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

Stepan Yu. Gatilov

• 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.