![]() You can see that initial guesses that are close to a root converge to the nearby root in five or fewer iterations. The height of the needle indicates the number of iterations required to converge. (Click to enlarge.) The color of the needle at x indicates the root to which x converges under Newton's method. If you apply Newton's method to 250 initial conditions on the interval, you get the results that are summarized in the needle plot to the left. You can ask the following question: For each point, x, to which root does Newton's method converge when x is the initial guess? You can also keep track of how many iterations it takes for Newton's method to converge to a root. Recall that Newton's method involves iteration of the rational function N( x) = x – f( x)/f'( x), which has singularities at the critical points of f. The polynomial has critical points (where the derivative vanishes) near -2.3, -0.2, 1.5, and 2.6. (You canĭownload the SAS/IML programs that I used to create the graphs in this article.) Consider the polynomial You can perform the same kind of computer experiments for Newton's method applied to a real function. ![]() The sensitivity of Newton's method to an initial guess Click To Tweet The sensitivity of Newton's method The points that eventually converge to a root are the Fatou set, whereas the points that do not converge form the Julia set. In the picture, each point in the complex plane is colored according to which root Newton's method converges to when it begins at that point. ![]() You might have seen pictures like the one at the beginning of this article, which show the domains of attraction for Newton's iteration for a cubic polynomial. If you owned a PC in the 80's and early 90's, you might have spent countless hours computing Mandelbrot sets and Julia sets. The dynamics of Newton iteration can be quite complex. Further iterations might converge to an arbitrary root, might endlessly cycle in a periodic or aperiodic manner, or might diverge to infinity. However, if your guess is near a critical point of the function, Newton's method will produce a "next guess" that is far away from the initial guess. If you provide a guess that is sufficiently close to a simple root, Newton's method will converge quadratically to the nearby root. The behavior of Newton's method depends on the initial guess. I have previously shown how to implement Newton's method in SAS. ![]() "might not converge or might converge to a root that is far away from the root that you wanted to find." A reader wanted more information about that statement. In my article about finding an initial guess for root-finding algorithms, I stated that Newton's root-finding method ![]()
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