Cubic Polynomial Complex Newton Method

Basic Operation

By selecting a value for ρ, either by typing in the value and clicking the Update button or clicking on the (left) Parameter Plane of ρ values window, the basins of attraction for Newton's method are then colored in the (right) Dynamic Plane (for Newton's method) of z values window. The picture is drawn in the same way as in the Complex Newton Method Applet (when a pre-defined function is selected). Each ρ in the parameter plane is colored according to the root which the free critical point (1+ρ)/3 finds. After clicking in the parameter plane, the ρ value can be moved by using the arrow keys on your keyboard.

In the center of the applet we have the following:

• Plot roots of z(z-1)(z-ρ) checkbox which when checked will plot in purple the point 0, 1, ρ, i.e., the roots of p_ρ(z) = z(z-1)(z-ρ).
• Show critical orbit checkbox which when checked will plot the critical orbit. Below the ρ value is the Period of the cycle (if any) detected by the critical orbit. The Connect critical orbit points checkbox, when checked, will draw line segments connecting orbit points (to make them easier to see).
• Show orbit 1 and Show orbit 2 checkboxes: By clicking in the Dynamic plane viewing window, a seed 1 value z_0 will be selected.  One can also type in the seed 1 value directly and then click the Update button. One can then iterate the Newton map F(z) by checking the Iterate orbits checkbox, and then clicking the + button that appears. The successive orbit values are then plotted and also displayed in a table under the Orbits 1 and 2 tab. (These values can be copied and pasted into another file in the usual way, if one wishes to.) By typing in a value for n and clicking on the + button next to Iterate orbits, the user can produce the first n orbit points z_1,..., z_n. The - button will allow the user to reverse the process, one step at a time. Also, by checking the box Show orbit 2, one can choose a second seed value to be iterated simultaneously. After checking the box Show orbit 1 or Show orbit 2, a Connect orbit points checkbox appears which when checked will draw line segments connecting orbit points (to make them easier to see).
• Update button will update all pictures and computations based on parameter settings.
• Show circles of convergence radius checkbox will show the circles (in the dynamic plane) centered at the roots of  p_ρ(z) = z(z-1)(z-ρ)   with valid radii of convergence (which were determined via an Additional Exercise in the text) .

Under the Settings tab, the user can adjust the following by entering in a new value and then clicking the Update button:

• Parameter plane max iterations value is the number points in the critical orbit that must be checked for convergence to a root of  p_ρ(z) = z(z-1)(z-ρ)  before a ρ value in the parameter plane is colored black.
• When a new ρ value is selected by clicking in the Parameter Plane of ρ values a straight line path of k steps is taken between the old ρ value and the new ρ value. At each step, the applet plots the corresponding picture in the dynamic plane. Here k takes on the value listed as the Number of path steps. Also, Delay between path steps is the number of milliseconds the applet pauses before computing the next Dynamic plane picture.
• Parameter plane color shades and Dynamic plane color shades determine the number of color shades in each picture.
• Root tolerance is a value such that the applet will treat a parameter ρ as 0 when it is within this distance of 0. Similarly, it will treat a parameter ρ as 1 when it is within this distance of 1. For technical reasons, the radii of convergence, and thus the basins of attraction, are calculated differently for such values (since there are only 2 distinct roots of p_ρ(z) = z(z-1)(z-ρ), instead of 3 when ρ is either 0 or 1.
• Dynamic plane max iterations value is the number of points in the orbit of each z value that must be checked for convergence to a root of  p_ρ(z) = z(z-1)(z-ρ) before that z value in the dynamic plane is colored black.
• Dynamic plane min iterations value is the number of iterations made before the convergence condition is checked (default is 0).
• Orbit max iterations is the largest number of orbit points that will be plotted for orbit 1, orbit 2, or the critical orbit.
• Orbit tolerance is the value used to determine if a periodic point has been found by the critical orbit. If two points on the critical orbit are within this much of each other, then it is deemed that the critical orbit is converging to a cycle.
• Parameter plane black/white plot and Dynamic plane black/white plot checkboxes will toggle between a color graph and a black/white graph (without needing to click the Update button).

Zooming on pictures

Zooming on any graph can be done by in a variety of ways.

• Depressing the mouse and dragging it across the graph zooms in.
• Right/left click on the graph zooms in/out (user will have to first check the Zoom with mouse click on).
• Scrolling the mouse wheel while the mouse is in the graphing window zooms in/out.
• Moving the slider on the zoom panel with a mouse drag or the right/left arrows on the keyboard zooms in/out.

The last three methods of zoom can either be centered at the cursor or at a preset default point. Ctrl+click in viewing window will reset zoom center to the cursor value.

For all zoom methods, the Default View button will return the viewing rectangle to the default setting. Default settings can be changed by selecting one from the drop down menu just above the viewing window. The user can also choose their own viewing window parameters by typing in the horizontal and vertical ranges manually. This view can then be set as the default by selecting Capture view in the drop down menu.

Thumbnail pictures

Thumbnail pictures at the bottom of the applet record previous pictures which can be restored (along with the corresponding applet parameters) by clicking on them. After creating 30 such thumbnails, the first created thumbnail will be overwritten (thus losing the picture that was previously there). The most recently created thumbnail has a white border (red border if picture is in black/white). The last enlarged thumbnail is surrounded by a yellow border. All thumbnails will be deleted when the Delete thumbnails button is pressed.

Exporting pictures and Applet Settings

At the top left of the applet are the following drop down menus:

• Export will allow the user to save a png, gif, or jpg picture of the either the whole applet or of the individual viewing windows (Parameter plane or Dynamic plane).
• Colors will allow the user to adjust the color of the text used in the headings of the various sections of the applet.

This material is based upon work supported by the National Science Foundation under Grant No. 0632976.

Version 1.0 (with menu bar).