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AN INFINITE FAMILY OF BOUNDS ON ZEROS OF ANALYTIC FUNCTIONS AND RELATIONSHIP TO SMALE’S BOUND
"... Abstract. Smale’s analysis of Newton’s iteration function induce a lower bound on the gap between two distinct zeros of a given complexvalued 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 o ..."
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Abstract. Smale’s analysis of Newton’s iteration function induce a lower bound on the gap between two distinct zeros of a given complexvalued 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,whereB2(z) coincides with Newton’s, our lower bound is more than twice as good as Smale’s bound or its improved version given by Blum, Cucker, Shub, and Smale. When f(z) isacomplex polynomial of degree n, for small m the corresponding bound is computable in O(n log n) arithmetic operations. For quadratic polynomials, as m increases the lower bounds converge to the actual gap. We show how to use these bounds to compute lower bounds on the distance between an arbitrary point and the nearest root of f(z). In particular, using the latter result, we show that, given a complex polynomial f(z) =anzn + ···+ a0, ana0 ̸ = 0,foreach m ≥ 2 we can compute upper and lower bounds Um and Lm such that the roots of f(z) lie in the annulus {z: Lm ≤z  ≤Um}. In particular, L2 =
Polynomiography: A New Intersection between Mathematics and Art 1
"... Polynomiography is defined to be “the art and science of visualization in approximation of the zeros of complex polynomials, via fractal and nonfractal images created using the mathematical convergence properties of iteration functions. ” An individual image is called a “polynomiograph.” The word p ..."
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Polynomiography is defined to be “the art and science of visualization in approximation of the zeros of complex polynomials, via fractal and nonfractal images created using the mathematical convergence properties of iteration functions. ” An individual image is called a “polynomiograph.” The word polynomiography is a combination of the word “polynomial ” and the suffix “graphy. ” It is meant to convey the idea that it represents a certain graph of polynomials, but not in the usual sense of graphing, say a parabola for a quadratic polynomial. Polynomiographs are obtained using algorithms requiring the manipulation of thousands of pixels on a computer monitor. Depending upon the degree of the underlying polynomial, it is possible to obtain beautiful images on a laptop computer in less time than a TV commercial. Polynomials form a fundamental class of mathematical objects with diverse applications; they arise in devising algorithms for such mundane task as multiplying two numbers, much faster than the ordinary way we have all learned to do this task (FFT). According to the Fundamental Theorem of Algebra, a polynomial of degree n, with real or complex coefficients, has n zeros (roots) which may or may not be distinct. The task of approximation of the zeros of polynomials is a problem that was known to Sumerians (third millennium B.C.). This problem
Can polynomiography be useful in computational geometry
 Rutgers University
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A COMBINATORIAL CONSTRUCTION OF HIGH ORDER ALGORITHMS FOR FINDING POLYNOMIAL ROOTS OF KNOWN MULTIPLICITY
"... Abstract. We construct a family of high order iteration functions for finding polynomial roots of a known multiplicity s. This family is a generalization of a fundamental family of high order algorithms for simple roots that dates back to Schröder’s 1870 paper. It starts with the well known variant ..."
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Abstract. We construct a family of high order iteration functions for finding polynomial roots of a known multiplicity s. This family is a generalization of a fundamental family of high order algorithms for simple roots that dates back to Schröder’s 1870 paper. It starts with the well known variant of Newton’s method ˆB2(x) =x − s · p(x)/p ′ (x) and the multiple root counterpart of Halley’s method derived by Hansen and Patrick. Our approach demonstrates the relevance and power of algebraic combinatorial techniques in studying rational rootfinding iteration functions. 1.