A numerical method for the solution of a problem based on repeated extrapolation, to produce increasingly more accurate results, from a sequence of basic approximations utilizing different values of a parameter, such as the stepsize (see finite-difference method).

Important examples are the Romberg method for numerical integration and Gragg’s extrapolation method for the solution of ordinary differential equations based on the midpoint rule. An essential requirement for such methods are theoretical results establishing the existence of an expansion for the error in (usually even) powers of the parameter, for sufficiently smooth problems.