Dual Algorithms

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. Keywords— Dual numbers, Derivatives, Runge–Kutta algorithm, Newton–Raphson algorithm Cubic spline interpolation.