Study of frozen Newton-like method in a Banach space with dynamics

M. K.  Singh; A. K. Singh

doi:10.37863/umzh.v74i2.6764

M. K. Singh Inst. Sci., Banaras Hindu Univ., Varanasi, India
A. K. Singh Inst. Sci., Banaras Hindu Univ., Varanasi, India

DOI: https://doi.org/10.37863/umzh.v74i2.6764

Keywords: Banach space Newton's method Semilocal convergence Recurrence relations Order of convergence Eciency index

Abstract

UDC 519.6

The main objective of this work is investigation of positives and negatives of the three steps iterative frozen-type Newtonlike method for solving nonlinear equations in a Banach space. We perform a local convergence analysis by Taylor’s expansion and semilocal convergence by recurrence relations technique under the conditions of Kantorovich theorem for the Newton’s method. The convergence results are examined by comparing the proposed method with the Newton’s method and the fourth order Jarratt’s method using some test functions. We discuss the corresponding conjugacy maps for quadratic polynomials along with the extraneous fixed points. Additionally, the theoretical and numerical results are examined by
using the dynamical analysis of a selected test function. It not only confirms the theoretical and numerical results, but also reveals some drawbacks of the frozen Newton-like method.

Author Biography

M. K. Singh, Inst. Sci., Banaras Hindu Univ., Varanasi, India

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