Using low-rank approximation of the Jacobian matrix in the Newton–Raphson method to solve certain singular equations

Abstract

It is well-known that the pseudoinverse Newton–Raphson method converges locally if the rank of the Jacobian matrix is constant.A weaker assumption is considered: a set of zeros Z is a smooth manifold of dimension k, and the rank of the Jacobian is exactly n−k at all zeros. Low-rank approximation of the Jacobian matrix is used.It is proved that Newton–Raphson quadratically converges in this case. Also, the predictor–corrector approach can be used to trace a curve of zeros if k=1.The application considered belongs to the field of computer-aided geometric design. The method is applied to trace a curve of tangential intersection of two parametric surfaces. Some experimental results are shown, suggesting that the method is stable.