Roots Of Polynomial Research Articles

Polynomials with small values in the neighborhoods of zeros in Archimedean and non-Archimedean metrics

For a positive integer 𝑄 > 0, let 𝐼 ⊂ R denote an interval of length 𝜇1𝐼 = 𝑄−𝛾1 (where 𝜇1 is the Lebesgue measure) and 𝜇2𝐾 = 𝑄−𝛾2 , 𝛾2 > 0 (where 𝜇2 is the Haar measure of a measurable cylinder 𝐾 ⊂ Q𝑝). Let us denote the set of polynomials of degree ≤ 𝑛 and height 𝐻 (𝑃) ≤ 𝑄 as 𝒫𝑛 (𝑄) = {𝑃 ∈ Z[𝑥] : deg 𝑃 ≥ 𝑛, 𝐻 (𝑃) ≤ 𝑄} . Let 𝒜(𝑛,𝑄) denote the set of real and 𝑝-adic roots of such polynomials 𝑃 (𝑥) lying in the space 𝑉 = 𝐼 ×𝐾. In this paper it is proved that the following inequality holds for a suitable constant 𝑐1 = 𝑐1 (𝑛) and 0 ≤ 𝑣1, 𝑣2 6 1 2 : #𝒜(𝑛,𝑄) > 𝑐1𝑄𝑛+1−𝛾1−𝛾2 . The proof relies on methods of metric theory of Diophantine approximation developed by V.G. Sprindzuk to prove Mahler’s conjecture and by V.I. Bernik to prove A. Baker’s conjecture.

Chebyshev Periodical Successive Over-Relaxation for Accelerating Fixed-Point Iterations

A novel method which is called the Chebyshev inertial iteration for accelerating the convergence speed of fixed-point iterations is presented. The Chebyshev inertial iteration can be regarded as a valiant of the successive over relaxation or Krasnosel'ski\v{\i}-Mann iteration utilizing the inverse of roots of a Chebyshev polynomial as iteration dependent inertial factors. One of the most notable features of the proposed method is that it can be applied to nonlinear fixed-point iterations in addition to linear fixed-point iterations. Linearization around the fixed point is the key for the analysis on the local convergence rate of the proposed method. The proposed method appears effective in particular for accelerating the proximal gradient methods such as ISTA. It is also proved that the proposed method can successfully accelerate almost any fixed-point iterations if all the eigenvalues of the Jacobian at the fixed point are real.

Rank-Structured QR for Chebyshev Rootfinding

We consider the computation of roots of polynomials expressed in the Chebyshev basis. We extend the QR iteration presented in [Y. Eidelman, L. Gemignani, and I. Gohberg, Numer. Algorithms, 47 (2008...

Hybrid Sparse Expansion and Separable Hybrid Prony Method

Signal sparsity plays an essential role in signal compression and reconstruction. Prony-like methods are widely used in the relevant applications e.g., the recovery of signals with finite rate of innovation (FRI). However, these methods are usually limited to the structured function expansions of a single family, i.e., the considered structured functions belonging to the eigenfunctions of a single operator. In this paper, we investigate hybrid Prony models (HPMs), wherein functions can be expanded sparsely by the combination of eigenfunctions of two different operators $\mathcal {L}_1$ and $\mathcal {L}_2$ . Although the Prony-like methods cannot directly deal with HPMs, we show that HPMs can be directly converted to the generalized Prony model recently considered by Peter and Plonka when the operator pair $\lbrace \mathcal {L}_1, \mathcal {L}_2\rbrace$ satisfies certain conditions; we refer to these functions as separable HPMs (SHPMs). Similar to the Prony method, the problem of nonlinear parameter estimation in SHPMs can be converted to the problem of finding the roots of polynomials. Examples and numerical results are given to illustrate SHPMs.

TOA and DOA Estimation for IR‐UWB Signals: An Efficient Algorithm by Improved Root‐MUSIC

The automatically paired time of arrival (TOA) and direction of arrival (DOA) can be jointly estimated via a high‐precision multidimensional spectral peak search‐ (SPS‐) based multiple signal classification (MUSIC) algorithm in the impulse radio ultrawideband (IR‐UWB) positioning system, while heavy computational burden is required. To tackle this issue, we propose an improved root‐MUSIC algorithm for joint TOA and DOA estimation. After modelling the frequency domain form of the received signal, the algorithm first uses the signal subspace to establish the relationship between the two antennas. Then, the MUSIC spatial spectrum function is reconstructed with this relation, which enables it to offer a spectrum function in regard to the one‐dimensional (1D) parameter of time delay. For further reducing the complexity, the TOA estimates of one antenna are obtained via 1D polynomial root finding instead of SPS, and the TOA estimates of the other antenna can be calculated by the established relationship. Finally, the DOA estimation can be achieved with the estimated TOAs. Due to the relationship between two antennas with signal subspace, the parameters estimated by the proposed algorithm are autopaired. Numerical simulations substantiate the superiority of the proposed algorithm.

Integrable quantum spin chains with free fermionic and parafermionic spectrum

We present a general study of the large family of exact integrable quantum chains with multispin interactions introduced recently in \cite{AP2020}. The exact integrability follows from the algebraic properties of the energy density operators defining the quantum chains. The Hamiltonians are characterized by a parameter $p=1,2,\dots$ related to the number of interacting spins in the multispin interaction. In the general case the quantum spins are of infinite dimension. In special cases, characterized by the parameter $N=2,3,\ldots$, the quantum chains describe the dynamics of $Z(N)$ quantum spin chains. The simplest case $p=1$ corresponds to the free fermionic quantum Ising chain ($N=2$) or the $Z(N)$ free parafermionic quantum chain. The eigenenergies of the quantum chains are given in terms of the roots of special polynomials, and for general values of $p$ the quantum chains are characterized by a free fermionic ($N=2$) or free parafermionic ($N>2$) eigenspectrum. The models have a special critical point when all coupling constants are equal. At this point the ground-state energy is exactly calculated in the bulk limit, and our analytical and numerical analyses indicate that the models belong to universality classes of critical behavior with dynamical critical exponent $z = (p+1)/N$ and specific-heat exponent $\alpha = \max\{0,1-(p+1)/N\}$.

On Efficiently Solvable Cases of Quantum k-SAT

Estimating ground state energies of local Hamiltonian models is a central problem in quantum physics. The question of whether a given local Hamiltonian is frustration-free, meaning the ground state is the simultaneous ground state of all local interaction terms, is known as the Quantum k-SAT (k-QSAT) problem. In analogy to its classical Boolean constraint satisfaction counterpart, the NP-complete problem k-SAT, Quantum k-SAT is hbox {QMA}_1-complete (for kge 3, and where hbox {QMA}_1 is a quantum generalization of NP with one-sided error), and thus likely intractable. But whereas k-SAT has been well-studied for special tractable cases, as well as from a “parameterized complexity” perspective, much less is known in similar settings for k-QSAT. Here, we study the open problem of computing satisfying assignments to k-QSAT instances which have a “dimer covering” or “matching”; such systems are known to be frustration-free, but it remains open whether one can efficiently compute a ground state. Our results fall into three directions, all of which relate to the “dimer covering” setting: (1) We give a polynomial-time classical algorithm for k-QSAT when all qubits occur in at most two interaction terms or clauses. (2) We give a “parameterized algorithm” for k-QSAT instances from a certain non-trivial class, which allows us to obtain exponential speedups over brute force methods in some cases. This is achieved by reducing the problem to solving for a single root of a single univariate polynomial. An explicit family of hypergraphs, denoted Crash, for which such a speedup is obtained is introduced. (3) We conduct a structural graph theoretic study of 3-QSAT interaction graphs which have a “dimer covering”. We remark that the results of (2), in particular, introduce a number of new tools to the study of Quantum SAT, including graph theoretic concepts such as transfer filtrations and blow-ups from algebraic geometry.

The MULTIPLISITAS NEWTON DAN TITIK TETAP ATRAKTIF DALAM MENENTUKAN KEKONVERGENAN

Abstract. Newton's method is one of the numerical methods used in finding polynomial roots. This method will be very effective to use, if the initial estimate of the roots for the Newton iteration function satisfies sufficient Newtonian convergence, [11]. In this article we will analyze the efficacy of this method by looking at the relationship between the fixed point method and Newton's iteration function. When the iteration of the function converges to the root, the velocity of convergence can also be determined. In terms of the speed of convergence, it turns out to be very dependent on the multiplicity of Newton's method itself.
 

Enumeration of various animals on the triangular lattice

In this paper, we consider various classes of polyiamonds that are animals residing on the triangular lattice. By careful analyses through certain layer-by-layer decompositions and cell pruning/growing arguments, we derive explicit forms for the generating functions of the number of nonempty translation-invariant baryiamonds (bargraphs in the triangular lattice), column-convex polyiamonds, and convex polyiamonds with respect to their perimeter. In particular, we show that the number of(A) baryiamonds of perimeter n are asymptotically (ξ+1)2ξ4+ξ3−2ξ+12πn3ξ−n−2,where ξ is a root of a certain explicit polynomial of degree 5.(B) column-convex polyiamonds of perimeter n are asymptotic to (1799780917+33⋅13⋅175463)9517−11927⋅432⋅8926πn33+172n−1.(C) convex polyiamonds of perimeter n is asymptotic to 12804413πn33n.

A Different Look at Polynomials within Man-Machine Interaction

Undoubtedly, one of the most powerful applications that allow symbolic computations is Wolfram Mathematica. However, it turns out that sometimes Mathematica does not give the desired result despite its continuous improvement. Moreover, these gaps are not filled by many authors of books and tutorials. For example, our attempts to obtain a compact symbolic description of the roots of polynomials or coefficients of a polynomial with known roots using Mathematica have often failed and they still fail. Years of our work with theory, computations, and different kinds of applications in the area of polynomials indicate that an application ‘offering’ the user alternative methods of solving a given problem would be extremely useful. Such an application would be valuable not only for people who look for solutions to very specific problems but also for people who need different descriptions of solutions to known problems than those given by classical methods. Therefore, we propose the development of an application that would be not only a program doing calculations but also containing an interactive database about polynomials. In this paper, we present examples of methods and information which could be included in the described project.

Nonlinear stress analysis of shell structures in buckling and snapping problems by exact geometry solid-shell elements through sampling surfaces formulation

In this paper, the nonlinear three-dimensional (3D) stress analysis of shell structures in buckling and snapping problems is presented. The exact geometry or geometrically exact (GeX) hybrid-mixed four-node solid-shell element is developed using a sampling surfaces (SaS) method. The SaS formulation is based on the choice of N SaS parallel to the middle surface to introduce the displacements of these surfaces as basic shell unknowns. The SaS are located at the Chebyshev polynomial nodes (roots of the Chebyshev polynomial of degree N), that is, the outer surfaces are not included into a set of SaS. Such choice of unknowns with the consequent use of Lagrange polynomials of degree N–1 in the through-the-thickness distributions of displacements, strains and stresses leads to an efficient higher-order shell formulation. The incremental equilibrium equations are solved by the Newton–Raphson method combined with the Crisfield arc-length algorithm. The tangent stiffness matrix is evaluated by effective 3D analytical integration. As a result, the proposed GeX hybrid-mixed solid-shell element exhibits superior performance for coarse meshes and allows much larger load increments than is possible with existing displacement-based solid-shell elements. This can be useful for the 3D stress analysis of thin and thick shells in different states such as pre-buckling, bifurcation and post-buckling.

A New Hybrid Method for Signal Estimation Based on Haar Transform and Prony Analysis

The signal estimation is very important in electrical and electronic engineering. In this study, it is shown that signal parameters' (frequency, amplitude, and phase) estimation can be realized with the implementation of Prony method on Haar transform coefficients. In order to accomplish this, mathematical relationship between roots of Prony polynomial which are found with original signal values and roots which are calculated with Haar approximation/detail coefficients is constructed. Frequency components of signal are estimated with this relationship. Next, the second part of Prony algorithm which constructs the matrix equation between roots and signal values in order to find the amplitude and phase values is implemented with Haar coefficients. In other words, a new matrix equation is derived for finding amplitudes and phases with the found roots in the first step and Haar coefficients. Thus, implementations of the first and second steps give signal parameters. Derived equations are valid for all degrees of Haar coefficients not just the first one. The use of Haar coefficients decreases the data size and increases the speed and accuracy. The proposed method is also more robust of selection of different Prony polynomial coefficient sizes.

Logarithmic A-hypergeometric series

The method of Frobenius is a standard technique to construct series solutions of an ordinary linear differential equation around a regular singular point. In the classical case, when the roots of the indicial polynomial are separated by an integer, logarithmic solutions can be constructed by means of perturbation of a root. The method for a regular [Formula: see text]-hypergeometric system is a theme of the book by Saito, Sturmfels and Takayama. Whereas they perturbed a parameter vector to obtain logarithmic [Formula: see text]-hypergeometric series solutions, we adopt a different perturbation in this paper.

Paths of Real Roots for Odd-Degree Polynomials with Parameters

In this paper, we construct a connected path of real roots of odd-degree real polynomials depending on parameters continuously, which joins with all parameters. Moreover, most of the points on the path are odd-multiple real roots except for at most countable even-multiple real roots. Furthermore, some geometrical features of the path can be seen easily from the construction.

Distribution and Properties of the Critical Values of Random Polynomials With Non-Independent and Non-Identically Distributed Roots

This paper considers the pairing between the distribution of the roots and the distribution of the critical values of random polynomials. The primary model of random polynomial considered in this paper consists of monic polynomials of degree n with a single complex variable z where the roots of the random polynomial are complex valued random variables that are chosen from two independent sequences of iid, complex valued random variables. The distribution of the random variables from each of the two sequences are different, producing roots of the random polynomial which have non-identical distributions. Furthermore, both the iid, complex valued random variables from one of the sequences of random variables and the complex conjugates of those random variables are roots of the random polynomial. Hence, this model of monic random polynomials of degree n has roots that are random variables which are not independent, due to the dependence based on complex conjugates, and the non-identical distributions which arise from the use of the two independent sequences. This paper also describes the relationship between the roots and critical values of monic random polynomials of degree n where the roots are chosen to be random variables and their complex conjugates where the random variables are from a sequence of iid complex valued random variables.

MODIFIED ALGORITHM FOR SEARCHING THE ROOTS OF THE ERROR LOCATORS POLYNOMINAL WHILE DECODING BCH CODES

Context. In telecommunications and information systems with an increased noise component the noise-resistant cyclic BCH and Reed-Solomon codes are used. The adjustment and correcting errors in a message require some effective decoding methods. One of the stages in the procedure of decoding RS and BCH codes to determine the position of distortions is the search for the roots of the error locator polynomial. The calculation of polynomial roots, especially for codes with significant correction capacity is a laborious task requiring high computational complexity. That is why the improvement of BCH and RS codes decoding methods providing to reduce the computational complexity is an urgent task.Objective. The investigation and synthesis of the accelerated roots search algorithm of the error locator polynomial presented as an affine polynomial with coefficients in the finite fields, which allows accelerating the process of BCH and RS code decoding.Method. The classical roots search method based on the Chan’s algorithm is performed using the arithmetic of the Galois finite fields and the laborious calculation, in this case depends on the number of addition and multiplication operations. For linearized polynomials, the roots search procedure based on binary arithmetic is performed taking into account the values obtained at the previous stages of the calculation, which provides the minimum number of arithmetic operations.Results. An accelerated algorithm for calculating the values of the error locator polynomial at all points of the GF(2m) finite field for linearized polynomials based on the Berlekamp-Massey method has been developed. The algorithm contains a minimum number of addition operations, due to the use at each stage of the calculations the values obtained at the previous step, as well as the addition in the finite field GF(2). A modified roots search method for affine polynomials over the finite fields has been proposed to determine error positions in the code word while decoding the cyclic BCH and RS codes.Conclusions. The scientific newness of the work is to improve the algorithm of calculating the roots of the error locator polynomial, which coefficients belong to the elements of the finite field. At the same time it simplifies the procedure for cyclic BCH and RS codes decoding, due to reducing the computational complexity of one of the decoding stages, especially finding the error positions using the modified Berlekamp-Massey algorithm. These facts are confirmed by the simulation program results of the roots search of the error locator polynomial algorithm. It is shown, that the application of the accelerated method permits to reach a gain on speed of 1.5 times.

Analysis of Blind Reconstruction of BCH Codes.

In this paper, the theoretical lower-bound on the success probability of blind reconstruction of Bose–Chaudhuri–Hocquenghem (BCH) codes is derived. In particular, the blind reconstruction method of BCH codes based on the consecutive roots of generator polynomials is mainly analyzed because this method shows the best blind reconstruction performance. In order to derive a performance lower-bound, the theoretical analysis of BCH codes on the aspects of blind reconstruction is performed. Furthermore, the analysis results can be applied not only to the binary BCH codes but also to the non-binary BCH codes including Reed–Solomon (RS) codes. By comparing the derived lower-bound with the simulation results, it is confirmed that the success probability of the blind reconstruction of BCH codes based on the consecutive roots of generator polynomials is well bounded by the proposed lower-bound.

Descartes' rule of signs, Newton polygons, and polynomials over hyperfields

In this note, we develop a theory of multiplicities of roots for polynomials over hyperfields and use this to provide a unified and conceptual proof of both Descartes' rule of signs and Newton's “polygon rule”.

Min-max elementwise backward error for roots of polynomials and a corresponding backward stable root finder

A new measure called min-max elementwise backward error is introduced for approximate roots of scalar polynomials p(z). Compared with the elementwise relative backward error, this new measure allows for larger relative perturbations on the coefficients of p(z) that do not participate much in the overall backward error. By how much these coefficients can be perturbed is determined via an associated max-times polynomial and its tropical roots. An algorithm is designed for computing the roots of p(z). It uses a companion linearization C(z)=A−zB of p(z) to which we added an extra zero leading coefficient, and an appropriate two-sided diagonal scaling that balances A and makes B graded in particular when there is variation in the magnitude of the coefficients of p(z). An implementation of the QZ algorithm with a strict deflation criterion for eigenvalues at infinity is then used to obtain approximations to the roots of p(z). Under the assumption that this implementation of the QZ algorithm exhibits a graded backward error when B is graded, we prove that our new algorithm is min-max elementwise backward stable. Several numerical experiments show the superior performance of the new algorithm compared with the MATLAB roots function. Extending the algorithm to polynomial eigenvalue problems leads to a new polynomial eigensolver that exhibits excellent numerical behaviour compared with other existing polynomial eigensolvers, as illustrated by many numerical tests.

Monostatic MIMO radar with nested L-shaped array: Configuration design, DOF and DOA estimation

In this paper, we consider the problem of two-dimensional (2-D) direction of arrival (DOA) estimation for L-shaped monostatic MIMO radar with two-level nested linear array (NLA). To obtain more degrees of freedom (DOFs) than traditional nested L-shaped (NLs) MIMO radar, we propose the extensional NLs (ENLs) MIMO radar by extending all of the sensor positions in transmitter array with an identical magnification compared with NLs MIMO radar. The ENLs MIMO radar can offer O(M1M2N1N2) DOFs with only O(M1+M2+N1+N2) sensors, where M1(N1) and M2(N2) respectively represents the numbers of the first-level and second-level sensors of NLA in transmitter (receiver) array. Furthermore, to avoid the eigenvalue decomposition, spatial spectral search and polynomial root finding technique in spatial smoothing multiple signals classification (SS-MUSIC) and SS-ROOT-MUSIC, and to improve the estimation accuracy simultaneously, we propose a search-free iterative Taylor expansion (SF-ITE) algorithm to perform DOA estimation, which employs the Discrete Fourier transform (DFT) method to get the coarse estimation, and then utilizes the iterative Taylor expansion technique to obtain fine estimation. SF-ITE can get more precise estimation by performing iteration only once than both SS-MUSIC and SS-ROOT-MUSIC. In addition, we derive an angle-pairing procedure to help to eventually resolve more targets than the total number of sensors. Finally, simulation results are provided to validate the superiority of the ENLs MIMO radar and SF-ITE algorithm.