New bivariate polynomial expansion with boundary data on the simplex

We introduce an extension to the two-dimensional simplex of the univariate two-point expansion formula for sufficiently smooth real functions introduced in [13]; it is a polynomial expansion with algebraic degree of exactness. This expansion is applied to obtain a new class of embedded boundary-type cubature formulae on the simplex.      

On the stability radius of a Schur polynomial

The robust Schur stability of a polynomial with uncertain coefficients will be investigated. The stability hypersphere for such polynomials will be determined in terms of Tshebyshev Polynomials.     

On the stability of a general polynomial in the space of Markov parameters

Markov parameters and the associated stability criterion were first introduced for continuous-time real polynomials. Recently, robust stability of such polynomials was considered in Markov parameters space, where efficient robust stability tests were obtained based on the Markov theorem. This has motivated the authors to extend the above idea to more general types of polynomials, and develop Markov parameters and the associated stability criterion for complex continuous-time as well as real and complex discrete-time polynomials. Moreover, for each polynomial type, we present compact relations in order to recover the coefficients of a polynomial corresponding to a given set of Markov parameters. The stability results presented here may be useful for 1-D and 2-D filtering and control applications.     

On the improved Newton-like methods for the inclusion of polynomial zeros

The aim of this paper is to present some modifications of Newton's type method for the simultaneous inclusion of all simple complex zeros of a polynomial. Using the concept of the R-order of convergence of mutually dependent sequences, the convergence analysis shows that the convergence rate of the basic method is increased from 3 to 6 using Jarratt's corrections. The proposed method possesses a great computational efficiency since the acceleration of convergence is attained with only few additional calculations. Numerical results are given to demonstrate convergence properties of the considered methods.     

The number of roots of a lacunary bivariate polynomial on a line

We prove that a polynomial f∈R[x,y]$f\in \mathbb{R}[x,y]$ with t$t$ non-zero terms, restricted to a real line y=ax+b$y=ax+b$, either has at most 6t−4$6t-4$ zeros or vanishes over the whole line. As a consequence, we derive an alternative algorithm for deciding whether a linear polynomial y−ax−b∈K[x,y]$y-ax-b\in K[x,y]$ divides a lacunary polynomial f∈K[x,y]$f\in K[x,y]$, where K$K$ is a real number field. The number of bit operations performed by the algorithm is polynomial in the number of non-zero terms of f$f$, in the logarithm of the degree of f$f$, in the degree of the extension K/Q$K/\mathbb{Q}$ and in the logarithmic height of a$a$, b$b$ and f$f$.     

Sorting-based localization and stable computation of zeros of a polynomial. I

A method of stable address sorting is designed for localization and approximate computation of real and complex zeros of a polynomial. In an arbitrary domain, a software implementation of the method allows one to localize and to calculate with high accuracy all zeros of a polynomial, including the case when they are ill-separated. The vicinity of each zero dynamically decreases during localizing the boundaries of the domains of all zeros. Algorithms developed are formally described in the Object Pascal language and are implemented in the Delphi 7 environment. Some upper-bound estimates of a parallel version of the method are given. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 165–182, January–February 2007.     

Computation of the stability radius of a Hurwitz polynomial with diamond-like uncertainties

This paper considers the stability radius problem of Hurwitz polynomials whose coefficients have Hölder 1-norm-bounded uncertainties. We show that the solution to this problem demands the computation of the minimum of a piece-wise real-rational function ρ(λ), called the stability radius function. It is then shown that the calculations of ρ(λ) at the intersection points where ρ(λ) changes its representation and at the stationary points where ρ′(λ)=0 can be reduced to two sets of eigenvalue problems of matrices of the form H_β⁻¹H_γ, where both H_β and H_γ are frequency-independent Hurwitz matrices. Using root locus technique, we analyze this function further and prove that, in some special cases, the minimum of this function can be achieved only at the intersection points. Extensions of the eigenvalue approach to cover other robust stability problems are also discussed.     

GF（2）上一类多项式因式分解及算法实现

循环码在信道编码中起着非常重要的作用，它构造简单，易于实现，可通过x^n-1的既约多项式构造出来，介绍了将GF（2）上，n^x-1型多项式分解为既约多项式的方法，并给出了具体的实现方法，实验表明此算法实现简单，有很高的实用价值。     

Higher-order simultaneous methods for the determination of polynomial multiple zeros

Starting from Laguerre's method and using Newton's and Halley's corrections for a multiple zero, new simultaneous methods of Laguerre's type for finding multiple (real or complex) zeros of polynomials are constructed. The convergence order of the proposed methods is five and six, respectively. By applying the Gauss–Seidel approach, these methods are further accelerated. The lower bounds of the R-order of convergence of the improved (single-step) methods are derived. Faster convergence of all proposed methods is attained with negligible number of additional operations, which provides a high computational efficiency of these methods. A detailed convergence analysis and numerical results are given.     

On the rank minimization problem over a positive semidefinitelinear matrix inequality

We consider the problem of minimizing the rank of a positive semidefinite matrix, subject to the constraint that an affine transformation of it is also positive semidefinite. Our method for solving this problem employs ideas from the ordered linear complementarity theory and the notion of the least element in a vector lattice. This problem is of importance in many contexts, for example in feedback synthesis problems, and such an example is also provided     

Taylor polynomial solutions of systems of linear differential equations with variable coefficients

A Taylor collocation method has been presented for numerically solving systems of high-order linear ordinary, differential equations with variable coefficients. Using the Taylor collocation points, this method transforms the ODE system and the given conditions to matrix equations with unknown Taylor coefficients. By means of the obtained matrix equation, a new system of equations corresponding to the system of linear algebraic equations is gained. Hence by finding the Taylor coefficients, the Taylor polynomial approach is obtained. Also, the method can be used for the linear systems in the normal form. To illustrate the pertinent features of the method, examples are presented and results are compared.     

On the immersion of a discrete-time polynomial analytic system into a polynomial affine one

The concept of immersion, just recently introduced by Fliess and Kupka, is related to the capability of reproducing the input-output maps of a given system by means of another one. In this paper we give necessary and sufficient conditions under which a polynomial analytic system, i.e. a system whose dynamics are polynomial with respect to the input and nonlinear analytic with respect to the state, is immersed into a polynomial affine system, i.e. a system whose dynamics are polynomial in the input and linear in the state. An application to the problem of the reproducibility of a linear analytic state feedback acting on a given system by means of a preprocessor suitably initialized is also performed.     

Improving the robustness of fractional polynomial models by preliminary covariate transformation: A pragmatic approach

Fractional polynomials have been found useful in univariate and multivariable non-linear regression analysis. As with many flexible regression models, they may be prone to distortion of the fitted function caused by values with high leverage at either extreme of the covariate distribution. Furthermore, fractional polynomial functions are not invariant to a change of origin of the covariate. A new approach, based on a preliminary, almost-linear transformation of a covariate, is proposed. The transformation is approximately linear within the bulk of the observations and tapers smoothly to a truncation of the extremes. It incorporates a predefined shift of the origin away from zero. Empirical studies show that this transformation is effective in reducing extreme leverages. In two real datasets, it is shown its use can result in more sensible final models.     

A statistical approach for economization of the polynomial functions

Due to having the minimax property, Chebyshev polynomials are used today to economize the arbitrary polynomial functions. In this work, we present a statistical approach to show that, contrary to current thought, the Chebyshev polynomials of the first kind are not appropriate for economizing these polynomials if one uses this statistical approach. In this way, a numerical results section is also given to clearly prove our claim.     

On the bounds for the roots of a polynomial

Some new sufficient conditions are presented which give bounds for the absolute values of the roots of a polynomial. These results extend certain earlier stability tests for linear discrete systems.     

On the minimum order of a robust servocompensator

In recent papers [1]-[4], Davison et al. have given the characterization of a minimal-order robust error-driven servocompensator which achieves asymptotic tracking and disturbance rejection. In this note, we establish this minimality property by frequency domain methods. We show that any right coprime factorization of the controller, sayN_{r}(S)D_{r}(S)^{-1}, must have all the elements ofD_{r}(s)divisible byPhi(s), the minimal polynomial of the tracking and disturbance signal generator. Hence, its order must be at leastn_{0}.d(phi)(n0=number of outputs,d(Phi)= degree ofPhi).     

On the minimum time function and the minimum energy problem; a nonlinear case

For a semilinear control system in general Banach spaces, we prove results on exact and approximate controllability, regularity of the minimum time function, connections between the minimum time function and the minimum energy.     

On the existence of a positive realization

The positive realization problem for linear systems is to find conditions, for a given transfer function with nonnegative impulse response, to have a realization such that the resulting system is a positive system. Recently, it has been shown that, under a mild assumption on the long-term behaviour of the impulse response, this problem is related to the maximum modulus poles only. In this paper necessary and sufficient conditions for positive realizability of discrete-time systems are given. They show that also nondominant poles play a role in the most general case. Positive realizability conditions for the continuous-time case are also given.