Unimodular polynomial matrix

Abstract: In this report we describe a subroutine that takes an arbitrary polynomial matrix P as input, and yields on output a unimodular polynomial matrix (I and a column reduced polynomial matrix R such that PU = R. The subroutine is hased 011 the algorithm described in the paper by Neven and Praagman. The subroutine was run on foUl" different computers, with comparable results. \Ve found examples iu which the routhle behaves well, a.s well as examples in which the routine performs poorly, if no precautions a.re takeu. \Ve provide both kinds of examples and discuss the cause of the beha.vior of the routine. From these considerations a guideline for the use of the routine is derived.

Abstract: A polynomial matrix is called column reduced if its column degrees are as low as possible in some sense. Two polynomial matrices P and R are called unimodularly equivalent if there exists a unimodular polynomial matrix U such that PU = R. Every polynomial matrix is unimodularly equivalent to a column-reduced polynomial matrix. In this article a subroutine is described that takes a polynomial matrix P as input and yields on output a unimodular matrix U and a column-reduced matrix R such that PU = R; actually PU - R is near zero. The subroutine is based on an algorithm, described in a paper by Neven and Praagman. The subroutine has been tested with a number of examples on different computers, with comparable results. The performance of the subroutine on every example tried is satisfactory in the sense that the magnitude of the elements of the residual matrix PU - R is about parallel to P parallel to parallel to U parallel to EPS, where EPS is the machine precision. To obtain these results a tolerance, used to determine the rank of some (sub)matrices, has to be set properly. The influence of this tolerance on the performance of the algorithm is discussed, from which a guideline for the usage of the subroutine is derived.

Abstract: A polynomial matrix is called column reduced if its column degrees are as low as possible in some sense. Two polynomial matrices P and R are called unimodularly equivalent if there exists a unimodular polynomial matrix U such that PU = R. Every polynomial matrix is unimodularly equivalent to a column reduced polynomial matrix. In this paper a subroutine is described that takes a polynomial matrix P as input and yields on output a unimodular matrix U and a column reduced matrix R such that PU = R, actually PU − R is near zero. The subroutine is based on an algorithm, described in a paper by Neven and Praagman. The subroutine has been tested with a number of examples on different computers, with comparable results. The performance of the subroutine on every example tried is satisfactory in the sense that the magnitude of the elements of the residual matrix PU − R is about ‖P‖‖U‖EPS, where EPS is the machine precision. To obtain these results a tolerance, used to determine the rank of some (sub)matrices, has to be set properly. The influence of this tolerance on the performance of the algorithm is discussed, from which a guideline for the usage of the subroutine is derived. AMS subject classification: 65F30, 15A23, 93B10, 15A22, 15A24, 15A33, 93B10, 93B17, 93B25 CR subject classification: Algorithms, Reliability, F 2.1 Computations on matrices, Computations on polynomials, G 1.3 Linear systems, G 4 Algorithm analysis