The name is absent

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2.2.5 Newton’s method

Newton’s method (also known as the Newton-Raphson method) is one of the classic
numeric analysis tools, named after Isaac Newton and Joseph Raphson. It is perhaps
the best known method for finding successively better approximations to the roots
of a real-valued function. Newton’s method can often converge remarkably quickly,
especially if the iteration begins “sufficiently near” the desired root.

Given a function f(x) and its derivative ʃ'(ɪ), we start from a first guess xq. A
better approximation xγ is

The iteration process of the method is as follows: one starts with an initial guess
which is reasonably close to the true root, then the function is approximated by its
tangent line, and one computes the х-intercept of this tangent line. This x-intercept
will typically be a better approximation to the function’s root than the original guess.
This process iterates until the new х-intercept converges to a value, which would be
the best estimate of the root. Figure 2.2 is an illustration of one iteration of Newton’s
method.

By definition, the derivative at a given point is the slope of a tangent at that
point. Furthermore, the Equation 2.2 can be extended to later iterations. That is,
for n = 0,1, 2,..., we have

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