Polynomial matrix

Abstract: Two notions of polynomml time reduclbihty, denoted here by ~ T e and <.~P, were defined by Cook and Karp, respectively The abstract propertms of these two relatmns on the domain of computable sets are investigated. Both relations prove to be dense and to have minimal pairs. Further , there is a strictly ascending sequence with a minimal pair of upper bounds to the sequence. Our method of showing density ymlds the result that if P ~ NP then there are members of NP -P that are not polynomml complete

Abstract: Many problems in electrical engineering, such as the synthesis of linear n ports and the detection and filtration of multivariable systems corrupted by stationary additive noise, depend for their successful solution upon the factorization of a matrix-valued function of a complex variable p . This paper presents several algorithms for affecting such decompositions for the class of rational matrices G(p) , i.e., matrices whose entries are ratios of polynomials in p . The methods employed are elementary in nature and center around the Smith canonic form of a polynomial matrix. Several nontrivial examples are worked out in detail to illustrate the theory.

Abstract: The parallel arithmetic complexities of matrix inversion, solving systems of linear equations, computing determinants and computing the characteristic polynomial of a matrix are shown to have the same growth rate. Algorithms are given that compute these problems in $O(\log ^2 n)$ steps using a number of processors polynomial in n. (n is the order of the matrix of the problem.)