Use of the Implicit Function Theorem in Solving Systems of Equations

摘要

Applications of the implicit function theorem in solving systems of n equations in (n + p) unknowns subject to p nonlinear 'constraints' are discussed. A generalized iterative technique that is defined for a sufficiently restricted 'region' is derived for solving the set of (n + p) equations. The derivation employs Newton's method and assumes that the unconstrained (n + p) system of equations has a local vector function solution that satisfies the p constraints at a unique point in its 'prescribed region' of definition. The implicit function theorem is further employed to derive mathematical formulas that characterize the conditions of the solutions of systems of linear equations and the zeros of polynomials with real coefficients. (Author)