Can someone explain to me what sensitivty refers to when it comes to root finding?
When we say it is sensitive to initial value for x, is because there are multiple roots, or because there might be a critical point nearby or what? I hade a trigonometric function where the minimi point was under x-axis, then it was very sensistive and would converge to the other one etc. It is the orange graph in the picture below and I want it to converge to the last root which is around 4.5.
Then when we lifted the function up a bit, and thus the last root has changed to being on the maxima which is around 2. Here I feel like even when I get a little far away, it still converges to 2 unlike the first one. Anyways is this what they are referring to when in the question they said to study and compare the sensitivty of the initial value for x?
The convergence of Newton's Method is sensitive to initial conditions (even if the initial point isn't obviously near a critical point) and can be difficult to predict. In many cases, applying the method to a function in the complex plane leads to intricate fractal regions of convergence for each zero. Pictured is the fractal for the polynomial $z^{3}-2z+2$ (red regions fail to converge to any zero)
Source: https://en.wikipedia.org/wiki/Newton_fractal?useskin=vector#/media/File:Newton_z3-2z+2.png