A general scheme of parameterized families of equations is
considered, and abstract results on the expansion of the
solutions and on the acceleration of their convergence in
terms of the parameter are presented. These results are
applied to fractional step approximations for linear parabolic
PDEs, systems of linear PDEs, and for nonlinear ordinary
differential equations. Applications to accelerating the
convergence of finite difference schemes for these equations
will be presented in a subsequent paper.