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Simplicial Algorithms on the Simplotope

  • Book
  • © 1988

Overview

Authors:
  1. Timothy Mark Doup

    1. Dept. of Mathematics and Systems Engineering, Koninklijke/Shell-Laboratorium, Amsterdam, Amsterdam, The Netherlands

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Part of the book series: Lecture Notes in Economics and Mathematical Systems (LNE, volume 318)

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Table of contents (12 chapters)

  1. Front Matter

    Pages N2-VIII
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  2. Introduction and Definitions

    1. Front Matter

      Pages 1-1
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    2. Introduction

      • Timothy Mark Doup
      Pages 3-13

    3. Definitions and Existence Theorems

      • Timothy Mark Doup
      Pages 15-44

    4. Triangulations of Sn and S

      • Timothy Mark Doup
      Pages 45-65

  3. Algorithms on the Unit Simplex

    1. Front Matter

      Pages 67-67
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    2. An Introduction to Simplicial Algorithms on the Unit Simplex

      • Timothy Mark Doup
      Pages 69-93

    3. The (2n+1 −2)-Ray Algorithm

      • Timothy Mark Doup
      Pages 95-110

    4. The 2-Ray Algorithm

      • Timothy Mark Doup
      Pages 111-126

    5. Comparisons and Computational Results

      • Timothy Mark Doup
      Pages 127-133

  4. Algorithms on the Simplotope

    1. Front Matter

      Pages 135-135
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    2. An Introduction to Simplicial Algorithms on the Simplotope

      • Timothy Mark Doup
      Pages 137-158

    3. The Product-Ray Algorithm

      • Timothy Mark Doup
      Pages 159-172

    4. The Exponent-Ray Algorithm

      • Timothy Mark Doup
      Pages 173-195

    5. Comparisons and Computational Results

      • Timothy Mark Doup
      Pages 197-203

  5. Continuous Deformation on the Simplotope

    1. Front Matter

      Pages 205-205
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    2. The Continuous Deformation Algorithm on the Simplotope

      • Timothy Mark Doup
      Pages 207-254

  6. Back Matter

    Pages 255-264
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Keywords

About this book

1.1. Introduction Solving systems of nonlinear equations has since long been of great interest to researchers in the field of economics, mathematics, en­ gineering, and many other professions. Many problems such as finding an equilibrium, a zero point, or a fixed point, can be formulated as the problem of finding a solution to a system of nonlinear equations. There are many methods to solve the nonlinear system such as Newton's method, the homotopy method, and the simplicial method. In this monograph we mainly consider the simplicial method. Traditionally, the zero point and fixed point problem have been solved by iterative methods such as Newton's method and modifications thereof. Among the difficulties which may cause an iterative method to perform inefficiently or even fail are: the lack of good starting points, slow convergence, and the lack of smoothness of the underlying function. These difficulties have been partly overcome by the introduction of homo­ topy methods.

Authors and Affiliations

  • Dept. of Mathematics and Systems Engineering, Koninklijke/Shell-Laboratorium, Amsterdam, Amsterdam, The Netherlands

    Timothy Mark Doup

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