Robinson Analysis and Computation of Fixed Points

Proceedings of a Symposium Conducted by the Mathematics Research Center, the University of Wisconsin-Madison, May 7-8, 1979
1. Auflage 2014
ISBN: 978-1-4832-6602-2
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 424 Seiten, Web PDF

Analysis and Computation of Fixed Points contains the proceedings of a Symposium on Analysis and Computation of Fixed Points, held at the University of Wisconsin-Madison on May 7-8, 1979. The papers focus on the analysis and computation of fixed points and cover topics ranging from paths generated by fixed point algorithms to strongly stable stationary solutions in nonlinear programs. A simple reliable numerical algorithm for following homotopy paths is also presented. Comprised of nine chapters, this book begins by describing the techniques of numerical linear algebra that possess attractive stability properties and exploit sparsity, and their application to the linear systems that arise in algorithms that solve equations by constructing piecewise-linear homotopies. The reader is then introduced to two triangulations for homotopy fixed point algorithms with an arbitrary grid refinement, followed by a discussion on some generic properties of paths generated by fixed point algorithms. Subsequent chapters deal with topological perturbations in the numerical study of nonlinear eigenvalue and bifurcation problems; general equilibrium analysis of taxation policy; and solving urban general equilibrium models by fixed point methods. The book concludes with an evaluation of economic equilibrium under deformation of the economy. This monograph should be of interest to students and specialists in the field of mathematics.

Autoren/Hrsg.

Robinson, Stephen M.

Inhaltsverzeichnis

1;Front Cover;1
2;Analysis and Computation of Fixed Points;4
3;Copyright Page;5
4;Table of Contents;6
5;Contributors;8
6;Preface;10
7;Chapter 1. Numerical Stability and Sparsity in Piecewise-Linear Algorithms;12
7.1;1. Introduction;12
7.2;2. The QR and LU Factorizations;15
7.3;3. Using factorizations in the General PL Algorithm;18
7.4;4. The Separable Case;25
7.5;5. The Sparse Case;27
7.6;Acknowledgements;32
7.7;REFERENCES;33
8;Chapter 2. Two New Triangulations for Homotopy Fixed Point Algorithms with an Arbitrary Grid Refinement;36
8.1;1. Introduction;36
8.2;2. The Triangulation Sa;38
8.3;3. The Dynamic Shift Algorithm;46
8.4;4. Properties of the Algorithms;54
8.5;5. Computational Experience;57
8.6;REFERENCES;66
9;Chapter 3. Some Generic Properties of Paths Generated by Fixed Point Algorithms;68
9.1;§1. Introduction;68
9.2;§2. Generic Properties of Paths;71
9.3;§3. An Application;74
9.4;§4. On minimizing smooth real valued functions;76
9.5;§5. Appendix;79
9.6;REFERENCES;79
10;Chapter 4. A Simple Reliable Numerical Algorithm for Following Homotopy Paths;84
10.1;§1. Introduction;84
10.2;§2. Proceeding along the curve;88
10.3;§3. Angle checking;89
10.4;§4. Speed-up the step size;90
10.5;§5. Termination;91
10.6;§6. Remarks;92
10.7;§7. Numerical results;93
10.8;§8. Continuation versus Newton Methods;99
10.9;References;100
11;Chapter 5. Strongly Stable Stationary Solutions in Nonlinear Programs;104
11.1;1. INTRODUCTION;104
11.2;2. PRELIMINARIES;111
11.3;3. LOCAL NONSINGULARITY OF THE MAP F : ¦K*¦. Rn+m;115
11.4;4. NECESSARY AND SUFFICIENT CONDITIONS FOR THE S-STABILITY;122
11.5;5. STATIONARY INDEX;126
11.6;6. S-STABLE LOCAL MINIMUM SOLUTIONS;130
11.7;7. DEGENERATE S-STABLE STATIONARY SOLUTIONS;132
11.8;8. AN APPLICATION TO A PARAMETRIC NONLINEAR PROGRAM;142
11.9;9. AN APPLICATION TO A CLASS OF CONTINUOUS DEFORMATION METHODS;144
11.10;10. CONCLUDING REMARKS;146
11.11;REFERENCES;146
12;Chapter 6. Topological Perturbations in the Numerical Study of Nonlinear Eigenvalue and Bifurcation Problems;150
12.1;1. INTRODUCTION;150
12.2;2. TOPOLOGICAL PERTURBATIONS I;156
12.3;3. TOPOLOGICAL PERTURBATIONS II;173
12.4;4. NUMERICAL EXPERIENCE;179
12.5;REFERENCES;189
13;Chapter 7. General Equilibrium Analysis of Taxation Policy;194
13.1;MODEL STRUCTURE;195
13.2;MODEL ESTIMATION;199
13.3;STRUCTURE OF COMPUTER CODE;199
13.4;AN EXAMPLE OF MODEL FINDINGS;200
13.5;CONCLUDING REMARKS;200
13.6;REFERENCES;206
14;Chapter 8. Solving Urban General Equilibrium Models by Fixed Point Methods;208
14.1;1. INTRODUCTION;208
14.2;2. URBAN GENERAL EQUILIBRIUM MODELS;209
14.3;3. REALISTIC COMPLICATIONS;213
14.4;4. CONCLUSION;222
14.5;REFERENCES;222
15;Chapter 9. Economic Equilibrium under Deformation of the Economy;224
15.1;ABSTRACT;224
15.2;PREFACE;226
15.3;INTRODUCTION;229
15.4;THE FUNDAMENTAL ALGORITHM;249
15.5;THE ECONOMIC MODEL;259
15.6;COMPUTATIONAL REFINEMENTS;315
15.7;EXAMPLES OF ECONOMIC DEFORMATIONS;348
15.8;COMPUTATIONAL EXPERIENCE;374
15.9;APPENDIX A: TECHNICAL LEMMAS;416
15.10;REFERENCES;420
16;Index;422