A note on a classical bound for the moduli of all zeros of a polynomial

Bounds for the Effective Interest Rate for Constant, Periodic Annuities

“…Recent treatments of problems around ours encompass those of Herzberger (et al) [3,4,5]. For the occurence of the root with largest modulus under all zeros, q n (a), in other contexts, see [1,2,5]. Proofs will be given elsewhere.…”

“…[8]. We propound an inititial enclosure [i n (a) such that ∆i (1) n (a) < 0.0007% at small numerical costs. The calculation procedure is designed for evaluation on (+, −, * , /, √ )-pocket calulators.…”

Generalizations of Gershgorin disks and polynomial zeros

“…316-319]), there exist several more modern methods, such as in [1], [5], [11], [12], [15], [20], [21], [22], [25], [26], [27], [28], [34], [37], [38], which were already mentioned in the introduction. Judging from the numerical examples in those references, these bounds are, by and large, comparable.…”

“…The method from [20] only computes an upper bound on the moduli of the zeros. To illustrate this additional information our bounds can sometimes provide, let us consider q 1 (z) = z 5 We conclude by mentioning that it may be possible to extend our techniques to compute bounds on the real and imaginary parts of polynomial zeros as well. Furthermore, more bounds can be obtained by considering powers of the companion matrix and of its transpose.…”

“…The literature on the computation of polynomial zeros and bounds on such zeros is extensive, and we refer to [1], [4], [5], [6], [11], [12], [13], [15], [16], [20], [21], [22], [25], [26], [27], [28], [31], [32], [34], [37], [38], and the references therein, to name but a few. Most of these take a linear algebra approach, but some do not.…”