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. 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.
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 to find roots of (Ypma 1995).
Ypma, Historical development of the Newton–Raphson method, SIAM Review 37 (4), 531–551, 1995.
Classics in Applied Mathematics, SIAM, 2000.
Kelley, Solving Nonlinear Equations with Newton's Method, no 1 in Fundamentals of Algorithms, SIAM, 2003.
Springer, Berlin, 2004.
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