A Numerical Study of Newton-Like Methods for Nonlinear Systems with a Singular Jacobian

摘要

In this paper we present some modifications of Newton's method for nonlinear systems with singular jacobian at the solution that converge quadratically. In the scalar case, if α is a root of f(χ) = 0 of multiplicity m, f(χ) = (χ-α)~mh(χ) with h(α) 6≠ 0, Newton's method is modified to χκ+1 = χκ - mf(χκ)/f′(χκ) in order to recover quadratic convergence. We first consider an extension of this method for nonlinear systems χ~(κ+1) = χ~(κ) - J_F (χ~(κ))~(-1)MF(χ~(κ)), Where M is a suitably chosen diagonal matrix. Quadratic convergence is proved under certain conditions on function F(χ) and on weight matrix M. This matrix is not easy to obtain analytically, so we compute a numerical estimation of it. We also generalize the method proposed in and to the case of systems with singular jacobian at the solution. We check the effectiveness of the modified methods by applying them to several nonlinear systems with singular jacobian at the solution, starting from different initial estimations. The numerical results confirm that quadratic convergence is reestablished.