A new family of trigonometric summation polynomials, Gn,r（f; θ）, of Bernstein type is constructed. In contrast to other trigonometric summation polynomials, the convergence properties of the new polynomials are superior to others. It is proved that Gn,r（f; θ） converges to arbitrary continuous functions with period 2π uniformly on （-∞ ＋∞） as n→ ∞. In particular, Gn,r（f; θ） has the best convergence order, and its saturation order is 1/n^2r＋4.
In this paper by means of generalized shooting method and homotopy technique a numerical method was given for computing free multipoint boundary value problem proposed in the intervention of exchange rate by Cadenillas and Femaado Zapatero. A numerical example was given for illustrating the validity of this method.
