This article introduces and constructively illustrates the concept of an adaptive homotopy for solving systems of nonlinear equations. Standard homotopy methods rely on a passive continuation parameter moving from 0 to 1 along the real line and are stymied if the homotopy Jacobian matrix becomes...

This article develops a complete system of ordinary differential equations for tracking a single eigenvalue of a parameterized matrix, together with one of its corresponding right or left eigenvectors, over parameter intervals. The feasibility and accuracy of the method are illustrated by...

The complete system of ordinary differential equations developed by Kalaba and Tesfatsion (1981) for tracking solution branches of parameterized nonlinear systems is tested using several illustrative examples. One example is the standard Ramsey optimal growth model, for which analytical...

This article appeared in Computer Science in Economics and Management (now Computational Economics). Parameter estimation problems for nonlinear systems are typically formulated as nonlinear optimization problems. For such problems, one has the usual difficulty that standard successive...

This article develops a complete system of ordinary differential equations for tracking the Frobenius-Perron root (largest eigenvalue) of a parameterized matrix, together with a unit-normalized right eigenvector, over parameter intervals. The feasibility and accuracy of the method are...

This article develops a complete system of ordinary differential equations for tracking the eigenvalues and the right and left eigenvectors of nonsymmetric parameterized matrices over parameter intervals. A simpler reduced form of the ODE system is then derived for tracking the eigenvalues and...

In Kalaba, Tesfatsion, and Wang (JMAA,1983) an algorithm was developed for the exact forward-mode automatic evaluation of higher-order partial derivatives of functions of many variables using derivative arrays. This algorithm was supported by a library of "calculus subroutines" for many standard...

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In Kalaba, Tesfatsion, and Wang (JMAA,1983) the FEED algorithm was introduced for the exact forward-mode automatic evaluation of higher-order partial derivatives of functions of many variables using derivative arrays. This study demonstrates that FEED can be applied to a much broader class of...

This article develops a complete system of ordinary differential equations for tracking solution branches x(b) of a parameterized system of nonlinear equations F(x,b) = 0 along arbitrary user-designated paths for the parameter vector b. The complete system is obtained by augmenting the usual...