. Interval Newton methods in conjunction with generalized bisection can form the basis of algorithms that find all real roots within a specified box X ae R n of a system of nonlinear equations F (X) = 0 with mathematical certainty, even in finite-precision arithmetic. In such methods, the system F (X) = 0 is transformed into a linear interval system 0 = F (M) +F 0 (X)( ~ X \Gamma M); if interval arithmetic is then used to bound the solutions of this system, the resulting box ~ X contains all roots of the nonlinear system. We may use the interval Gauss--Seidel method to find these solution bounds. In order to increase the overall efficiency of the interval Newton / generalized bisection algorithm, the linear interval system is multiplied by a preconditioner matrix Y before the interval Gauss--Seidel method is applied. Here, we review results we have obtained over the past few years concerning computation of such preconditioners. We emphasize importance and connecting relationships,...

This paper is the first to introduce an algorithm to compute stationary equilibria in stochastic games, and shows convergence of the algorithm for almost all such games. Moreover, since in general the number of stationary equilibria is overwhelming, we payattention to the issue of equilibrium selection. We do this by extending the linear tracing procedure to the class of stochastic games, called the stochastic tracing procedure. From a computational point of view, the class of stochastic games possesses substantial difficulties compared to normal form games. Apart from technical difficulties, there are also conceptual difficulties, for instance the question how to extend the linear tracing procedure to the environment of stochastic games. We prove that there is a generic subclass of the class of stochastic games for which the stochastic tracing procedure is a compact one-dimensional piecewise differentiable manifold with boundary. Furthermore, we prove that the stochastic tracing procedure generates a unique path leading from any exogenously specified prior belief, to a stationary equilibrium. A well-chosen transformation of variables is used to formulate an everywhere differentiable homotopy function, whose zeros describe the (unique) path generated by the stochastic tracing procedure. Because of differentiability we are able to follow this path using standard path-following techniques. This yields a globally convergent algorithm that is easily and robustly implemented on a computer using existing software routines. As a by-product of our results, we extend a recent result on the generic finiteness of stationary equilibria in stochastic games to oddness of equilibria.

This paper develops an algorithm for solving mixed complementarity problems which is based upon probability one homotopy methods. After the complementarity problem is reformulated as a system of nonsmooth equations, a homotopy method is used to solve a sequence of smooth approximations to this system of equations. The global convergence properties of this approach are considerably stronger than other recent algorithms, depending on very weak assumptions about the problem. To improve efficiency, the homotopy algorithm is embedded in a generalized Newton-method. Keywords: Complementarity problems, homotopy methods, smoothing. 1 Introduction This paper discusses a robust method for solving mixed complementarity problems, which is based upon the probability one homotopy methods of [13, 31, 33]. The idea is to reformulate the mixed complementarity problem as a system of equations, and then solve smooth approximations of this system with a homotopy method. While extremely robust, the homot...

E-mail is welcome anytime! Prerequisite: Undergraduate-level knowledge of numerical analysis: linear equations, nonlinear equations, integration, interpolation. Programming assignments will be in Matlab. Text: None. See reference list. Topics: Monte Carlo simulation, numerical linear algebra, nonlinear systems and continuation method, optimization, ordinary differential equations. Fundamental techniques in scientific computation with an introduction to the theory and software for each topic. News: Assignments, course notes, answers to homeworks and quizzes, and announcements will be posted on the course’s homepage. You are responsible for checking this site before each class. Final Exam: None. Grading: Grading will be on a curve, except that you will be guaranteed an A if your average is 90 % or better, a B if your average is 80 % or better,

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In this paper we argue that in realistically calibrated two period general equilibrium models with incomplete markets CAPM-pricing provides a good benchmark for equilibrium prices even when agents are not mean-variance optimizers and returns are not normally distributed. We numerically approximate equilibria for a variety of di erent speci cations for preferences, endowments and dividends and compare the equilibrium prices and portfolio-holdings to the predictions of the CAPM. While the CAPM does not hold exactly for the chosen speci cation, it turns out that pricingerrors are extremely small. Furthermore, two-fund separation holds approximately.

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