Parallel computation of determinants of matrices with polynomial entries for robust control design

Abstract

In this paper we consider computing determinants of polynomial matrices symbolically. Determinant computation of matrices with polynomial entries in a small number of variables is of particular interest since it commonly appears in solving engineering design problems. A parallel algorithm based on multivariate Newton polynomial interpolation with "cut-surface" (total degree bound) is presented and its efficiency is demonstrated by showing computational results for some practical examples from control system design.

Parallel computation of determinants of matrices with multivariate polynomial entries

“…Note that the numbers of terms of f i , i = −1, 0, 1, 2 are 10808, 1867, 10, 10 respectively, and their degrees with respect to L, a, b, t 1 , t 2 , t 3 can be written as: [8,10,10,9,12,12], [8,6,6,8,8,8] , [0, 2, 2, 1, 4, 1], and [0, 2, 2, 1, 1, 4]. By using the built-in functions Groebner Basis, Resultant of Maple 13 on the IBM 3950M2 cluster as mentioned in the Example 1, we were not able to reach the final point in one month calculation.…”

“…in multivariate interpolation, it is not easy for this method to avoid too-large integers in the interpolation, and it is known that the performance for computing determinants of dense matrices with integer entries closely depends on the average length of integers. In 2010, Kumura et al [10] presented a parallel algorithm for determinant computation based on multivariate Newton polynomial interpolation, but their method is also restricted to polynomials with two and three variables. This paper is devoted to an extension of Bjöck and Pereyra's algorithm to general case.…”