Kalantari, Bahman

Polynomial root-finding and polynomiography by Bahman Kalantari ( Book )
9 editions published between 2007 and 2009 in English and held by 130 libraries worldwide
This book offers fascinating and modern perspectives into the theory and practice of the historical subject of polynomial root-finding, rejuvenating the field via polynomiography, a creative and novel computer visualization that renders spectacular images of a polynomial equation. Polynomiography will not only pave the way for new applications of polynomials in science and mathematics, but also in art and education. The book presents a thorough development of the basic family, arguably the most fundamental family of iteration functions, deriving many surprising and novel theoretical and practi.

High order iterative methods for approximating square and cube roots by Bahman Kalantari ( Book )
2 editions published in 1993 in English and held by 6 libraries worldwide
Abstract: "In this paper we describe high order iterative methods for computing the q-th root of a given [alpha]> 0, q = 2,3. Given any natural number m [> or =] 2, and any initial approximation x₀> [q root of alpha], we construct a rational function g[subscript m](x) so that the fixed-point iteration x[subscript k+1] = g[subscript m](x[subscript k]), has an m-th order monotonic rate of convergence to [alpha]. In particular for m=2, g₂(x) coincides with Newton's function N(x) = x - p(x)/p'(x), where p(x) = x[superscript q] - [alpha]. For approximation of square roots, parallel implementation of these high order methods allow an asymptotic speedup factor of 3 over Newton's method."

High order iterative methods for approximating roots of polynomials by Bahman Kalantari ( Book )
2 editions published in 1993 in English and held by 6 libraries worldwide
Abstract: "Let p(x) be a polynomial of degree n [> or =] 2 with real coefficients, and K the field of rationals with the adjunction of the coefficients of p. For any natural number m [> or =] 2, we prove the existence of n rational functions g[subscript m](x), and [gamma][subscript i](x), i = m ..., (m+n-2), with coefficients in K so that for all roots [theta] of p(x) we have, [formula]. Furthermore, if p(x) is irreducible over K, g[subscript m](x) is unique. The corresponding fixed-point iteration, x[subscript k+1] = g[subscript m](x[subscript k]), results in high order of convergence to roots of p(x). In particular, for m = 2, we obtain an explicit formula for g₂(x) and prove that it coincides with Newton's function N(x) = x - p(x)/p'(x). This formula which is derived algebraically gives a new representation of N(x), as well as new interpretation of Newton's method. Except possibly for a finite set of points whose characterization will be given, the fixed-point iteration is well-defined. Using this characterization we also prove the uniqueness of g[subscript m](x), given that p(x) has n distinct real roots."

A simple polynomial time algorithm for a convex hull problem equivalent to linear programming by Bahman Kalantari ( Book )
2 editions published in 1993 in English and held by 6 libraries worldwide
Abstract: "Over the rationals, the general linear programming problem is equivalent to the convex hull problem of determining if a given m x n matrix H has a nontrivial nonnegative zero. We give a polynomial time algorithm that either finds a nontrivial nonnegative zero of H, or it obtains a hyperplane separating the column vectors of H from the origin. In particular, the algorithm provides an alternate proof of a strengthened version of Gordan's duality theorem, previously proved by the author. The algorithm which is motivated by this duality theorem is analogous to Karmarkar's algorithm but its analysis is much simpler."

On the rate of convergence of the RAS method by Bahman Kalantari ( Book )
2 editions published in 1992 in English and held by 6 libraries worldwide
Abstract: "It is well-known that given an n x n matrix with positive entries, there exists two positive diagonal matrices X and Y such that XAY is doubly stochastic. One of the best known algorithms for computing the scaling factors X and Y is the so called RAS algorithm which alternatively normalizes rows and columns of the matrix. In this paper we prove that the RAS is a fully-polynomial time approximation scheme and give a bound of O((1/[epsilon] + ln n) [squareroot n]ln 1/v), on the number of iterations of the RAS for scaling A to an accuracy of [epsilon], where v is the ratio of the least entry of A to its largest."

A generalized hypergreedy algorithm for perfect matching by Celina Imielinska ( Book )
2 editions published in 1991 in English and held by 6 libraries worldwide
Abstract: "We give a generalization of the hypergreedy algorithm for minimum weight perfect matching on a complete edge weighted graph K(V) satisfying the triangle inequality, where V is a set of an even number, n, of vertices. We define a heuristic, called the t-hypergreedy, where t is an integer parameter satisfying [formula], which produces an approximate solution whose edges weight is bounded above by (2t + 1) times the optimal weight. Its time complexity is [formula]. For points in the Euclidean plane, the t-hypergreedy can be modified to produce an approximate solution, whose weight is bounded above by [squareroot 10](2t + 1) times the optimal weight.

On the complexity of nonnegative matrix scaling by Bahman Kalantari ( Book )
2 editions published in 1990 in English and held by 6 libraries worldwide
Abstract: "An n x n nonnegative matrix is said to be (doubly stochastic) scalable if there exists two positive diagonal matrices X and Y such that XAY is doubly stochastic. We derive an upper bound on the norms of the scaling factors X and Y and give a polynomial time complexity bound on the problem of computing the scaling factors with prescribed accuracies."

Generalization of Karmarkar's algorithm to convex homogeneous functions by Bahman Kalantari ( Book )
2 editions published in 1989 in English and held by 5 libraries worldwide

A theorem of the alternative for bihomogeneous functions and its relationship to diagonal scaling of positive matrices by Bahman Kalantari ( Book )
2 editions published in 1989 in English and held by 5 libraries worldwide

A theorem of the alternative for multihomogeneous functions and its relationship to diagonal scaling of matrices by Bahman Kalantari ( Book )
1 edition published in 1993 in English and held by 5 libraries worldwide
Abstract: "Let [phi](x) be a continuously differentiable real- valued function defined over the nonnegative points of a subspace W = W₁ x ... x W[subscript m] of R[superscript n] = R[superscript n₁] x ... x R[superscript n[subscript m]], and for each j = 1, ..., m, homogeneous of positive degree K[subscript j] with respect to nonnegative points of W[subscript j]. A vector [formula] is defined to be admissible if it is positive and the ratios [formula] are identical. Assume that [phi](d₀)> 0, for some positive d₀ [epsilon] W.