On minor prime factorizations for multivariate polynomial matrices

Multidimensional Systems and Signal Processing - Multivariate polynomial matrix factorizations have been widely investigated during the past years due to the fundamental importance in the areas of...     

On factor left prime factorization problems for multivariate polynomial matrices

This paper is concerned with factor left prime factorization problems for multivariate polynomial matrices without full row rank. We propose a necessary and sufficient condition for the existence of factor left prime factorizations of a class of multivariate polynomial matrices, and then design an algorithm to compute all factor left prime factorizations if they exist. We implement the algorithm on the computer algebra system Maple, and two examples are given to illustrate the effectiveness of the algorithm. The results presented in this paper are also true for the existence of factor right prime factorizations of multivariate polynomial matrices without full column rank.

     

Factorizations for a class of multivariate polynomial matrices

Multidimensional Systems and Signal Processing - This paper investigates how to factorize a class of multivariate polynomial matrices. We prove that an $$l\times m$$ multivariate polynomial matrix...     

Direct synthesis of discrete-time feedback systems by equivalence transformations of polynomial matrices

It is shown that the synthesis of closed-loop linear multivariable discrete-time systems can be directly effected by performing equivalence transformations on appropriate polynomial matrices. These polynomial matrices are the Smith canonical forms of the closed-loop characteristic matrices of such systems subject to the constraints imposed by the fundamental theorem of linear state-variable feedback.     

Relatively prime polynomial matrices

Relatively prime polynomial matrices are defined, and some of their properties are deduced and applied. The problem of extending these properties is posed.     

Factorizations fornD polynomial matrices

In this paper, a constructive general matrix factorization scheme is developed for extracting a nontrivial factor from a givennD (n>2) polynomial matrix whose maximal order minors satisfy certain conditions. It is shown that three classes ofnD polynomial matrices admit this kind of general matrix factorization. It turns out that minor prime factorization and determinantal factorization are two interesting special cases of the proposal general factorization. As a consequence, the paper provides a partial solution to an open problem of minor prime factorization as well as to a recent conjecture on minor prime factorizability fornD polynomial matrices. Three illustrative examples are worked out in detail.     

Generation of polynomial system matrices

It is shown that certain polynomial system matrices giving rise to a given transfer-function matrix can all be generated by a defined transformation from a representation of least order.     

Inversion of rational polynomial matrices

The letter is concerned with automatic computational procedures for the inversion of a matrix where every element is a rational function. Two possible methods are considered: one based on the Gaussian elimination algorithm and the other based on the Faddeev algorithm.     

Detecting infinite zeros in polynomial matrices

A simple necessary and sufficient algebraic condition is given to detect zeros at infinity in a polynomial matrix.     

Some properties of relatively prime polynomial matrices

Some necessary and sufficient conditions are given for the polynomial matrices sIn, ? A and B to be relatively prime.     

Spectral factorization of two-variable para-Hermitian polynomial matrices

This paper presents an algorithm for the so-called spectral factorization of two-variable para-Hermitian polynomial matrices which are nonnegative definite on thej axis, arising in the synthesis of two-dimensional (2-D)passive multiports, Wiener filtering of 2-D vector signals, and 2-D control systems design. First, this problem is considered in the scalar case, that is, the spectral factorization of polynomials is treated, where the decomposition of a two-variable nonnegative definite real polynomial in a sum of squares of polynomials in one of the two variables having rational coefficients in the other variable plays an important role (cf. Section 4). Second, by using these results, the matrix case can be accomplished, where in a first step the problem is reduced to the factorization of anunimodular para-Hermitian polynomial matrix which is nonnegative definite forp=j , and in a second step this simplified problem is solved by using so-called elementary row and column operations which are based on the Euclidian division algorithm. The matrices considered may be regular or singular and no restrictions are made concerning the coefficients of their polynomial entries; they may be either real or complex.     

Generalized inverses of two-variable polynomial matrices and applications

The main contribution of this paper is to present (a) an algorithm for the computation of the generalized inverse of a not necessarily square two-variable polynomial matrix and (b) some applications of the proposed algorithm to the solution of Diophantine equations.This work is supported by the Greek General Secretariat of Industry, Research and Technology.     

隐私保护的可验证多元多项式外包计算方案

随着云计算的发展和大数据时代的到来,如何对隐私数据进行外包计算且有效验证计算结果具有重要的现实意义。基于多线性映射和同态加密方案,提出了可验证的多元多项式外包计算方案,用户可准确验证外包计算结果的正确性。方案在标准模型中可证安全,且多项式函数和用户输入对于服务器都是保密的。分析表明,用户计算量远小于服务器的计算代价以及直接计算多项式函数。     

Bounds for the degrees of the entries of left inverses of polynomial matrices

We provide explicit bounds for the degrees of the polynomials which appear as the entries of the left inverse of a polynomial matrixF. When such an inverse does not exist, bounds can be given for the entries of the (1)-inverse ofF.     

On the ranks of certain matrices

A few basic results regarding the ranks of certain types of matrices are presented. Also, some bounds on the ranks of the real and imaginary parts of a positive semi-definite Hermitian matrix are worked out. Applications of particular cases of these results in the areas like passive network synthesis, generation of two-variable reactance matrices, etc., are pointed out.     

On the computation of discrete Legendre polynomial coefficients

A new and fast method to find the discrete Legendre polynomial (DLP) coefficients is presented. The method is based on forming a simple matrix using addition only and then multiplying two elements of the matrix to compute the DLP coefficients.     

On the equivalence between SOVA and max-log-MAP decodings

It is shown that after a proper simple modification, the soft-output Viterbi algorithm (SOVA) proposed by Hagenauer and Hoeher (1989) becomes equivalent to the max-log-maximum a posteriori (MAP) decoding algorithm. Consequently, this modified SOVA allows to implement the max-log-MAP decoding algorithm by simply adjusting the conventional Viterbi algorithm. Hence, it provides an attractive solution to achieve low-complexity near-optimum soft-input soft-output decoding     

On the equivalence of PDA algorithm and SIC-MMSE algorithm

In this letter, by employing Gaussian distribution to approximate the probability density function （pdf） of the extrinsic information at the output of the multiuser detector as a function of the pdf of the input extrinsic messages, it is concluded that the Probabilistic Data Association （PDA） algorithm is equivalent to the Soft Interference Cancellation plus Minimum Mean Square Error algorithm （SIC-MMSE）.     

On the equivalence of set-theoretic and maxent MAP estimation

We establish an equivalence between two conceptually different methods of signal estimation under modeling uncertainty, viz., set-theoretic (ST) estimation and maximum entropy (maxent) MAP estimation. The first method assumes constraints on the signal to be estimated, and the second assumes constraints on a probability distribution for the signal. We provide broad conditions under which these two estimation paradigms produce the same signal estimate. We also show how the maxent formalism can be used to provide solutions to three important problems: how to select sizes of constraint sets in ST estimation (the analysis highlights the role of shrinkage); how to choose the values of parameters in regularized restoration when using multiple regularization functionals; how to trade off model complexity and goodness of fit in a model selection problem.