Geodesic
How to increase convergence order of the Newton method to $2\times m$?
Applications of Mathematics, Tome 59 (2014) no. 1, pp. 15-24.

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We present a simple and effective scheme for forming iterative methods of various convergence orders. In this scheme, methods of various convergence orders, such as four, six, eight and ten, are formed through a modest modification of the classical Newton method. Since the scheme considered is a simple modification of the Newton method, it can be easily implemented in existing software packages, which is also suggested by the presented pseudocodes. Finally some problems are solved, to very high precision, through the proposed scheme. Numerical work suggests that the presented scheme requires less number of function evaluations for convergence and it may be suitable in high precision computing.
DOI : 10.1007/s10492-014-0038-6
Classification : 41A25, 65D99, 65H05
Keywords: iterative method; fourth order convergent method; eighth order convergent method; quadrature; Newton method; convergence; nonlinear equation; optimal choice 
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Khattri, Sanjay Kumar. How to increase convergence order of the Newton method to $2\times m$?. Applications of Mathematics, Tome 59 (2014) no. 1, pp. 15-24. doi : 10.1007/s10492-014-0038-6. http://geodesic.mathdoc.fr/articles/10.1007/s10492-014-0038-6/

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