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Section B

Constructing third-order derivative-free iterative methods

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Pages 1509-1518 | Received 07 Mar 2010, Accepted 17 Aug 2010, Published online: 11 Mar 2011

References

  • Amat , S. , Busquier , S. and Gutiérrez , J. M. 2003 . Geometric constructions of iterative functions to solve nonlinear equations . J. Comput. Appl. Math. , 157 : 197 – 205 .
  • Amat , S. , Bermúdez , C. , Busquier , S. and Mestiri , D. 2009 . A family of Halley–Chebyshev iterative schemes for non-Frétchet differentiable operators . J. Comput. Appl. Math. , 228 : 486 – 493 .
  • Argyros , I. K. 1997 . The super-Halley method using divided differences . Appl. Math. Lett. , 10 : 91 – 95 .
  • Argyros , I. K. 2007 . “ More results on Newton's method ” . In Computational Theory of Iterative Methods , Edited by: Chui , C. K. and Wuytack , L. 187 – 244 . New York : Elsevier Publishing Company . Studies in Computational Mathematics, Vol. 15
  • ARPREC . “ C++/Fortran-90 arbitrary precision package ” . software available at http://crd.lbl.gov/~dhbailey/mpdist/
  • Chun , C. 2007 . Some second-derivative-free variants of Chebyshev–Halley methods . Appl. Math. Comput. , 191 : 410 – 414 . MR2385542
  • Ezquerro , J. A. and Hernández , M. A. 1998 . Avoiding the computation of the second Frechet-derivative in the convex acceleration of Newton's method . J. Comput. Appl. Math. , 96 : 1 – 12 . MR1649466
  • Ezquerro , J. A. and Hernández , M. A. 2000 . A modification of the super-Halley method under mild differentiability conditions . J. Comput. Appl. Math. , 114 : 405 – 409 .
  • Ezquerro , J. A. and Hernández , M. A. 2004 . On Halley-type iterations with free second derivative . J. Comput. Appl. Math , 170 ( 2 ) : 455 – 459 .
  • Ezquerro , J. A. and Hernández , M. A. 2007 . Halley's method for operators with unbounded second derivative . Appl. Numer. Math. , 57 : 354 – 360 .
  • Frontini , M. and Sormani , E. 2004 . Third-order methods from quadrature formulae for solving systems of nonlinear equations . Appl. Math. Comput. , 149 : 771 – 782 . MR2033160
  • Grau , M. and Díaz-Barrero , J. L. 2006 . An improvement of the Euler–Chebyshev iterative method . J. Math. Anal. Appl. , 315 : 1 – 7 . MR2196525
  • Grau , M. and Noguera , M. 2004 . A variant of Cauchy's method with accelerated fifth-order convergence . Appl. Math. Lett. , 17 : 509 – 517 .
  • Gutiérrez , J. M. and Hernández , M. A. 2001 . An acceleration of Newton's method: Super-Halley method . Appl. Math. Comput. , 117 : 223 – 239 .
  • Hernández , M. A. 2000 . Second-derivative-free variant of the Chebyshev method for nonlinear equations . J. Optim. Theory Appl , 104 ( 3 ) : 501 – 515 . MR1760735
  • Hernández , M. A. 2001 . Chebyshev's approximation algorithms and applications . Comput. Math. Appl. , 41 : 433 – 445 . MR1822563
  • Kanwar , V. and Tomar , S. K. 2007 . Modified families of Newton, Halley and Chebyshev methods . Appl. Math. Comput. , 192 : 20 – 26 .
  • Khattri , S. K. 2006 . Newton–Krylov algorithm with adaptive error correction for the Poisson–Boltzmann equation . MATCH Commun. Math. Comput. Chem. , 56 : 197 – 208 . MR2312481
  • Kou , J. 2008 . Some variants of Cauchy's method with accelerated fourth-order convergence . J. Comput. Appl. Math. , 213 : 71 – 78 .
  • Kou , J. and Li , Y. 2007 . Modified Chebyshev method free from second derivative for non-linear equations . Appl. Math. Comput. , 187 : 1027 – 1032 . MR2323110
  • Kou , J. and Li , Y. 2008 . On Chebyshev-type methods free from second derivative . Commun. Numer. Methods Eng. , 24 : 1219 – 1225 . MR2474681
  • Kou , J. , Li , J. Y. and Wang , X. 2006 . A uniparametric Chebyshev-type method free from second derivatives . Appl. Math. Comput. , 179 : 296 – 300 . MR2260878
  • Kou , J. , Li , Y. and Wang , X. 2006 . A modification of Newton method with third-order convergence . Appl. Math. Comput. , 181 : 1106 – 1111 . MR2269989
  • Kou , J. , Li , Y. and Wang , X. 2006 . On a family of second-derivative-free variants of Chebyshev's method . Appl. Math. Comput. , 181 : 982 – 987 . MR2269977
  • Kou , J. , Li , Y. and Wang , X. 2006 . Modified Halley's method free from second derivative . Appl. Math. Comput. , 183 : 704 – 708 . MR2286232
  • Kou , J. , Li , Y. and Wang , X. 2007 . Fourth-order iterative methods free from second derivative . Appl. Math. Comput. , 184 : 880 – 885 . MR2294954
  • Li , S. , Li , H. and Cheng , L. 2009 . Some second-derivative-free variants of Halley's method for multiple roots . Appl. Math. Comput. , 215 : 2192 – 2198 .
  • Noor , K. I. and Noor , M. A. 2007 . Predictor–corrector Halley method for nonlinear equations . Appl. Math. Comput. , 188 : 1587 – 1591 .
  • Noor , M. A. , Khan , W. A. and Hussaina , A. 2007 . A new modified Halley method without second derivatives for nonlinear equation . Appl. Math. Comput. , 189 : 1268 – 1273 .
  • Osada , N. 2008 . Chebyshev–Halley methods for analytic functions . J. Comput. Appl. Math. , 216 : 585 – 599 .
  • Özban , A. Y. 2004 . Some new variants of Newton's method . Appl. Math. Lett. , 17 : 677 – 682 . MR2064180
  • Potra , F. A. and Pták , V. 1984 . Nondiscrete induction and iterative processes . Res. Notes Math. , 103 : 121 – 150 . MR0754338
  • Rafiq , A. , Awais , M. and Zafar , Z. 2007 . Modified efficient variant of super-Halley method . Appl. Math. Comput. , 189 : 2004 – 2010 .
  • Reeves , R. 1991 . A note on Halley's method . Comput. Graph. , 15 : 89 – 90 .
  • Traub , J. F. 1964 . “ Iterative Methods for Solution of Equations ” . Englewood Cliffs, NJ : Prentice-Hall .
  • Wu , Q. B. and Zhao , Y. Q. 2006 . The convergence theorem for a family deformed Chebyshev method in Banach space . Appl. Math. Comput. , 182 : 1369 – 1376 . MR2282579
  • Xiaojian , Z. 2008 . Modified Chebyshev–Halley methods term free from second derivative . Appl. Math. Comput. , 203 : 824 – 827 .
  • Xu , X. and Ling , Y. 2009 . Semilocal convergence for Halley's method under weak Lipschitz condition . Appl. Math. Comput. , 215 : 3057 – 3067 .
  • Ye , X. and Li , C. 2006 . Convergence of the family of the deformed Euler Halley iterations under the Hölder condition of the second derivative . J. Comput. Appl. Math. , 194 : 294 – 308 .
  • Yueqing , Z. and Wu , Q. 2008 . Newton–Kantorovich theorem for a family of modified Halley's method under Hölder continuity conditions in Banach space . Appl. Math. Comput. , 202 : 243 – 251 . MR2437155

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