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In nonlinear regression, the sum of squared errors (SSE) is only "close to" parabolic in the region of the final parameter estimates.
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function.
The most basic version starts with a single-variable function until a sufficiently precise value is reached.
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.
In these situations, it may be appropriate to approximate the derivative by using the slope of a line through two nearby points on the function.
Using this approximation would result in something like the secant method whose convergence is slower than that of Newton's method.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 is an extremely powerful technique—in general the convergence is quadratic: as the method converges on the root, the difference between the root and the approximation is squared (the number of accurate digits roughly doubles) at each step.However, there are some difficulties with the method.An example of a function with one root, for which the derivative is not well behaved in the neighborhood of the root, is the root will not be overshot at all.In some cases, Newton's method can be stabilized by using successive over-relaxation, or the speed of convergence can be increased by using the same method.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 .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 .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.