A hybrid approach to some stability problems for bivariate polynomials

Frequently, in control system design, we are asked to locate the roots of a bivariate polynomial of the following form \begin{eqnarray*} H(s,k)=\sum_{i=0}^n Q_i(s) k^i\in {\bf Z}[k,s]={\bf Z}[k][s]={\bf D}[s], \end{eqnarray*} where $Q_i(s)\in {\bf Z}[s]$ for each $i$ and, moreover, $k$ is a free parameter ranging in some real interval. For a fixed value $\bar k$ of $k$, the zero distribution of the univariate polynomial $p(s)=H(s,\bar k)$ with respect to the imaginary axis can be found by determining the inertia of a Bezout matrix $B=B(\bar k)$ whose entries are expressed in terms of the coefficients of $p(s)$. This evaluation is usually accomplished by computing a block factorization of $B$, namely, $U^TBU=D$ where $D$ is a block diagonal matrix with lower triangular blocks with respect to the antidiagonal. It is intended in this paper to propose an efficient hybrid approach for determining the zero-distribution of $H(s,k)$ with respect to the imaginary axis for any value of $k$. We develop a fast fraction-free method for factoring the Bezout matrix $B(k)$ with entries over ${\bf D}$ determined by $H(s,k)\in {\bf D}[s]$. In this way, we easily compute the sequence $\{\phi_i(k)\}$ of the trailing principal minors of $B(k)$. For almost any value $\bar k$ of $k$ the associated sign sequence $\{sign(\phi_i(\bar k))\}$ specifies the inertia of $B(\bar k)$ and, therefore, the zero-distribution of $H(s,\bar k)$. The function $sign(\phi_i( k))$ is finally obtained by numerically computing rational approximations of the real zeros of $\phi_i(k)\in {\bf Z}[k]$.