On a Numerical Treatment for the Curve-Tracing of the Homotopy Method

where H: I R " x N ~ R " is a C 2 function. We would like to numerically trace this curve F from the point (Xo, to) to the point (x*, t*). For the simplicity of discussion we shall assume that 0EIt" is a regular value for H, i.e., the n x (n+ 1) Jacobian matrix DH(x, t) has full rank at every point on F. in recent years a number of approaches have been proposed for the numerical computation of this curve-tracing problem [1, 2, 5, 6-9, 11, 13]. Essentially all these techniques are of the predictor-corrector type. A prototype algorithm usually takes steps of the following form: (i) Let Y~=(xi, tl)MR "+1 he a point being accepted as an approximating point for E Choose a predictor for the next approximating point. Usually this is done by setting Z o = Y~ + h i T~ (1.2)