# Complex Newton Method Applet

This applet requires the use of a digital signature/certificate in order to save screenshots of graphs generated by the applet. When asked for your approval, you should run/trust the certificate.

This applet visualizes the complex-valued Newton's method. The user can operate this applet much like the Real Newton Method Applet; however, we note a few important changes to keep in mind.

## Instructions

When a function f(z) has been typed in (thus NOT selected by using the Pre-selected f(z) checkbox and corresponding drop down menu), this applet works identically to the Real Newton Method Applet. This process is computationally intensive, and so the applet will sometimes take a few moments to produce a picture when using the Graph basins of attraction feature. However, when using the Pre-defined f(z) checkbox (and corresponding drop down menu) to select f(z), the Graph basins of attraction feature works differently. In this case, since the roots of f(z) are known, the Radius of convergence is used to determine convergence to the roots. Specifically, a seed value is colored when its Newton iterate (up to the number of iterates in the box Max iterations) falls inside a Circle of convergence of one of the roots. This makes this process much faster and so is the preferred method. The Show circles of convergence checkbox makes these circles appear in the graph. The Radius of convergence, i.e., radius of the circles of convergence, can be changed by typing in a new value, and then hitting the Graph button (using the enter key will NOT effect this change). The Plot roots checkbox will make the roots appear. Also, by changing the # color shades value (and then clicking Graph/Update), the basins of attraction will exhibit more shading to help indicate how many iterates are required for a given seed value to enter the circle of convergence. For example, when # color shades is 4, then seed values of the same color and shade will take the same number of iterates mod 4 to enter the circle of convergence.

## 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. The most recently created thumbnail has a white border. All thumbnails will be deleted when the Delete thumbnails button is pressed or a new function is selected.

## 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 the viewing window (Graph).
• Graph settings will allow the user to adjust various settings such as axes style and color.
• 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.