Computing the Invariant Polynomials of a Polynomial Matrix. I

An algorithm for computing the invariant polynomials and the canonical triangular (trapezoidal) matrix for a polynomial matrix of full column rank is suggested. The algorithm is based on the Δ W-1 rank-factorization method for solving algebraic problems for polynomial matrices, previously suggested by the author. Bibliography: 3 titles.     

Computing Derivatives of Jacobi Polynomials Using Bernstein Transformation and Differentiation Matrix

In this paper, we give a new, simple, and efficient method for evaluating the pth derivative of the Jacobi polynomial of degree n. The Jacobi polynomial is written in terms of the Bernstein basis, and then the pth derivative is obtained. The results are given in terms of both Bernstein basis of degree n ? p and Jacobi basis form of degree n ? p and presented in a matrix form. Numerical examples and comparisons with other well-known methods are presented.     

在基本计算系统中实现切比雪夫多项式的算法

提出了一个仅使用基本运算加、减和移位计算切比雪夫多项式的坐标旋转算法,证明了收敛性,讨论了误差估计.算法编码占用空间很小,适合在微计算系统中使用.     

多项式零点保持线性映射

设H是维数大于2的复Hilbert空间,β(H)代表H上所有有界线性算子全体．假定Φ是从β(H)到其自身的弱连续线性双射．我们证明了映射Φ满足对所有的A,B∈β(H),AB=BA~*蕴涵Φ(A)Φ(B)=Φ(B)Φ(A)~*当且仅当存在非零实数c和酉算子U∈(?)(H),使得Φ(A)=cUAU~*对所有的A∈β(H)成立．     

Symmetric Equivalence of Matrix Polynomials and Isolation of a Common Unital Divisor in Matrix Polynomials

We find necessary and sufficient conditions for the existence of common unital divisors with given Smith forms of nonsingular matrix polynomials and common factorization of symmetric matrices over rings of polynomials with involution. We obtain conditions for the symmetric equivalence of such matrices.     

On a Compression of Normal Matrix Polynomials

In this article, we study a compression of normal matrices and matrix polynomials with respect to a given vector and its orthogonal complement. The numerical range of this compression satisfies special boundary properties, which are investigated in detail. The characteristic polynomial of the compression is also considered.     

Invariant Operators Between Spaces of h-Monogenic Polynomials

In this paper we study properties of invariant differential operators acting between spaces of h-monogenic functions. In this way one obtains differential operators acting between invariant (m)-modules which can be seen as the Hermitean analogues of the classical Rarita-Schwinger operators. Received: October, 2007. Accepted: February, 2008.     

On Brown Polynomials. II

The results of the author’s previous article are improved. We also establish that each irreducible polynomial over a Henselian valued field is a Brown polynomial.

     

Conversion of Polynomials between Different Polynomial Bases

The purpose of this paper is to derive a recursive scheme forthe evaluation of the coefficients in the expansion , in terms of the coefficients in the expansion , where both q_k(x) and Q_k(x) are polynomials in xof degree k, and where both q_k(x) and Q_k{x} satisfy recursionformulae of the type satisfied by orthogonal polynomials. Thesets {Q_k(x)} and {q_k(x)} need not be orthogonal polynomials,though they usually are in the applications. An applicationis made to the evaluation of integrals with oscillatory andsingular integrands.     

On Computing Zeros of a Bivariate Bernstein Polynomial

1.IntroductionIn[6]and[4],theproblemoffindingtheintersectionofacubicB6zierpatchandaplanewasconsidered.[6]consideredarectangular,and[41atriangularpatch.SincetheBernsteinoperatorB.:f-Bn(f)preserveslinearfunctions,theproblemwassimplifiedtothecomputationofzerosofabivariateBernsteinpolynomialB.(f).BothpaPersproducedsimpleandefficientcomputationalalgorithms.Itisbaseduponthefollowingidea:determinethepointswhereinsidethesupportthetopologyofzerosofB.(f)changes.Thiswasdonebyrestrictingthebivariatepo…     

一类矩阵多项式秩的恒等式

得到了一类矩阵多项式秩的恒等式,改进和推广了近期的一些结果.     

单纯形上的Stancu多项式与最佳多项式逼近

作为Bernstein多项式的推广,本文定义单纯形上的多元Stancu多项式.以最佳多项式逼近为度量,建立Stancu多项式对连续函数的逼近定理与逼近阶估计,给出Stancu多项式的一个逼近逆定理,从而用最佳多项式逼近刻划Stancu多项式的逼近特征.     

关于与给定矩阵可交换的矩阵的多项式表示

文[1]得到:若矩阵A的Jordan标准形中没有纯量矩阵的Jordan块,那么AB=BA的充要条件为B可以化为A的n-1次多项式.本文指出这个结论是错误的.在已有相关文献的基础上,得到了与给定矩阵A可交换的矩阵B是A的多项式的十个等价刻划.     

Dividing Polynomials Using the Resultant Matrix

Let f, g be polynomials over a Noetherian ring A. We use the matrix coming from the resultant of f and g to get a criterion for divisibility of f by g in terms of Fitting invariants as well as a method of dividing polynomials once we know g divides f. Further we show that this is equivalent to that cokernel or the image of the multiplication-by-g map on A[X]/(f) is free. As an application we show one can test irreducibility of an integral polynomial by computing minors of a matrix.     

Invariant symmetric restrictions of a self-adjoint operator. II

We give a criterion of invariance and symmetry of the restriction of an arbitrary unbounded self-adjoint operator in the space L ₂(ℝⁿ, dx) by using the introduced notion of support of an arbitrary vector and the notion of capacity of a subspace N ⊂ ℝⁿ. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 6, pp. 781–791, June, 1998. This work was partially supported by the Foundation for Fundamental Research of the Ministry of Science and Technology of the Ukraine (grant No. 1/238 “Operator”).     

On the Nontriviality of the Polynomial Invariant of Two-Component Link

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Random Perturbations of Matrix Polynomials

Journal of Theoretical Probability - A sum of a large-dimensional random matrix polynomial and a fixed low-rank matrix polynomial is considered. The main assumption is that the resolvent of the...     

On the Stability Radius of Matrix Polynomials

The stability radius of a matrix polynomial P ( λ) relative to an open region Ωof the complex plane and its relation to the numerical range of P ( λ) are investigated. Using an expression of the stability radius in terms of λon the boundary of Ωand ‖P ( λ) -1 ‖2 , a lower bound is obtained. This bound for the stability radius involves the distances of Ωto the connected components of the numerical range of P ( λ) and can be applied in conjunction with polygonal approximations of the numerical range. The special case of hyperbolic matrix polynomials is also considered.     

Computing the Matrix Cosine

An algorithm is developed for computing the matrix cosine, building on a proposal of Serbin and Blalock. The algorithm scales the matrix by a power of 2 to make the -norm less than or equal to 1, evaluates a Padé approximant, and then uses the double angle formula cos(2A)=2cos(A)²–I to recover the cosine of the original matrix. In addition, argument reduction and balancing is used initially to decrease the norm. We give truncation and rounding error analyses to show that an [8,8] Padé approximant produces the cosine of the scaled matrix correct to machine accuracy in IEEE double precision arithmetic, and we show that this Padé approximant can be more efficiently evaluated than a corresponding Taylor series approximation. We also provide error analysis to bound the propagation of errors in the double angle recurrence. Numerical experiments show that our algorithm is competitive in accuracy with the Schur–Parlett method of Davies and Higham, which is designed for general matrix functions, and it is substantially less expensive than that method for matrices of -norm of order 1. The dominant computational kernels in the algorithm are matrix multiplication and solution of a linear system with multiple right-hand sides, so the algorithm is well suited to modern computer architectures.