The new family of four-step methods

We consider the following family of four-step methods to integrate w = f(x,w)  

The above method is one of the families of symmetric four-step methods for the numerical solution of problems of the form w = f x,w). In the above general form the coefficient a and bj,j=0(1)2 are free parameters. In the same formula, h is the step size of the integration and n is the number of steps, i.e. is the approximation of the solution in the point x and Xn = XQ + n h and Xq is the initial value point. 

1 The first optimized new four-step method of the family with vanished phase-lag and its first derivative 

Application of the method (8) to the scalar test equation leads to the dilference equation (5) with k = 2 and  

The characteristic equation associated with the above difference eqnation is given by  

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