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.
Newton's method was used by 17th-century Japanese mathematician Seki Kōwa to solve single-variable equations, though the connection with calculus was missing.
instead of the more complicated sequence of polynomials used by Newton.
Arthur Cayley in 1879 in The Newton–Fourier imaginary problem was the first to notice the difficulties in generalizing Newton's method to complex roots of polynomials with degree greater than 2 and complex initial values.
This opened the way to the study of the theory of iterations of rational functions.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 .It is only here that the Hessian matrix of the SSE is positive and the first derivative of the SSE is close to zero.In a robust implementation of Newton's method, it is common to place limits on the number of iterations, bound the solution to an interval known to contain the root, and combine the method with a more robust root finding method.Finally, in 1740, Thomas Simpson described Newton's method as an iterative method for solving general nonlinear equations using calculus, essentially giving the description above.In the same publication, Simpson also gives the generalization to systems of two equations and notes that Newton's method can be used for solving optimization problems by setting the gradient to zero.If a stationary point of the function is encountered, the derivative is zero and the method will terminate due to division by zero.A large error in the initial estimate can contribute to non-convergence of the algorithm.Newton's method requires that the derivative can be calculated directly.An analytical expression for the derivative may not be easily obtainable or could be expensive to evaluate.