# Newton Raphson C3 Coursework

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.

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 .

## Newton Raphson C3 Coursework

The method can also be extended to complex functions and to systems of equations. Furthermore, for a zero of multiplicity 1, the convergence is at least quadratic (see rate of convergence) in a neighbourhood of the zero, which intuitively means that the number of correct digits roughly doubles in every step.

In nonlinear regression, the sum of squared errors (SSE) is only "close to" parabolic in the region of the final parameter estimates.

Initial estimates found here will allow the Newton–Raphson method to quickly converge.

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 .

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