The secant method is an iterative technique for numerically solving equations that is very similar to Newton's method. For details, see the Technical page on this. It converges more slowly to the solution than Newton's method does. That may sound undesirable, and when used for its intended purpose (solving equations) it is. It does have the advantage of not requiring knowledge of the derivative of the function that you're trying to solve.
The main benefit of this method from the standpoint of fractal art is that it makes much more interesting pictures than Newton's method or the more sophisticated variations.
Two videos have been made based on the secant method. SecantAnimation1 uses the cosine function, and SecantAnimation2 uses (z+1)(z-1)(z+2)(z-2).
Here we apply the secant method to the cosine function. This fractal was used in SecantAnimation1. The top image shows an overview of this amazingly complex fractal. It is centered at (0,0), and each of the points surrounded by a violet ring is a multiple of 2 pi. If you look very closely you will notice that they are all different, unlike most fractals in this family. This means there is great potential for deep zooming. The second image is a magnification to a size of 1 vertically. The third and fourth images are progressively deeper zooms off to the side, around -12 pi. Note in the third image that all the detail is different in each period. Look at how complex this thing is! Exploring all this is going to be a lot of fun. |