Newton's method for singular nonlinear equations using approximate left and right nullspaces of the Jacobian

The convergence of Newton's method to a solution x∗ of f(x)=0 may be unsatisfactory if the Jacobian matrix f′(x∗) is singular. When the rank deficiency is one, and a simple regularity condition is satisfied at x∗, it is possible to define a bordered system for which Newton's 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 f′(x¯) for some point x¯ near x∗ can be used to form a bordered system for which Newton's method converges quadratically. The theory and method are illustrated by several examples.