Showing papers on "Homotopy analysis method published in 1984"

Solving systems of polynomial equations by bounded and real homotopy

Abstract: The homotopy method for solving systems of polynomial equations proposed by Chow, Mallet-Paret and Yorke is modified in two ways. The first modification allows to keep the homotopy solution curves bounded, the second one to work with real polynomials when solving a system of real equations. For the first method numerical results are presented.

On the cost of computing roots of polynomials

Abstract: Recently Smale has obtained probabilistic estimates of the cost of computing a zero of a polynomial using a global version of Newton's method. Roughly speaking, his result says that, with the exception of a set of polynomials where the method fails or is very slow, the cost grows as a polynomial in the degree. He also asked whether similar results hold for PL homotopy methods. This paper gives such a result for a special algorithm of the PL homotopy type devised by Kuhn. Its main result asserts that the cost of computing some zero of a polynomial of degreen to an accuracy of e (measured by the number of evaluations of the polynomial) grows no faster than O(n 3 log2(n/e)). This is a worst case analysis and holds for all polynomials without exception.

Computational complexity of a piecewise linear homotopy algorithm

Abstract: We give a bound on the number of steps required by the piecewise linear algorithm based on component wise homotopy (proposed by the author for structured problems) when solving a linear problem. When the coefficient matrix is symmetric and positive definite, this bound is polynomial inn and linear in the condition number of the matrix. We also investigate the expected value of the bound for a particular distribution of such matrices.

Exact sequences in stable homotopy pair theory

Abstract: A cylinder-web diagram with associated diagonal sequences is described in stable homotopy pair theory. The diagram may be used to compute stable homotopy pair groups and also stable track groups of two-cell complexes. For the stable Hopf class i1 the stable homotopy pair groups Gk (t. i) ( k 8) ae computed together with some of the additive structure of the stable homotopy ring of the complex projective plane. 0. Introduction. Let a E Gr,8 E/e Gm denote stable classes of maps between spheres and let 7T = DirLim [sn+k U en+r+k+l Sn U en+m+lI n -400 denote the corresponding stable track group. The groups 7Tr have been studied by N. Yamamoto [13] in the case a = X = p # 2 and by J. Mukai [8, 9] in the case a = /3 = 2. In these papers rather extensive computations are given, but they use special information concerning the stable homotopy of Moore spaces. Here we develop a technique for computation which relies on the (stable) Puppe and dual-Puppe sequences passing through 7Tk but which attempts to resolve the problems of group extension through the additional information contained in a cylinderweb diagram. Besides the rectangular mesh of Puppe and dual-Puppe sequences the cylinder-web diagram has three diagonal sequences (Theorem 3.3) that pass through the stable homotopy pair groups Gk(a, /). These groups are the natural "home" for (stable) Toda brackets of the form K<, y, a) in the sense that the brackets live here with zero indeterminacy. In ?6 the technique is applied to compute, for the stable Hopf class 7, the stable homotopy pair groups Gk(, 7q) (k < 8) and the stable track groups 7Tk (k < 8) of the complex projective plane. It is interesting that besides knowledge of the stable homotopy groups of spheres, including composition and secondary composition operations, the method also requires information concerning the third order composition (quaternary Toda brackets). The computation presented as an illustration encountered a difficulty at 7T9; however, it seems reasonable to expect that a better understanding of the quaternary bracket will enable it to be continued. Received by the editors June 6, 1983 and, in revised form, December 7, 1983. 1980 Mathematics Subject Classification. Primary 55Q10; Secondary 55Q05. 'Grants to the Topology Research Group from the University of Cape Town and the South African Council for Scientific and Industrial Research are acknowledged. (1 984 American Mathematical Society 0002-9947/84 $1.00 + $.25 per page 803 This content downloaded from 157.55.39.35 on Mon, 29 Aug 2016 06:24:12 UTC All use subject to http://about.jstor.org/terms 804 K. A. HARDIE AND A V. JANSEN 1. Preliminaries. Letf: X -* Y, g: E -B be pointed continuous maps (i.e. pairs in the sense of Eckmann and Hilton [1]). We recall that the morphism set IT(f, g) in the category HPC of homotopy pairs and homotopy pair classes [3, 4] is obtained from the set of tracks from f to g by factoring out by the equivalence relation:

A homotopy method for solving an equation of the type − Δu = F(u)

Abstract: We describe a homotopy algorithm for solving the equation − Δu = F(u). To this end, we define a pseudo-inverse and a pseudo-determinant with sufficient regularity properties, for operators of Laplacian type.