Sufficient condition of a priori estimation for computational complexity of the homotopy method

A priori estimation is presented for a computational complexity of the homotopy method applying to a certain class of strongly monotone nonlinear equations. In the present paper, a condition is presented for a certain class of uniquely solvable equations, under which an upper bound of a computational complexity of the Newton type homotype method can be a priori estimated. In this paper, a condition is considered in a case of linear homotopy equations including the Newton type homotopy equations. In the first place the homotopy algorithm based on the simplified Newton method is introduced. Then by using Urabe type theorem, which gives a sufficient condition guaranteeing the convergence of the simplified Newton method, a condition is presented under which an upper bound of a computational complexity of the algorithm can be a priori estimated, when it is applied to a certain class of strongly monotone nonlinear equations. The presented condition is demonstrated by numerical experiments.