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 hav e no restriction on sign. Naturally positive variables arise frequently whe n modelling the behaviour of engineering systems, such as physical dimensio n, concentration of a chemical species, duration of an event, etc. When mod elling systems of this type, it is also common to introduce additional vari ables that are not restricted in sign, such as stresses, displacements, vel ocities, accelerations, etc. Many numerical methods may experience performa nce difficulties due to the existence of spurious solutions which have nega tive components for one or more of the positive variables. Recently, the mo nomial method has been developed as an effective tool for systems with vari ables that are all strictly positive. This paper presents a hybrid method, combining the monomial method and Newton's method, for systems containing b oth types of variables. It is demonstrated that this hybrid method can be m ore effective in solving systems of equations with both positive and free v ariables than either method alone. Basins of attraction constructions are p resented as a demonstration of the effectiveness of the hybrid method as ap plied to the design of a civil engineering frame structure. (C) 1999 John W iley & Sons, Ltd.