An algorithm for targeted convergence of Euler or Newton iterations

The concept of multistationarity has become essential for understanding cell differentiation. For this reason theoretical biologists have more and more frequently to determine the steady values, often multiple, of systems of non-linear differential equations. It is well known that iteration processes of current use converge or not towards a fixed point depending on the absolute value of the slope of the iteration function in the vicinity of the considered fixed point. A number of methods have been developed to obtain or accelerate convergence. As biologists, we do not pretend to review these works. Rather, we propose here a simple algorithm which permits to converge at will towards a chosen type of steady state. Others and we have used this procedure extensively for years for the analysis of complex biological systems. A compact program (using Mathematica) is available.