A scalar homotopy method for solving an Over/Under-determined system of non-linear algebraic equations

Iterative algorithms for solving a system of nonlinear algebraic equations (NAEs): Fi(xj) = 0, i, j= 1,. . . ,n date back to the seminal work of Issac Newton. 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 algorithm is sensitive to the initial guess of solution, and is expensive in terms of the computations of the Jacobian matrix ∂F_i= ∂×_j and its inverse at each iterative step. In addition, the Newton-like methods restrict one to construct an iteration procedure 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 Equations (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 integrating the resultant ODEs, which does not need the inverse of ∂F_i=∂×_j. 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 matrix. Numerical examples given confirm that this Scalar Homotopy Method (SHM) is highly efficient to find the true solutions with residual errors being much smaller.