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Finally, Newton views the method as purely algebraic and makes no mention of the connection with calculus.Newton may have derived his method from a similar but less precise method by Vieta.In nonlinear regression, the sum of squared errors (SSE) is only "close to" parabolic in the region of the final parameter estimates.
However, the extra computations required for each step can slow down the overall performance relative to Newton's method, particularly if or its derivatives are computationally expensive to evaluate.
The name "Newton's method" is derived from Isaac Newton's description of a special case of the method in De analysi per aequationes numero terminorum infinitas (written in 1669, published in 1711 by William Jones) and in De metodis fluxionum et serierum infinitarum (written in 1671, translated and published as Method of Fluxions in 1736 by John Colson).
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 .
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.
For situations where the method fails to converge, it is because the assumptions made in this proof are not met.
If the first derivative is not well behaved in the neighborhood of a particular root, the method may overshoot, and diverge from that root.
More details can be found in the analysis section below.
Householder's methods are similar but have higher order for even faster convergence.