Numerical search for algebraically stable two-step almost collocation methods

We investigate algebraic stability of the new class of two-step almost collocation methods for ordinary differential equations. These continuous methods are obtained by relaxing some of the interpolation and collocation conditions to achieve strong stability properties together with uniform order of convergence on the whole interval of integration. We describe the search for algebraically stable methods using the criterion based on the Nyquist stability function proposed recently by Hill. This criterion leads to a minimization problem in one variable which is solved using the subroutine fminsearch from MATLAB. Examples of algebraically stable methods in this class are also presented.