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The Computation of Fixed Points and Applications M. J. Todd

The Computation of Fixed Points and Applications By M. J. Todd

The Computation of Fixed Points and Applications by M. J. Todd


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Summary

Fixed-point algorithms have diverse applications in economics, optimization, game theory and the numerical solution of boundary-value problems. Much of this work is available only in research papers, although Scarf's book [58] gives a remarkably clear exposition of the power of fixed-point methods.

The Computation of Fixed Points and Applications Summary

The Computation of Fixed Points and Applications by M. J. Todd

Fixed-point algorithms have diverse applications in economics, optimization, game theory and the numerical solution of boundary-value problems. Since Scarf's pioneering work [56,57] on obtaining approximate fixed points of continuous mappings, a great deal of research has been done in extending the applicability and improving the efficiency of fixed-point methods. Much of this work is available only in research papers, although Scarf's book [58] gives a remarkably clear exposition of the power of fixed-point methods. However, the algorithms described by Scarf have been super~eded by the more sophisticated restart and homotopy techniques of Merrill [~8,~9] and Eaves and Saigal [1~,16]. To understand the more efficient algorithms one must become familiar with the notions of triangulation and simplicial approxi- tion, whereas Scarf stresses the concept of primitive set. These notes are intended to introduce to a wider audience the most recent fixed-point methods and their applications. Our approach is therefore via triangu- tions. For this reason, Scarf is cited less in this manuscript than his contri- tions would otherwise warrant. We have also confined our treatment of applications to the computation of economic equilibria and the solution of optimization problems. Hansen and Koopmans [28] apply fixed-point methods to the computation of an invariant optimal capital stock in an economic growth model. Applications to game theory are discussed in Scarf [56,58], Shapley [59], and Garcia, Lemke and Luethi [24]. Allgower [1] and Jeppson [31] use fixed-point algorithms to find many solutions to boundary-value problems.

Table of Contents

I Brouwer's Theorem.- II Some Applications of Brouwer's Theorem.- III Triangulations.- IV Algorithms to Find Completely-Labelled Simplices.- V Extensions of Brouwer's Theorem.- VI Applications of Kakutani's Theorem and its Extensions.- VII Eaves' First Algorithm.- VIII Merrill's Algorithm.- IX Homotopy Algorithms.- X Triangulations with Continuous Refinement of Grid Size.- XI Measures of Efficiency for Triangulations.- References.

Additional information

NLS9783540076858
9783540076858
3540076859
The Computation of Fixed Points and Applications by M. J. Todd
New
Paperback
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
1976-05-01
132
N/A
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