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ON EFFICIENT COMPUTATION AND ASYMPTOTIC SHARPNESS OF KALANTARI’S BOUNDS FOR ZEROS OF POLYNOMIALS
"... Abstract. We study an infinite family of lower and upper bounds on the modulus of zeros of complex polynomials derived by Kalantari. We first give a simple characterization of these bounds which leads to an efficient algorithm for their computation. For a polynomial of degree n our algorithm compute ..."
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Abstract. We study an infinite family of lower and upper bounds on the modulus of zeros of complex polynomials derived by Kalantari. We first give a simple characterization of these bounds which leads to an efficient algorithm for their computation. For a polynomial of degree n our algorithm computes the first m bounds in Kalantari’s family in O(mn) operations. We further prove that for every complex polynomial these lower and upper bounds converge to the tightest annulus containing the roots, and thus settle a problem raised in Kalantari’s paper. 1.
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.
NewtonEllipsoid Method and its Polynomiography∗
"... We introduce a new iterative rootfinding method for complex polynomials, dubbed NewtonEllipsoid method. It is inspired by the Ellipsoid method, a classical method in optimization, and a property of Newton’s Method derived in [7], according to which at each complex number a halfspace can be found ..."
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We introduce a new iterative rootfinding method for complex polynomials, dubbed NewtonEllipsoid method. It is inspired by the Ellipsoid method, a classical method in optimization, and a property of Newton’s Method derived in [7], according to which at each complex number a halfspace can be found containing a root. NewtonEllipsoid method combines this property, bounds on zeros, together with the planecutting properties of the Ellipsoid Method. We present computational results for several examples, as well as corresponding polynomiography. Polynomiography refers to algorithmic visualization of rootfinding. Newton’s method is the first member of the infinite family of iterations, the basic family. We also consider general versions of this ellipsoid approach where Newton’s method is replaced by a higherorder member of the family such as Halley’s method.
The Quality of Zero Bounds for Complex Polynomials
, 2012
"... In this paper, we evaluate the quality of zero bounds on the moduli of univariate complex polynomials. We select classical and recently developed bounds and evaluate their quality by using several sets of complex polynomials. As the quality of priori bounds has not been investigated thoroughly, our ..."
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In this paper, we evaluate the quality of zero bounds on the moduli of univariate complex polynomials. We select classical and recently developed bounds and evaluate their quality by using several sets of complex polynomials. As the quality of priori bounds has not been investigated thoroughly, our results can be useful to find optimal bounds to locate the zeros of complex polynomials.
Numerical Evaluation and Comparison of Kalantari’s Zero Bounds for Complex Polynomials
, 2014
"... In this paper, we investigate the performance of zero bounds due to Kalantari and Dehmer by using special classes of polynomials. Our findings are evidenced by numerical as well as analytical results. ..."
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In this paper, we investigate the performance of zero bounds due to Kalantari and Dehmer by using special classes of polynomials. Our findings are evidenced by numerical as well as analytical results.