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

• Autores: Stepan Yu. Gatilov
• Localización: Journal of computational and applied mathematics, ISSN 0377-0427, Vol. 272, Nº 1, 2014, págs. 8-24
• Idioma: inglés
• DOI: 10.1016/j.cam.2014.04.024
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• Resumen

  • 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.