Improved Bounds on Polynomial Spectral Radius with Applications to Stability: A Recent Study

In this work we assumed that f(z) is a monic complex polynomial of degree n whose roots zi, i=1, ..., n satisfy |z1| ... |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:= 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:= . 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 , we will show how to overcome this restriction by considering the polynomial f(z ; ):= n f(z / ) whose roots are zi, i=1, ..., n and its spectral radius is R. We identify the upper bounds of f(z ; ) by R1( ) and RM( ), respectively. Hence, R R1( ):= R1( ) / and R \leq R_M( ):=RM( ) / . We call R1( ) and RM( ) the -method upper bounds on R which for >1 are at least 1 / , thus removing the restriction that R1, RM 1. We give a specific example in which R<1, R1( 1), RM( 2)<1, 1 2, and 1, 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 {R1, RC, RM} associated with h(z):=gm(z) f (z) depends explicitly on xm, i.e., Ru 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 , say . For we could show that and or vice versa. We will present three examples (i) demonstrating the improvements obtained by the LP-method; (ii) an explicit solution of the LP problem for ; and, (iii) an example for which the LP-method does not improve for . We will also show how to further improve the LP-method by combining it with the -method. Notice that pertains to the LP-method, where and without the brackets pertains to the -method. A special case of the LP-method is the beautiful Kakeya Theorem, i.e., if and then is a global minimum of . We will also show that if and then . Next, if we obtain another Kakeya-type Theorem, where is another global minimum of . We will show that this case is associated with whose zeros preserve the spectral radius of . We will also talk about Kakeya's Theorem stability problems when it is applicable for -dimensional (N ) discrete shift-invariant linear systems. For , Jury's cited book on the -transform states that if then and consequently is stable. For we could sharpen Rudin's Theorem and use it in conjunction with Kakeya's Theorem when the latter is applicable. Finally, we present lower bounds, say , on by using Vieta's formulas which imply that if then the 1-D linear time-invariant discrete system corresponding to is not stable. Simulations show that if increases the condition is more often satisfied. The investigation into Vieta's formulas led us to a disturbing occurrence when we computed the coefficients of a high degree stable polynomial from its given roots. Specifically, we obtained that many of 's trailing coefficients where in the interval [-eps, eps], where eps is Matlab's constant. Furthermore, we arrived at two hypotheses based on simulation results.