Newtonʹs method for singular nonlinear equations using approximate left and right nullspaces of the Jacobian Original Research Article

The convergence of Newtonʹs method to a solution x∗x∗ of f(x)=0f(x)=0 may be unsatisfactory if the Jacobian matrix f′(x∗)f′(x∗) is singular. When the rank deficiency is one, and a simple regularity condition is satisfied at x∗x∗, it is possible to define a bordered system for which Newtonʹs method converges quadratically [Griewank, SIAM Rev. 27 (1985) 537]. In this paper we extend this technique to the case of higher rank deficiencies. We show that if a generalized regular singularity condition is satisfied then one singular value decomposition of View the MathML sourcef′(x¯) for some point View the MathML sourcex¯ near x∗x∗ can be used to form a bordered system for which Newtonʹs method converges quadratically. The theory and method are illustrated by several examples.