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Efficient families of Newton’s method and its variants suitable for non-convergent cases

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Abstract

In this paper, we derive new one-parameter family of Newton’s method, Schröder’s method, super-Halley method and Halley’s method respectively for finding simple zeros of nonlinear functions, permitting \(f^{\prime }(x_n)=0\) at some points in the vicinity of required root. Using the newly derived family of super-Halley method, we further obtain new interesting families of famous quartically convergent Traub–Ostrowski’s and Jarratt’s methods respectively. Further, the approach has been extended to solve a system of nonlinear equations. It is found by way of illustration that the proposed methods are very useful in high-precision computing environment and non-convergent cases.

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Authors and Affiliations

  1. University Institute of Engineering and Technology, Panjab University, Chandigarh, 160 014, India

    V. Kanwar & Munish Kansal

  2. School of Mathematics and Computer Applications, Thapar University, Patiala, Punjab, 147 004, India

    Sanjeev Kumar

  3. D. A. V. College, Sector 10, Chandigarh, 160 010, India

    Arvind Garg

Authors
  1. V. Kanwar
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  2. Sanjeev Kumar
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  3. Munish Kansal
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  4. Arvind Garg
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Correspondence to V. Kanwar.

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Kanwar, V., Kumar, S., Kansal, M. et al. Efficient families of Newton’s method and its variants suitable for non-convergent cases. Afr. Mat. 27, 767–779 (2016). https://doi.org/10.1007/s13370-015-0376-x

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  • DOI: https://doi.org/10.1007/s13370-015-0376-x

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