Some issues in solving large sparse systems of equations (Q919749)

The author discusses some problems concerning the solution of large sparse systems of equations. They are argued to be solved efficiently if the equations are ordered in such a way that the set of feedback variables is small. Questions of applying the technology for reordering the equations when the presented systems are solved are discussed. These results can be applied to econometric models which are described by non normalized equations, implicit equations and vector equations. It is proved that a good normalization does not always exist (even if a bad one does). Note, that one can rewrite the system in an alternative (but mathematically equivalent) way which allows a good normalization. In spite of its large size the systems can often be solved efficiently by a Newton-type algorithm. It is noted that this procedure can be used to enchance the convergence of the Gauss-Seidel algorithm. Numerical examples are presented.