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## A simple algorithm for high order Newton iteration formulae and some new variants

#### Mustafa TURKYİLMAZOGLU [1]

The high order Newton iteration formulas are revisited in this paper. Translating the nonlinear root finding problem into a fixed point iteration involving an unknown general function whose root is searched, a double Taylor series is undertaken regarding the root and the root finding function. Based on the error analysis of the expansion, a simple algorithm is later proposed to construct Newton iteration formulae of any order commencing from the traditional linearly convergent fixed point iteration method and quadratically convergent Newton-Raphson method of frequently at the disposal of the scientific community. It is shown that the well-known variants like the Halley's method or Haouseholder's methods of high order can be reproduced from the general case outlined here. Some further rare single-step classes of any order are shown to be derivable from the presented algorithm. Finally, some new higher order accurate variants are also offered taking into account multi-step compositions which demand less computation of higher derivatives. The efficiency, accuracy and performance of the proposed methods and also their potential advantages over the classical ones are numerically demonstrated and discussed on some well-documented examples from the open literature.
Newton iterations, High order methods, Convergence, New singlestep variants
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Primary Language en Mathematics Mathematics Orcid: 0000-0003-0412-4580Author: Mustafa TURKYİLMAZOGLU (Primary Author)Institution: HACETTEPE UNIVERSITYCountry: Turkey Publication Date : February 6, 2020
 Bibtex @research article { hujms459810, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {425 - 438}, doi = {10.15672/hujms.459810}, title = {A simple algorithm for high order Newton iteration formulae and some new variants}, key = {cite}, author = {TURKYİLMAZOGLU, Mustafa} } APA TURKYİLMAZOGLU, M . (2020). A simple algorithm for high order Newton iteration formulae and some new variants. Hacettepe Journal of Mathematics and Statistics , 49 (1) , 425-438 . DOI: 10.15672/hujms.459810 MLA TURKYİLMAZOGLU, M . "A simple algorithm for high order Newton iteration formulae and some new variants". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 425-438 Chicago TURKYİLMAZOGLU, M . "A simple algorithm for high order Newton iteration formulae and some new variants". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 425-438 RIS TY - JOUR T1 - A simple algorithm for high order Newton iteration formulae and some new variants AU - Mustafa TURKYİLMAZOGLU Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.459810 DO - 10.15672/hujms.459810 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 425 EP - 438 VL - 49 IS - 1 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.459810 UR - https://doi.org/10.15672/hujms.459810 Y2 - 2018 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics A simple algorithm for high order Newton iteration formulae and some new variants %A Mustafa TURKYİLMAZOGLU %T A simple algorithm for high order Newton iteration formulae and some new variants %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 1 %R doi: 10.15672/hujms.459810 %U 10.15672/hujms.459810 ISNAD TURKYİLMAZOGLU, Mustafa . "A simple algorithm for high order Newton iteration formulae and some new variants". Hacettepe Journal of Mathematics and Statistics 49 / 1 (February 2020): 425-438 . https://doi.org/10.15672/hujms.459810 AMA TURKYİLMAZOGLU M . A simple algorithm for high order Newton iteration formulae and some new variants. Hacettepe Journal of Mathematics and Statistics. 2020; 49(1): 425-438. Vancouver TURKYİLMAZOGLU M . A simple algorithm for high order Newton iteration formulae and some new variants. Hacettepe Journal of Mathematics and Statistics. 2020; 49(1): 438-425.

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