On Newton-type methods with cubic convergence

  • Authors:

  • H. H. H. Homeier

  • Affiliations:

  • Science + Computing Ag, IT Services Muenchen, Ingolstädter Str. 22, D80807 München, Germany and Institut für Physikalische und Theoretische Chemie, Universität Regensburg, 9304 ...

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2005

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Abstract

Recently, there has been some progress on Newton-type methods with cubic convergence that do not require the computation of second derivatives. Weerakoon and Fernando (Appl. Math. Lett. 13 (2000) 87) derived the Newton method and a cubically convergent variant by rectangular and trapezoidal approximations to Newton's theorem, while Frontini and Sormani (J. Comput. Appl. Math. 156 (2003) 345; 140 (2003) 419 derived further cubically convergent variants by using different approximations to Newton's theorem. Homeier (J. Comput. Appl. Math. 157 (2003) 227; 169 (2004) 161) independently derived one of the latter variants and extended it to the multivariate case. Here, we show that one can modify the Werrakoon-Fernando approach by using Newton's theorem for the inverse function and derive a new class of cubically convergent Newton-type methods.