دانلود کتاب Pathways to Solutions, Fixed Points, and Equilibria

Numerical methods for the solution of systems of nonlinear equations have long been available. Until quite recently, however, they were local in character in the sense of requiring an accurate estimate of the correct solution in order to guarantee convergence of the algorithm. Global methods were developed approximately two decades ago in an attempt to solve a class of problems which were central to economics, though surely peripheral to the concerns of most numerical analysts. The economic problem was to find a vector of prices which would equate supply and demand in a system of interrelated markets. We are, of course, all familiar with the elementary economic considerations which suggest the way in which prices move toward equilibrium. If, at a particular price, the demand for a specific commodity exceeds its supply, then an increase in the price of that commodity will presumably decrease its demand, call forth an increased supply, and narrow the gap between the two sides of the market. If such a price adjustment mechanism is to have general economic validity, it should be capable of providing an algorithm for the solution of the equilibrium equations which arise when the economic problem is given mathematical form. The most natural translation of this market adjustment mechanism is to a system of nonlinear, first-order differential equations in which the rate of change of price is proportional to the discrepancy between supply and demand. But economic intuition and mathematical technique come to an abrupt conflict at this point. There is no reason for these differential equations to converge and, from a mathematical point of view, the complete disregard of information conveyed by first-order derivatives would seem to lead to particularly inefficient algorithms. The difficulties of numerical calculation were avoided, for many years, by alternative developments in mathematical economics. The central concern became that of demonstrating the existence of equilibrium prices. This involved making use of non-constructive methods, such as the Brouwer and Kakutani fixed point theorems. The shift in point of view which allowed fixed point techniques to be used as effective computational algorithms took place in the 1960s. Since then the field has developed substantially. Not only has there been an enormous improvement in the numerical methods themselves, and an extension of the range of applications to problem areas far from mathematical economics, but there has also been a substantial increase in our understanding of how fixed point methods relate to other branches of mathematics. In these pages, Garcia and Zangwill bring this material together in an elegant and lucid presentation, which makes these important developments available to the general reader, provides a wealth of fascinating examples, and bears their unique intellectual signatures.