A new type of Hermite matrix polynomial series

Conventional Hermite polynomials emerge in a great diversity of applications in mathematical physics, engineering, and related fields. However, in physical systems with higher degrees of freedom it will be of practical interest to extend the scalar Hermite functions to their matrix analogue. This work introduces various new generating functions for Hermite matrix polynomials and examines existence and convergence of their associated series expansion by using Mehler’s formula for the general matrix case. Moreover, we derive interesting new relations for even- and odd-power summation in the generating-function expansion containing Hermite matrix polynomials. Some new results for the scalar case are also presented.     

Bezout equations over bivariate polynomial matrices related by an entire function

This study addresses Bezout equations over bivariate polynomial matrices, where the relationship between two variables is described by a real entire function. This paper proposes a solution method that makes optimal use of minor primeness to reduce such Bezout equations to simple equations over univariate scalar polynomials. The proposed solution method requires only matrix calculations, thus making it very useful, especially in the absence of modern computer algebra systems.     

To solving multiparameter problems of algebra. 10. Computing zeros of a scalar polynomial

The paper considers the problem of computing zeros of scalar polynomials in several variables. The zeros of a polynomial are subdivided into the regular (eigen-and mixed) zeros and the singular ones. An algorithm for computing regular zeros, based on a decomposition of a given polynomial into a product of primitive polynomials, is suggested. The algorithm is applied to solving systems of nonlinear algebraic equations. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 346, 2007, pp. 119–130.     

Is the polynomial so perfidious?

Summary. Wilkinson, in [1], has given a comprehensive account of the numerical difficulties of working with polynomials on a floating point computer. The object of this note is to attempt to rehabilitate the polynomial to a certain extent. In particular it is shown here that polynomial deflation can be performed satisfactorily by a method akin to `backward recursion'. Error analyses and examples are given to illustrate the stability of the process. Received October 4, 1993 / Revised version received July 14, 1993     

Note on roots location of a symmetric polynomial with respect to the imaginary axis

In the theory of the separation of roots of algebraic equations, the well-known Routh–Hurwitz–Fujiwara theorem enables us to separate the complex roots of a polynomial with complex coefficients in terms of the inertia of a related Hermitian matrix. Unfortunately, it fails if the polynomial has a nontrivial factor which is symmetric with respect to the imaginary axis. In this article, we present a method to overcome the fault and formulate the inertia of a scalar polynomial with complex coefficients in terms of the inertia of several Hermitian matrices based on a factorization of a monic symmetric polynomial into products of monic symmetric polynomials with only simple roots in the complex plane and on computing the inertia of each factor by means of a subtle perturbation.     

Jordan structures of alternating matrix polynomials

Alternating matrix polynomials, that is, polynomials whose coefficients alternate between symmetric and skew-symmetric matrices, generalize the notions of even and odd scalar polynomials. We investigate the Smith forms of alternating matrix polynomials, showing that each invariant factor is an even or odd scalar polynomial. Necessary and sufficient conditions are derived for a given Smith form to be that of an alternating matrix polynomial. These conditions allow a characterization of the possible Jordan structures of alternating matrix polynomials, and also lead to necessary and sufficient conditions for the existence of structure-preserving strong linearizations. Most of the results are applicable to singular as well as regular matrix polynomials.     

Simultaneous stabilization of linear plants by a low order controller

We consider simultaneous stabilization of linear scalar stationary plants of second order and derive an easily verified necessary condition of simultaneous stabilization, as well as a sufficient condition that leads to a constructive algorithm for a stabilizing controller. The existence of a stabilizing controller is established in the parametric polynomial space by a method that finds the stability radii of polynomials.     

Operations over scalar polynomials and their computer realization

In this paper, algorithms which realize some operations over scalar polynomials in one and two variables and their computer realization are suggested. The following operations are considered: 1) the computation of the GCD for given scalar polynomials and the decomposition of each polynomial into a product of two factors: the first factor is the GCD, and the second factors form a sequence of relatively prime polynomials; 2) the division of polynomials by their common divisor; 3) the decomposition of polynomials in two variables into irreducible factors; 4) the computation of the LCM for given scalar polynomials. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 219, 1994, pp. 158–175. This work was supported by the Russian Foundation of Fundamental Research (grant 94-01-00919). Translated by V. N. Kublanovskaya.     

A scaling algorithm for polynomial constraint satisfaction problems

Good scaling is an essential requirement for the good behavior of many numerical algorithms. In particular, for problems involving multivariate polynomials, a change of scale in one or more variable may have drastic effects on the robustness of subsequent calculations. This paper surveys scaling algorithms for systems of polynomials from the literature, and discusses some new ones, applicable to arbitrary polynomial constraint satisfaction problems.     

Multiplication of a Schubert polynomial by a Schur polynomial

Schur polynomials are a special case of Schubert polynomials. In this paper, we give an algorithm to compute the product of a Schubert polynomial with a Schur polynomial on the basis of Schubert polynomials. This is a special case of the general problem of the multiplication of two Schubert polynomials, where the corresponding algorithm is still missing. The main tools for the given algorithm is a factorization property of a special class of Schubert polynomials and the transition formula for Schubert polynomials.     

On the orthogonality of residual polynomials\newline of minimax polynomial preconditioning

Summary. This paper studies polynomials used in polynomial preconditioning for solving linear systems of equations. Optimum preconditioning polynomials are obtained by solving some constrained minimax approximation problems. The resulting residual polynomials are referred to as the de Boor-Rice and Grcar polynomials. It will be shown in this paper that the de Boor-Rice and Grcar polynomials are orthogonal polynomials over several intervals. More specifically, each de Boor-Rice or Grcar polynomial belongs to an orthogonal family, but the orthogonal family varies with the polynomial. This orthogonality property is important, because it enables one to generate the minimax preconditioning polynomials by three-term recursive relations. Some results on the convergence properties of certain preconditioning polynomials are also presented. Received February 1, 1992/Revised version received July 7, 1993     

Convergence of polynomial ergodic averages

We prove theL ² convergence for an ergodic average of a product of functions evaluated along polynomial times in a totally ergodic system. For each set of polynomials, we show that there is a particular factor, which is an inverse limit of nilsystems, that controls the limit behavior of the average. For a general system, we prove the convergence for certain families of polynomials. Dedicated to Hillel Furstenberg upon his retirement The second author was partially supported by NSF grant DMS-0244994.     

Complexity of solving parametric polynomial systems

In this paper, we present three algorithms: the first one solves zero-dimensional parametric homogeneous polynomial systems within single exponential time in the number n of unknowns; it decomposes the parameter space into a finite number of constructible sets and computes the finite number of solutions by parametric rational representations uniformly in each constructible set. The second algorithm factirizes absolutely multivariate parametic polynomials within single exponential time in n and in the upper bound d on the degree of the factorized polynomials. The third algorithm decomposes algebraic varieties defined by parametric polynomial systems of positive dimension into absolutely irreducible components uniformly in the values of the parameters. The complexity bound for this algorithm is double exponential in n. On the other hand, the lower bound on the complexity of the problem of resolution of parametric polynomial systems is double exponential in n. Bibliography: 72 titles.     

A polynomial approach for generating a monoparametric family of chaotic attractors via switched linear systems

In this paper, we consider characteristic polynomials of n-dimensional systems that determine a segment of polynomials. One parameter is used to characterize this segment of polynomials in order to determine the maximal interval of dissipativity and unstability. Then we apply this result to the generation of a family of attractors based on a class of unstable dissipative systems (UDS) of type affine linear systems. This class of systems is comprised of switched linear systems yielding strange attractors. A family of these chaotic switched systems is determined by the maximal interval of perturbation of the matrix that governs the dynamics for still having scroll attractors.     

Sharp precision in Hensel lifting for bivariate polynomial factorization

Popularized by Zassenhaus in the seventies, several algorithms for factoring polynomials use a so-called lifting and recombination scheme. Concerning bivariate polynomials, we present a new algorithm for the recombination stage that requires a lifting up to precision twice the total degree of the polynomial to be factored. Its cost is dominated by the computation of reduced echelon solution bases of linear systems. We show that our bound on precision is asymptotically optimal.

     

Laguerre matrix polynomial series expansion: Theory and computer applications

This paper deals with the expansion of matrix functions in series of Laguerre matrix polynomials of a complex matrix parameter. A new asymptotic expression for positive and large x and n is obtained for this family of matrix polynomials that improves previous existing expressions. This expression is applied to improving the conditions imposed on the matrix function that are to be expanded in a series of non-hermitian Laguerre matrix polynomials, providing a new series expansion theorem that is more general than previous works. An application to solving linear differential systems without matrix exponentials is included.     

Shape preserving polynomial curves

We introduce particular systems of functions and study the properties of the associated Bézier-type curve for families of data points in the real affine space. The systems of functions are defined with the help of some linear and positive operators, which have specific properties: total positivity, nullity diminishing property and which are similar to the Bernstein polynomial operator. When the operators are polynomial, the curves are polynomial and their degrees are independent of the number of data points. Examples built with classical polynomial operators give algebraic curves written with the Jacobi polynomials, and trigonometric curves if the first and the last data points are identical.     

Smooth polynomial approximation of spiral arcs

Constructing fair curve segments using parametric polynomials is difficult due to the oscillatory nature of polynomials. Even NURBS curves can exhibit unsatisfactory curvature profiles. Curve segments with monotonic curvature profiles, for example spiral arcs, exist but are intrinsically non-polynomial in nature and thus difficult to integrate into existing CAD systems. A method of constructing an approximation to a generalised Cornu spiral (GCS) arc using non-rational quintic Bézier curves matching end points, end slopes and end curvatures is presented. By defining an objective function based on the relative error between the curvature profiles of the GCS and its Bézier approximation, a curve segment is constructed that has a monotonic curvature profile within a specified tolerance.     

Reverse generalized Bessel matrix differential equation,polynomial solutions,and their properties

This paper is devoted to the study of reverse generalized Bessel matrix polynomials (RGBMPs) within complex analysis. This study is assumed to be a generalization and improvement of the scalar case into the matrix setting. We give a definition of the reverse generalized Bessel matrix polynomials Θ_n(A; B; z), , for parameter (square) matrices A and B, and provide a second‐order matrix differential equations satisfied by these polynomials. Subsequently, a Rodrigues‐type formula, a matrix recurrence relationship, and a pseudo‐generating function are then developed for RGBMPs. © 2013 The Authors Mathematical Methods in the Applied Sciences Published by John Wiley & Sons, Ltd.     

On the distribution of solutions to polynomial congruences

We use a result of é. Fouvry about the distribution of solutions to systems of congruences with multivariate polynomials in small cubic boxes and some ideas of W. Schmidt to derive an asymptotic formula for the number of such solutions in very general domains.