In essence, the homotopy method for solving a system of nonlinear equations f ( x ) = 0 consists of choosing a function h ( x , t ) such that h ( x , 1 ) = f ( x ) and h ( x0 , 0 ) = 0 for known x0, then tracing the path x ( t ) defined by h ( x ( t ) , t ) = 0 and x ( 0 ) = x0 to a solution x = x ( 1 ) of the original system. Assuming that such a path exists, it suffices to follow it closely enough to arrive in a neighborhood of ( x , 1 ) such that an end game iteration will converge rapidly to the solution. Difficulties arise if the Jacobian of h is rank deficient at one or more points on the path, especially at ( x , 1).
Watson participated in the development of a well-known code package, HOMPACK, implementing a number of path-following techniques. The current work examined several candidate end game iterations through experiments on a published suite of challenging test problems. When all were successful, these candidates were comparable, so their performance was ranked based on cases in which one but not all of them failed to converge. A preferred candidate emerged and is recommended as a replacement for that used in HOMPACK. The comparison appears to be indicative but not definitive.