Nonlocal Automated Sensitivity Analysis

Previously a complete ordinary differential equation (ODE) system was developed for tracking the solution x(~) of a parameterized system of nonlinear equations 0 = ~'(x,~) over an ~-interval [~0, ~]. This paper develops a two-phase complex homotopy continuation method for obtaining the required initial conditions at ~t °. An initial "short" artificial continuation is followed by a continuation which essentially proceeds through the physically meaningful function ~, which can ameliorate the problem of artificially induced singularities. Also, the path in the complex plane followed by the continuation parameter in each phase evolves sequentially in an attempt to keep the path both short (minimal number of integration steps) and numerically stable (avoidance of singular points). A FORTRAN program Nasa has been developed for solving the complete ODE system, starting with a two-phase initialization. The program incorporates a fast and efficient procedure for the automatic evaluation of partial derivatives. Numerical experiments are reported which illustrate the program's effectiveness.