Dual Algorithms

TL;DR

This paper extends classical numerical algorithms to dual versions using dual numbers, enabling derivative calculations of complex function compositions without closed-form expressions, demonstrated through various practical examples.

Contribution

It introduces dual algorithms for interpolation, root-finding, and differential equation solving, expanding their applicability to functions without explicit derivatives.

Findings

Dual Newton-Raphson method for derivatives of output angles in mechanisms.

Dual cubic spline interpolation for thermal diffusivity estimation.

Dual Runge-Kutta method for derivatives in differential equations.

Abstract

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.