A modified newton method for solving non-linear algebraic equations

The Newton algorithm based on the "continuation" method may be written as being governed by the equation x_j(t) + B_ij ^-1F_i(x_j) = 0, where F_i(x _j) = 0, i, j = 1, ...n are nonlinear algebraic equations (NAEs) to be solved, and B_ij = ∂F_iJ∂x_j is the corresponding Jacobian matrix. It is known that the Newton's algorithm is quadratically convergent; however, it has some drawbacks, such as being sensitive to the initial guess of solution, and being expensive in the computation of the inverse of B_ij at each iterative step. How to preserve the convergence speed, and to remove the drawbacks is a very important issue in the solutions of NAEs. In this paper we discretize the above equation being written as B_ijx_j(t) + F_i(X_j) = 0, by a backward difference scheme in a new time scale of s = 1 - e ^-t, and an ODEs system is derived by introducing a fictitious time-like variable. The new algorithm is obtained by applying a numerical integration scheme to the resultant ODEs. The new algorithm does not need the inverse of By, and is thus resulting in a significant reduction in computational time than the Newton's algorithm. A similar technique is also used to modify the homotopy method. Numerical examples given confirm that the modified Newton method is highly efficient, insensitive to the initial condition, to find the solutions with a very small the residual error.