  The cubic spline interpolation method, the Runge--Kutta method, and the
Newton-Raphson method are extended to dual versions (developed in the context
of dual numbers). This extension allows the calculation of the derivatives of
complicated compositions of functions which are not necessarily defined by a
closed form expression. The code for the algorithms has been written in Fortran
and some examples are presented. Among them, we use the dual Newton--Raphson
method to obtain the derivatives of the output angle in the RRRCR spatial
mechanism; we use the dual normal cubic spline interpolation algorithm to
obtain the thermal diffusivity using photothermal techniques; and we use the
dual Runge--Kutta method to obtain the derivatives of functions depending on
the solution of the Duffing equation.

Computational Engineering, Finance, and Science