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This algorithm is first in the class of Householder's methods, succeeded by Halley's method. But, in the absence of any intuition about where the zero might lie, a "guess and check" method might narrow the possibilities to a reasonably small interval by appealing to the intermediate value theorem.) The method will usually converge, provided this initial guess is close enough to the unknown zero, and that .
It is important to review the proof of quadratic convergence of Newton's Method before implementing it.
Specifically, one should review the assumptions made in the proof.
To overcome this problem one can often linearise the function that is being optimized using calculus, logs, differentials, or even using evolutionary algorithms, such as the stochastic funnel algorithm.
Good initial estimates lie close to the final globally optimal parameter estimate.
The essence of Vieta's method can be found in the work of the Persian mathematician Sharaf al-Din al-Tusi, while his successor Jamshīd al-Kāshī used a form of Newton's method to solve (Ypma 1995).
A special case of Newton's method for calculating square roots was known since ancient times and is often called the Babylonian method.
More details can be found in the analysis section below.
Householder's methods are similar but have higher order for even faster convergence.
However, his method differs substantially from the modern method given above: Newton applies the method only to polynomials.
He does not compute the successive approximations .