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Smale's analysis of Newton's iteration function induce a lower bound on the gap between two distinct zeros of a given complex-valued analytic function f (z). In this paper we make use of a fundamental family of iteration functions Bm(z), m ≥ 2, to derive an infinite family of lower bounds on the above gap. However, even for m = 2, where B 2 (z) coincides… (More)

Given a set P with n points in R li, its diameter d, is the maximum of the Euclidean distances between its points. We describe an algorithm that in m < n iterations obtains r, < rs <. .). For k fixed, the cost of each iteration is O(n). In particular, the first approximation r, is within fi of dp, independent of the dimension k.

Let p(z) be a complex polynomial of degree n. To each complex number a we associate a sequence called the Basic Sequence {Bm(a) = a − p(a)D m−2 (a)/D m−1 (a)}, where Dm(a) is defined via a homogeneous linear recurrence relation and depends only on the normalized derivatives p (i) (a)/i!. Each Dm(a) is also representable as a Toeplitz determinant. Except… (More)

Polynomiography is the art and science of visualization in approximation of zeros of complex polynomials. Informally speaking polynomiography allows one to take colorful pictures of polynomials. These images can subsequently be recolored in many ways using one's own creativity and artistry. It has tremendous applications in visual arts, education, and… (More)