An efficient division algorithm for polynomial matrices

A new algorithm for carrying out the division of two polynomial matrices is presented. The quotient and remainder are directly computed in a way similar to scalar polynomial division. This method is much simpler than earlier algorithms.     

Inverse of polynomial matrices in the irreducible form

An algorithm is developed for finding the inverse of polynomial matrices in the irreducible form. The computational method involves the use of the left (right) matrix division method and the determination of linearly dependent vectors of the remainders. The obtained transfer function matrix has no nontrivial common factor between the elements of the numerator polynomial matrix and the denominator polynomial.     

Stabilisation of matrix polynomials

A state feedback is proposed to analyse the stability of a matrix polynomial in closed loop. First, it is shown that a matrix polynomial is stable if and only if a state space realisation of a ladder form of certain transfer matrix is stable. Following the ideas of the Routh–Hurwitz stability procedure for scalar polynomials, certain continued-fraction expansions of polynomial matrices are carrying out by unimodular matrices to achieve the Euclid’s division algorithm which leads to an extension of the well-known Routh–Hurwitz stability criteria but this time in terms of matrix coefficients. After that, stability of the closed-loop matrix polynomial is guaranteed based on a Corollary of a Lyapunov Theorem. The sufficient stability conditions are: (i) The matrices of one column of the presented array must be symmetric and positive definite and (ii) the matrices of the cascade realisation must satisfy a commutative condition. These stability conditions are also necessary for matrix polynomial of second order. The results are illustrated through examples.     

An algorithm for the division of two polynomial matrices

This note presents a computational procedure for the division of two polynomial matrices. It is achieved by solving a set of linear algebraic equations. This method is much simpler than the existing one.     

Matrix structures in parallel matrix computations

We survey certain techniques such as approximation, interpolation and embedding in matrix algebra, which are particularly useful to devise fast parallel algorithms for some matrix-structured problems. The problems we consider are triangular Toeplitz matrix inversion and its algebraic counterpart polynomial division, matrix multiplication, and some computations concerning banded Toeplitz matrices.     

n-D polynomial matrix equations

Linear matrix equations in the ring of polynomials in n indeterminates (n-D) are studied. General- and minimum-degree solutions are discussed. Simple and constructive, necessary and sufficient solvability conditions are derived. An algorithm to solve the equations with general n-D polynomial matrices is presented. It is based on elementary reductions in a greater ring of polynomials in one indeterminate, having as coefficients polynomial fractions in the other n-1 indeterminates, which makes the use of Euclidean division possible     

A Region-Dividing Technique for Constructing the Sum-of-Squares Approximations to Robust Semidefinite Programs

In this technical note, we present a novel approach to robust semidefinite programs, of which coefficient matrices depend polynomially on uncertain parameters. The approach is based on approximation with the sum-of-squares polynomials, but, in contrast to the conventional sum-of-squares approach, the quality of approximation is improved by dividing the parameter region into several subregions. The optimal value of the approximate problem converges to that of the original problem as the resolution of the division becomes finer. An advantage of this approach is that an upper bound on the approximation error can be explicitly obtained in terms of the resolution of the division. A numerical example on polynomial optimization is presented to show usefulness of the present approach.      

Novel theory for polynomial and rational matrices at infinity. Part1. Polynomial matrices

A new approach for dealing with the structural properties of polynomial and rational matrices at infinity is presented. In this first part of the development, the attention is focused on polynomial matrices. Some fundamental notions and definitions such as the notion of homogeneous form for polynomial matrices are introduced. These enable us to define important concepts such as the cancellation at infinity, coprimeness at infinity, etc. for polynomial matrices in precisely the same way as for their finite counterparts. We demonstrate that our approach allows interesting generalizations of some important conventional concepts.     

Division-free computation of subresultants using Bezout matrices

We present an algorithm to compute the subresultant sequence of two polynomials that completely avoids division in the ground domain, generalizing an algorithm given by Abdeljaoued et al. [J. Abdeljaoued, G. Diaz-Toca, and L. Gonzalez-Vega, Minors of Bezout matrices, subresultants and the parameterization of the degree of the polynomial greatest common divisor, Int. J. Comput. Math. 81 (2004), pp. 1223–1238]. We evaluate determinants of slightly manipulated Bezout matrices using the algorithm of Berkowitz. Although the algorithm gives worse complexity bounds than pseudo-division approaches, our experiments show that our approach is superior for input polynomials with moderate degrees if the ground domain contains indeterminates.     

An algebraic method to determine the common divisor,poles and transmission zeros of matrix transfer functions

A purely algebraic method which uses the matrix Routh algorithm and its reverse process of the algorithm is presented to decompose a matrix transfer function into a pair of right co-prime polynomial matrices or left co-primo polynomial matrices. The poles and transmission zeros of the matrix transfer function are determined from a pair of relatively prime polynomial matrices. Also, the common divisor of two matrix polynomials can be obtained by using the matrix Routh algorithm and the matrix Routh array.     

A note on the Wolovich method of extraction of a greatest common divisor of two polynomial matrices

In this note, a modification of the Wolovich method of extraction of a greatest common divisor of two noncoprime polynomial matrices is proposed, which demands less computational operations and requires dealing with matrices of lower dimensions. The modified method allows us to establish the whole class of greatest common divisors of the given pair of polynomial matrices.     

New Algorithms for Polynomial J-Spectral Factorization

In this paper new algorithms are developed for J-spectral factorization of polynomial matrices. These algorithms are based on the calculus of two-variable polynomial matrices and associated quadratic differential forms, and share the common feature that the problem is lifted from the original one-variable polynomial context to a two-variable polynomial context. The problem of polynomial J-spectral factorization is thus reduced to a problem of factoring a constant matrix obtained from the coefficient matrices of the polynomial matrix to be factored. In the second part of the paper, we specifically address the problem of computing polynomial J-spectral factors in the context of H _∞ control. For this, we propose an algorithm that uses the notion of a Pick matrix associated with a given two-variable polynomial matrix. Date received: January 1, 1998. Date revised: October 15, 1998.     

Algorithm of J-spectral factorization of polynomial matrices

Numerical algorithms for the canonical J-factorization of polynomial matrices with respect to the imaginary axis are given. The factorization problems for the non-regular polynomial matrices and for polynomials, whose determinants have zero roots, are considered. The constructed algorithms are not connected to the finding of roots of the polynomial. The basic calculation procedure is the construction of the stabilizing solution of the matrix algebraic Riccati equation.     

Inversion of polynomial matrices by interpolation

Known polynomial interpolation methods to polynomial matrices are generalized to obtain new algorithms for the computation of the inverse of such matrices. The algorithms use numerically stabilizable manipulations of constant matrices. Among the three methods investigated Lagrange's interpolation seems especially suitable for the purpose     

两个多项式阵最大公因子的计算

本文给出了用对多项式阵的系数阵进行初等变换的方法确定两个多项式阵的最大公因 子及其互质部分的一种算法.     

Regulator problem with internal stability: A frequency domain solution

This paper considers the general regulator problem with internal stability where the measured outputs are not necessarily the same as the regulated outputs. Using polynomial matrix techniques, necessary and sufficient conditions are obtained in terms of skew-primeness of two polynomial matrices; one of these polynomial matrices represents the disturbance modes, whereas the other is the polynomial system matrix representing the system zeros. Various special cases considered in the literature are also analyzed in terms of these necessary and sufficient conditions.