# Real 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 real-valued Newton method.  The screen shot and text below will guide you through how to use the applet.

## Basic Operation

By entering a function f(x) (either by typing it in 3 (with help in 1) or by selecting one from the drop down menu of pre-selected functions 2) and then clicking the Update F(x) button 5, the user will see the graph of f(x) appear in the Viewing window 17 as well as see that the Newton Map F(x) has been calculated 4. By clicking on the graph of f(x), a seed value for Newton's method will be selected and the corresponding tangent line will appear. One can also type in the initial value x_0 directly 6 and then click the Update button 7. One can then iterate the Newton map by clicking on the + button next to Iterate F(x) see 8 (or iterate 10 or 50 steps at a time with the corresponding buttons 10). The successive Newton method approximations are then plotted and also displayed in a table 11, along with the corresponding f(x) values. (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 in 9 the user can produce the first n Newton approximations x_n. The - button 8 will allow the user to reverse the process, one step at a time.

## 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 15).
• Scrolling the mouse wheel while the mouse is in the graphing window zooms in/out.
• Moving the slider on the zoom panel 14 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 13 (set to (0,0) in the above screen shot). Ctrl+click in viewing window will reset zoom center to the cursor value.

For all zoom methods, the Default View button 20 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 18. The user can also choose their own viewing window parameters by typing in the horizontal and vertical ranges manually 16. This view can then be set as the default by selecting Capture view in the drop down menu 18. Crossed or boxed axes may be selected with the buttons in 19.

## Graphing basins of attraction

Do not use the Graph basins of attraction feature until told to do so in the text. To color the basins of attraction check the Graph basins of attraction checkbox and then click on the Graph button that appears. For each seed value in the domain in the current viewing window, the Newton Iterates are computed (up to the number of iterates in the box Max iterations) until the iterates are within Step tolerance of each other, and it is deemed that a root of f(x) has been found. The seed value is then colored according to which root of f(x) it has found, or colored black if no root has been found. This process can be tweaked by the user when new parameters for Max iterations and Steptolerance are entered.

## 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.