Solving systems of non-linear equations with both free and positive variables

A numerical method is presented for solving systems of non-linear equations that contain some variables that are strictly positive and others that have no restriction on sign. Naturally positive variables arise frequently when modelling the behaviour of engineering systems, such as physical dimension, concentration of a chemical species, duration of an event, etc. When modelling systems of this type, it is also common to introduce additional variables that are not restricted in sign, such as stresses, displacements, velocities, accelerations, etc. Many numerical methods may experience performance di$culties due to the existence of spurious solutions which have negative components for one or more of the positive variables. Recently, the monomial method has been developed as an e!ective tool for systems with variables that are all strictly positive. This paper presents a hybrid method, combining the monomial method and Newtonʹs method, for systems containing both types of variables. It is demonstrated that this hybrid method can be more e!ective in solving systems of equations with both positive and free variables than either method alone. Basins of attraction constructions are presented as a demonstration of the e!ectiveness of the hybrid method as applied to the design of a civil engineering frame structure