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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. 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 ...
Newton's method, in its original version, has several caveats: It does not work if the Hessian is not invertible. This is clear from the very definition of Newton's method, which requires taking the inverse of the Hessian. It may not converge at all, but can enter a cycle having more than 1 point. See the Newton's method § Failure analysis.
The Newton fractal is a boundary set in the complex plane which is characterized by Newton's method applied to a fixed polynomial p(z) ∈ [z] or transcendental function. It is the Julia set of the meromorphic function z ↦ z − p ( z ) / p′ ( z ) which is given by Newton's method.
Newton's form has the simplicity that the new points are always added at one end: Newton's forward formula can add new points to the right, and Newton's backward formula can add new points to the left. The accuracy of polynomial interpolation depends on how close the interpolated point is to the middle of the x values of the set of points used ...
Julia set (in white) for the rational function associated to Newton's method for f : z → z 3 −1. Coloring of Fatou set in red, green and blue tones according to the three attractors (the three roots of f). For some functions f(z) we can say beforehand that the Julia set is a fractal and not a simple curve. This is because of the following ...
Calculation of the Hessian adds to the complexity of the algorithm. This method is not in general use. Davidon–Fletcher–Powell method. This method, a form of pseudo-Newton method, is similar to the one above but calculates the Hessian by successive approximation, to avoid having to use analytical expressions for the second derivatives.
In numerical analysis, a quasi-Newton method is an iterative numerical method used either to find zeroes or to find local maxima and minima of functions via an iterative recurrence formula much like the one for Newton's method, except using approximations of the derivatives of the functions in place of exact derivatives.
One can use linear algebra, by taking the coefficients of the interpolating polynomial as unknowns, and writing as linear equations the constraints that the interpolating polynomial must satisfy. For another method, see Chinese remainder theorem § Hermite interpolation. For yet another method, see, [1] which uses contour integration.