for k > 1, and the right-hand side Rm is dependent only upon the known results u0, u1, ..., um − 1 and can be obtained easily using computer algebra software. In this way, the original nonlinear equation is transferred into an infinite number of linear ones, but without the assumption of any small/large physical parameters. Since the HAM is based on a homotopy, one has great freedom to choose the initial guess u0(x), the auxiliary linear operator , and the convergence-control parameter c0 in the zeroth-order deformation equation. Thus, the HAM provides the mathematician freedom to choose the equation-type of the high-order deformation equation and the base functions of its solution. The optimal value of the convergence-control parameter c0 is determined by the minimum of the squared residual error of governing equations and/or boundary conditions after the general form has been solved for the chosen initial guess and linear operator. Thus, the convergence-control parameter c0 is a simple way to guarantee the convergence of the homotopy series solution and differentiates the HAM from other analytic approximation methods. The method overall gives a useful generalization of the concept of homotopy. The HAM and computer algebra The HAM is an analytic approximation method designed for the computer era with the goal of "computing with functions instead of numbers." In conjunction with a computer algebra system such as Mathematica or Maple, one can gain analytic approximations of a highly nonlinear problem to arbitrarily high order by means of the HAM in only a few seconds. Inspired by the recent successful applications of the HAM in different fields, a Mathematica package based on the HAM, called BVPh, has been made available online for solving nonlinear boundary-value problems . BVPh is a solver package for highly nonlinear ODEs with singularities, multiple solutions, and multipoint boundary conditions in either a finite or an infinite interval, and includes support for certain types of nonlinear PDEs. Another HAM-based Mathematica code, APOh, has been produced to solve for an explicit analytic approximation of the optimal exercise boundary of American put option, which is also available online . Frequency response analysis for nonlinear oscillators The HAM has recently been reported to be useful for obtaining analytical solutions for nonlinear frequency response equations. Such solutions are able to capture various nonlinear behaviors such as hardening-type, softening-type or mixed behaviors of the oscillator,. These analytical equations are also useful in prediction