Algorithm of simplification of nonlinear equations systems

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Abstract
The recent years have seen the development of automatic methods for the resolution of nonlinear equations systems. The standard numerical approach, when faced with a nonlinear equation system, is to solve it by Newton-Raphson method, that requires the expensive calculation of the Jacobian matrix of the system. Since inversion of a N*N matrix is O ( N 3 ) expensive, reducing the size of the system is therefore critical.We present here a simple, graph-theoretical algorithm for automatically reducing a nonlinear equation system beforehand, by use of successive substitutions. The algorithm treats the equation system as a graph, is heuristic, and automatically chooses what substitutions are likely to yield, at the end, the highest size reduction. This system must not necessarily be polynomial, can even be ill defined (then the algorithm just performs reduction) and can be used even in the presence of transcendental or a priori unknown functions.Two implementations have been made, one in FORTRAN, quicker but less efficient, and one in the computer algebra language MACSYMA, slower but with better reduction. The promises of computer algebra in this field are investigated. Applications are then presented to the reduction of equations systems simulating the steady state behavior of HVAC systems in buildings.

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