Volume 5, Issue 4
Quadrature Based Optimal Iterative Methods with Applications in High-Precision Computing

Numer. Math. Theor. Meth. Appl., 5 (2012), pp. 592-601.

Published online: 2012-05

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• Abstract

We present a simple yet effective and applicable scheme, based on quadrature, for constructing optimal iterative methods. According to the, still unproved, Kung-Traub conjecture an optimal iterative method based on $n+1$ evaluations could achieve a maximum convergence order of $2^n$. Through quadrature, we develop optimal iterative methods of orders four and eight. The scheme can further be applied to develop iterative methods of even higher orders. Computational results demonstrate that the developed methods are efficient as compared with many well known methods.

• Keywords

Iterative methods, fourth order, eighth order, quadrature, Newton, convergence, nonlinear, optimal.

65H05, 65D99, 41A25

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@Article{NMTMA-5-592, author = {}, title = {Quadrature Based Optimal Iterative Methods with Applications in High-Precision Computing}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2012}, volume = {5}, number = {4}, pages = {592--601}, abstract = {

We present a simple yet effective and applicable scheme, based on quadrature, for constructing optimal iterative methods. According to the, still unproved, Kung-Traub conjecture an optimal iterative method based on $n+1$ evaluations could achieve a maximum convergence order of $2^n$. Through quadrature, we develop optimal iterative methods of orders four and eight. The scheme can further be applied to develop iterative methods of even higher orders. Computational results demonstrate that the developed methods are efficient as compared with many well known methods.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2012.m1114}, url = {http://global-sci.org/intro/article_detail/nmtma/5951.html} }
TY - JOUR T1 - Quadrature Based Optimal Iterative Methods with Applications in High-Precision Computing JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 592 EP - 601 PY - 2012 DA - 2012/05 SN - 5 DO - http://doi.org/10.4208/nmtma.2012.m1114 UR - https://global-sci.org/intro/article_detail/nmtma/5951.html KW - Iterative methods, fourth order, eighth order, quadrature, Newton, convergence, nonlinear, optimal. AB -

We present a simple yet effective and applicable scheme, based on quadrature, for constructing optimal iterative methods. According to the, still unproved, Kung-Traub conjecture an optimal iterative method based on $n+1$ evaluations could achieve a maximum convergence order of $2^n$. Through quadrature, we develop optimal iterative methods of orders four and eight. The scheme can further be applied to develop iterative methods of even higher orders. Computational results demonstrate that the developed methods are efficient as compared with many well known methods.

Sanjay Kumar Khattri. (2020). Quadrature Based Optimal Iterative Methods with Applications in High-Precision Computing. Numerical Mathematics: Theory, Methods and Applications. 5 (4). 592-601. doi:10.4208/nmtma.2012.m1114
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