Results 1  10
of
52
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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
Newton's Method and Generation of a Determinantal Family of Iteration Functions
, 1998
"... It is wellknown that Halley's method can be obtained by applying Newton's method to the function f/ # f # . Gerlach [3], gives a generalization of this approach, and for each m # 2, recursively defines an iteration function Gm (x) having order m. Kalantari et al. [6], and Kalantari [8] ..."
Abstract

Cited by 7 (7 self)
 Add to MetaCart
It is wellknown that Halley's method can be obtained by applying Newton's method to the function f/ # f # . Gerlach [3], gives a generalization of this approach, and for each m # 2, recursively defines an iteration function Gm (x) having order m. Kalantari et al. [6], and Kalantari [8
A Note on a Boundedness Property of Normal Barriers of Convex Cones
"... Let K be a closed convex pointed cone in a finite dimensional Hilbert space E. Assume K has a nonempty interior K # . Let r = inf{#x, y# : x, y # K, #x# = #y# = 1}, where #·, ·# denotes the inner product and # · # the corresponding induced norm. The quantity r gives a measure of obtusenes ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
, then # # 1 + # 2. We prove the above using a property from the theory of selfconcordance of Nesterov and Nemirovskii [3], and a result in Kalantari [2]. The existence of this bound implies that the polynomialtime potentialreduction and pathfollowing algorithms which were described in [2], for self
MATRIX SCALING DUALITIES IN CONVEX PROGRAMMING
, 2005
"... We consider convex programming problems in a canonical homogeneous format, a very general form of Karmarkar’s canonical linear programming problem. More specifically, by homogeneous programming we shall refer to the problem of testing if a homogeneous convex function has a nontrivial zero over a s ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
of Khachiyan and Kalantari for linear programming, as well as for quasi doubly stochastic scaling of Q, i.e. computing D such that DQDe = e. Our general results here give nontrivial extensions of our previous work on the role of matrix scaling in linear or
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 ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
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 =
On Homogeneous Linear Recurrence Relations and Approximation of Zeros of Complex Polynomials
 Department of Computer Science, Rutgers University
, 2000
"... . 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)Dm2 (a)/Dm1 (a)}, where Dm (a) is defined via a homogeneous linear recurrence relation and depends only on the normalized derivatives p (i) (a)/i!. Each ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
. 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)Dm2 (a)/Dm1 (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 possibly for the locus of points equidistant to two distinct roots, given any input a, the Basic Sequence converges to a root of p. The roots of p partition the Euclidean plane into Voronoi regions. Under some regularity assumption (e.g. simplicity of the roots), for almost all inputs within the Voronoi polygon of a root, the corresponding Basic Sequence converges to that root. The discovery of the Basic Sequence, its error estimates, and several of its properties are consequences of our previous analysis of a fundamental family of iteration functions {Bm(z)}, called the Basic Family. Given any fixed m # 2 and an appropriat...
Scaling Dualities And SelfConcordant Homogeneous Programming In Finite Dimensional Spaces
, 1998
"... . In this paper first we prove four fundamental theorems of the alternative, called scaling dualities, characterizing exact and approximate solvability of four significant conic problems in finite dimensional spaces, defined as: homogeneous programming (HP), scaling problem (SP), homogeneous scali ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
. In this paper first we prove four fundamental theorems of the alternative, called scaling dualities, characterizing exact and approximate solvability of four significant conic problems in finite dimensional spaces, defined as: homogeneous programming (HP), scaling problem (SP), homogeneous scaling problem (HSP), and algebraic scaling problem (ASP). Let # be a homogeneous function of degree p > 0, K a pointed closed convex cone, W a subspace, and F a #logarithmically homogeneous barrier for K # . HP tests the existence of a nontrivial zero of # over W # K. SP, and HSP test the existence of the minimizer of # = # + F , and X = #/ exp(pF/#) over W # K # , respectively. ASP tests the solvability of the scaling equation (SE), a fundamental equation inherited from properties of homogeneity and those of the operatorcone, T (K) = {D # F ## (d) 1/2 : d # K # }. Each D induces a scaling of # # (d) (or # ## (d)), and SE is solvable if and only if there exists a fixedpoint under...
A Simple PathFollowing Algorithm for the Feasibility Problem in Semidefinite Programming and for Matrix Scaling over the Semidefinite Cone
"... Let E be the Hilbert space of symmetric matrices of the form diag(A, M ), where A is n n, and M is an l l diagonal matrix, and the inner product #x, y# # T race(xy). Given x # E, we write x # 0 (x > 0) if it is positive semidefinite (positive definite). Let Q : E # E be a symmetri ..."
Abstract
 Add to MetaCart
Let E be the Hilbert space of symmetric matrices of the form diag(A, M ), where A is n n, and M is an l l diagonal matrix, and the inner product #x, y# # T race(xy). Given x # E, we write x # 0 (x > 0) if it is positive semidefinite (positive definite). Let Q : E # E be a symmetric positive semidefinite linear operator, and = min{#(x) = 0.5T race(xQx) : #x# = 1, x # 0}. The feasibility problem in SDP can be formulated as the problem of testing if = 0 for some Q. Let # # (0, 1) be a given accuracy, u = Qe  e, e the identity matrix in E, and N = n + l. We describe a simple pathfollowing algorithm that in case = 0, in O( # N ln[N#u#/#]) Newton iterations produces x # 0, #x# = 1 such that T race(xQx) # #. If > 0, in O( # N ln[N#u#/#]) Newton iterations the algorithm produces d > 0 such that #DQDe  e# # #, where D is the operator that maps w # E to d 1/2 wd 1/2 . Moreover, we use the algorithm to prove: > 0, if and only if there exists d > 0...
unknown title
"... Polynomials are present in every branch and mathematics and the sciences with numerous fundamental applications and new ones continue to emerge. However, surprisingly there are seldom courses dedicated to their study. Even so, it would be impossible to exhaust all possible directions. This seminar o ..."
Abstract
 Add to MetaCart
Polynomials are present in every branch and mathematics and the sciences with numerous fundamental applications and new ones continue to emerge. However, surprisingly there are seldom courses dedicated to their study. Even so, it would be impossible to exhaust all possible directions. This seminar offers a novel and modern point of view into theoretical and practical properties of polynomials and their applications in computer science, mathematics, education, art and more. In particular, Polynomiography, algorithmic visualization of polynomial rootfinding, gives rise to many interesting interdisciplinary applications. We will study a subset of a wide range of topics on polynomials and polynomiography and their applications, such as: geometric properties of complex polynomials algorithms for the computation of polynomial roots, including bounds on roots local and global behavior of iteration functions, such as Newton’s method topics in dynamical systems, such as Fatou and Julia sets and fractals in iteration functions negative results on polynomial rootfinding: unsolvability, undecidability, and nonconvergence connections to computational geometry, e.g. Voronoi diagrams combinatorics
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 ..."
Abstract
 Add to MetaCart
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.
Results 1  10
of
52