affect its roots. No root finding iterations or any numerical approximations are used and no equations of degree higher than two are solved.

Self-interaction in a cosmic dark fluid: The four-kernel rheological extension of the equations of state

We establish a new self-consistent model of coupling between the cosmic dark energy and dark matter in the framework of the rheological approach, which is based on the representation of the equations of state in terms of integral operators of the Volterra-type. We elaborate the so-called four-kernel model, in the framework of which both the dark energy and dark matter pressures are presented by two integrals containing the energy densities of the dark energy and dark matter. For the Volterra operators, the kernels of which are associated with the effects of fading memory, the corresponding isotropic homogeneous cosmological model is shown to be exactly integrable. We consider the classification of the model exact solutions, based on the analysis of roots of the characteristic polynomial associated with the key equation of the presented model. The scalars of the pressure and energy-density of the dark energy and dark matter, the Hubble function and acceleration parameter are presented explicitly as the functions of the dimensionless scale factor. The scale factor as the function of the cosmological time is found in quadratures and is described analytically, qualitatively and numerically. Asymptotic analysis allowed us to classify the models with respect to behavior typical for the Big Rip, Little Rip and Pseudo Rip (de Sitter type). Two intriguing exact cosmological solutions are discussed, which describe the super-exponential expansion and the symmetric bounce. New solutions are presented, which correspond to the quasi-periodic behavior of the state functions of the dark fluid and of the geometric characteristics of the Universe.

Classification of KPI lumps

A large family of nonsingular rational solutions of the Kadomtsev–Petviashvili (KP) I equation are investigated. These solutions are constructed via the Gramian method and are identified as points in a complex Grassmannian. Each solution is a traveling wave moving with a uniform background velocity but have multiple peaks which evolve at a slower time scale in the co-moving frame. For large times, these peaks separate and form well-defined wave patterns in the xy-plane. The pattern formation are described by the roots of well-known polynomials arising in the study of rational solutions of Painlevé II and IV equations. This family of solutions are shown to be described by the classical Schur functions associated with partitions of integers and irreducible representations of the symmetric group of N objects. It is then shown that there exists a one-to-one correspondence between the KPI rational solutions considered in this article and partitions of a positive integer N.

Rationalizability of field extensions with a view towards Feynman integrals

In 2021, Marco Besier and the first author introduced the concept of rationalizability of square roots to simplify arguments of Feynman integrals. In this work, we generalize the definition of rationalizability to field extensions. We then show that the rationalizability of a set of quadratic field extensions is equivalent to the rationalizability of the compositum of the field extensions, providing a new strategy to prove rationalizability of sets of square roots of polynomials.