From the Publisher:
Introduction to Numerical Continuation Methods continues to be useful for researchers and graduate students in mathematics, sciences, engineering, economics, and business looking for an introduction to computational methods for solving a large variety of nonlinear systems of equations. A background in elementary analysis and linear algebra is adequate preparation for reading this book; some knowledge from a first course in numerical analysis may also be helpful.

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Introduction to Numerical Continuation Methods

This paper surveys in a tutorial fashion theoretical and applications aspects of the continuation method for the solution of large scale system engineering problems. The continuation method is motivated and defined. The existence of the method is formulated in terms of degree theory a la Algebraic Topology. Examples are given throughout the paper and applications to engineering and economic problems are cited and described.

Continuation methods: Theory and applications

General equilibrium models and homotopy methods

Our aim here is to provide a brief introductory overview of numerical homotopy methods for approximating a solution of a system of nonlinear equations 

$$G(x) = 0$$

(1)

where the map G: ℝ N → ℝ N satisfies some appropriate conditions.

Predictor-Corrector and Simplicial Methods for Approximating Fixed Points and Zero Points of Nonlinear Mappings

A continuous deformation algorithm is introduced on S × [1, ∞, where S denotes the product space of unit simplices, with arbitrary grid refinement between two subsequent levels. The set S × [1, ∞ is triangulated in such a way that for each m, m = 1, 2,..., S × {m} is triangulated by the so-called V-triangulation. The algorithm starts by applying a variable dimension algorithm on S until an approximating simplex has been found on level 1. Then the algorithm follows a path of approximating simplices in S × [1, ∞, starting on level 1, until a certain level or a certain accuracy of a solution of the underlying problem has been reached. If the algorithm returns to level 1, then we again apply the variable dimension algorithm until a new approximating simplex is found on level 1, etc. We allow solutions to lie on the boundary of S × [1, ∞ in which case the algorithm, in general, will follow a path on the boundary of S × [1, ∞.

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A continuous deformation algorithm on the product space of unit simplices

A philosophical problem arises when one attempts to predict a competitive economy's response to a fundamental change in its structure with the aid of a competitive equilibrium model. Unless the model is known to admit unique solutions, there is little basis for assuming that the computed equilibrium will be attained, even though the model accurately describes the economy's structure and the behavior of its agents. If, however, one is able to arrive at the new model by continuously deforming the old one, then the two versions generally admit solutions which are connected by a path of equilibria arising from the continuum of intermediate economies. By ascribing a suitable dynamic interpretation to the deformation, one obtains a rationale for expecting the path-connected solutions to be mutually attained. The description of economic deformations and the computation of equilibrium paths is the central theme of this study. A general mathematical framework for modeling economies under deformation is developed by expanding Herbert Scarf's original activity analysis formulation to include uncountable unit activity sets, unbounded multi-valued demand correspondences, and tax and revenue systems similar to those introduced by John Shoven and John Whalley. Deformations of virtually all economic constructs are allowed in this general model. The computation of equilibrium paths is accomplished by a simplicial pivot algorithm designed along the lines of the homotopy-type fixed point techniques pioneered by Curtis Eaves. The dimension normally used to refine piecewise linear approximations now serves as the index of the economic deformation. To make this approach viable in practice, a new family of triangulations of Euclidean space is fashioned out of two conventional triangulations invented by Michael Todd. The geometry of these triangulations can be dynamically altered by the algorithm as it attempts to maintain uniform approximation error along the equilibrium path. The economic model and computational algorithm are translated into a set of computer routines which generate explicit numerical approximations to equilibrium paths for a variety of examples. Due to the vast amount of information embodied in an equilibrium path, problems of this type require a great deal of computational effort. A detailed analysis of the behavior of the algorithm on a series of test problems is presented in the final chapter.

Charles R. Engles

Papers

Economic equilibrium under deformation of the economy