| Abstract: | A monomial-based method for solving systems of algebraic non-linear equations is presented. The method uses the arithmetic-geometric mean inequality to construct a system of monomial equations that approximates the system of non-linear equations. A change of variables transforms the monomial system into a system of linear equations, which is readily solved. Special properties of the monomial method are identified and their significance is discussed. Invariance properties of the monomial method produce a built-in, self-adjusting scaling of the variables and equilibration of the equations of the linear system. Other special properties can lead to useful bounds on, and invariances of, the conditioning of the linear system. An invariance to uniform scaling is responsible for extremely rapid convergence to the equation surfaces in the initial iterations. An invariance to multiplication of the algebraic equations by a certain class of functions leads to a useful insensitivity to form of the algebraic system. Insensitivity of the monomial method to solutions with negative components avoids meaningless solutiuons of the algebraic system that appear as undesirable by-products of the formulation. A practical engineering design problem is solved to demonstrate the special properties of the monomial method. |
