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Why do you think your paper is
highly cited?
|
“In this paper we provide a
new interpretation of the classical Newton
method...” |
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In this paper we provide a new interpretation of the
classical Newton method, showing that it is a particular
case of a general class of iterative methods obtained by the
use of quadrature formulas.
Does it describe a new discovery, methodology, or synthesis of
knowledge?
We use a non-classical interpretation of Newton’s method
to introduce a new class of iterative methods with order of
convergence three, while Newton’s method is only of order
two.
Would you summarize the significance of your paper in layman’s
terms?
Newton’s method may be obtained by many different
approaches: geometric interpretation, Taylor expansion,
fixed-point iteration of order two, or by using a quadrature
formula of order zero to approximate the integral arising
from Newton’s theorem. We have proved that it is possible to
obtain a class of methods, with order of convergence three,
by using a quadrature formula of order higher than zero.
How did you become involved in this research and were there any
particular problems encountered along the way?
While we were studying different methods to compute the
zeros of a nonlinear equation and looking for high-order
methods involving only the first derivative, we obtained the
new class.
Where do you see your research leading in the future?
This interpretation of Newton’s method as a particular
one in a new class has provided an impetus toward the study
of iterative methods with higher convergence order and lower
derivatives.
Marco Frontini, Ph.D.
Professor
Dipartimento di Matematica
Politecnico di Milano
Milano, Italy |
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