Keywords: nonlinear system, Newton's method, Quadratic convergence.

In this paper, we consider the problem of finding a real solution alpha of the nonlinear system F(x)=0, with n equations and n unknowns. The most used iterative method is the classical Newton's method. It is well known that this method requires the Jacobian matrix of F to be nonsingular in the neighborhood of alpha in order to obtain quadratic convergence. This condition restricts to some extent the application of Newton's method.

In [1,2] some variants of Newton's method which converge quadratically in spite of Jacobian matrix being singular in some iterations are proposed.

In this paper, we present several modified Newton methods for nonlinear systems F(x)=0, allowing the Jacobian matrix to be singular at the solution. Specifically, we consider the iterative method obtained from Newton's method by replacing the inverse of the Jacobian matrix by the product of this inverse and a suitably chosen diagonal matrix M=diag(m₁,m₂,...,m_n), called the weight matrix. Quadratic convergence is proved under certain conditions on the function F(x) and on the weight matrix M. We present two modifications of Newton's method: the modified Newton's method (MN), obtained by taking particular values for the weights m_i, i=1,2,...,n, and the generalized Newton's method obtained by considering a equivalent nonlinear system that depends on n parameters which are chosen in order to obtain a nonsingular Jacobian in the solution.

As matrix M is sometimes not easy to obtain analytically, we have devised an alternative way to numerically estimate the diagonal entries of M. So we obtain the modified Newton's method with estimated weights (MNE) and the generalized Newton's method with estimated weight (GNE).

We check the effectiveness of the modified methods by applying it to several nonlinear systems with singular Jacobian at the solution. The numerical results confirm that quadratic convergence is re-established and enables us to compare these variants with Newton's method. For each example, we run each method starting from 25 points randomly chosen at distance 0.5 from the solution and plot the logarithm of the distance between two consecutive iterations against the iteration number. The result is a straight line for Newton's method and a parabolic-like curve for the other methods, which reaches before the tolerance condition, graphically showing the convergence order.

1

K. Jisheng, L. Yitian, W. Xiuhua, "Efficient continuation Newton-like method for solving systems of non-linear equations", Applied Mathematics and Computation, 174, 846-853, 2006. doi:10.1016/j.amc.2005.05.031

2

X. Wu, " Note on the improvement of Newton's method for systems of nonlinear equations", Applied Mathematics and Computation, 189, 1476-1479, 2007. doi:10.1016/j.amc.2006.12.035

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