Improved Bounds on Polynomial Spectral Radius with Applications to Stability: A Recent Study
Résumé
In this work we assumed that f(z) is a monic complex polynomial of degree n whose roots zi, i=1, ..., n satisfy |z1| \(\le\) ... \(\le\)|zn|. We define the spectral radius of f(z) by R:=|zn| and we let r:=|z1|. We will introduce upper and lower bounds on R and r in this self-contained article. The upper bounds coincide with two induced matrix norms of f(z)'s companion matrix, say C. I.e., the norm-one bound, say R1:=\(\|c\|\)1; Cauchy's bound, say RC, which is a trivial upper bound on R1; and, the norm-infinity bound also called Montel's bound RM:= \(\|c\|\)\(\infty\). The significance of Cauchy's bound lies in the proof technique it implies the sharper bound R < RC. In order to create new proofs, we shall enhance Cauchy's proof method that sharpen the other two bounds R1 and RM. Noting that these bounds are unnaturally restricted by R1, RC, RM \(\ge\), we will show how to overcome this restriction by considering the polynomial f(z ; \(\beta\)):= \(\beta\)n f(z / \(\beta\)) whose roots are \(\beta\)zi, i=1, ..., n and its spectral radius is \(\beta\)R. We identify the upper bounds of f(z ; \(\beta\)) by R1(\(\beta\)) and RM(\(\beta\)), respectively. Hence, R \(\le\) R1(\(\beta\)):= R1(\(\beta\)) / \(\beta\) and R \leq R_M(\(\beta\)):=RM(\(\beta\)) / \(\beta\). We call R1(\(\beta\)) and RM(\(\beta\)) the \(\beta\)-method upper bounds on R which for\(\beta\)>1 are at least 1 / \(\beta\), thus removing the restriction that R1, RM \(\ge\) 1. We give a specific example in which R<1, R1(\(\beta\)1), RM(\(\beta\)2)<1, \(\beta\)1 \(\neq\) \(\beta\)2, and \(\beta\)1, \(\beta\)2>1.
Next, We'll concentrate on real monic polynomials and demonstrate how to strengthen the bounds R1, RC, and RM. The idea is to multiply f(z) = zn+an-1 zn-1+ ...+a1z+a0 by a series of monic real polynomials, say gm(z)=zm+xm-1 zm-1+ ...+x1z + x0, m=1,2, ...., M. We let xm:= [x0, x1 ...., xm-1]T, thus Ru \(\in\) {R1, RC, RM} associated with h(z):=gm(z) f (z) depends explicitly on xm, i.e., Ru \(\equiv\) Ru {xm). By reducing, we shall acquire the enhanced boundaries Ru(xm) with respect to xm which turns out to be a set of linear programming (LP) problems. The justification to minimize Ru(xm) is because the feasible solution xm=0 for which h(z)=zm f (z) preserves R. As a result, we arrive at a series of LP problems with progressively higher complexity that produce a series of nonincreasing bounds on \(R_u \in\left\{R_C, R_M\right\}\), say \(R_u \geq R_u^{(1)} \geq R_u^{(2)} \geq \cdots \geq R_u^{(M)}\). For \(R_1\) we could show that \(R_1^{(1)} \geq R_1^{(2)} \geq \cdots \geq R_1^{(M)}\) and \(R_1