A Scalar Homotopy Method for Solving an Over/Under-Determined System of Non-Linear Algebraic Equations

Abstract

Iterative algorithms for solving a system of nonlinear algebraic equa- tions (NAEs): Fi(xj) = 0, i, j= 1,. . . ,n date back to the seminal work of Issac New- ton. Nowadays a Newton-like algorithm is still the most popular one to solve the NAEs, due to the ease of its numerical implementation. However, this type of al- gorithm is sensitive to the initial guess of solution, and is expensive in terms of the computations of the Jacobian matrixFi=¶xj and its inverse at each iterative step. In addition, the Newton-like methods restrict one to construct an iteration proce- dure for n-variables by using n-equations, which is not a necessary condition for the existence of a solution for underdetermined or overdetermined system of equations. In this paper, a natural system of first-order nonlinear Ordinary Differential Equa- tions (ODEs) is derived from the given system of Nonlinear Algebraic Equations (NAEs), by introducing a scalar homotopy function gauging the total residual error of the system of equations. The iterative equations are obtained by numerically in- tegrating the resultant ODEs, which does not need the inverse ofFi=¶xj. The new method keeps the merit of homotopy method, such as the global convergence, but it does not involve the complicated computation of the inverse of the Jacobian ma- trix. Numerical examples given confirm that this Scalar Homotopy Method (SHM) is highly efficient to find the true solutions with residual errors being much smaller.