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Newton Basins

Section 3.9 of Edwards & Penney (CALCULUS 5/e, Prentice Hall, 1998) includes a sub-section on "Newton's method and computer graphics". The pictures above are successive magnifications of the "Newton basins" for the 12th degree equation

that

• Encodes a flower
• With 10 leaves
• That have buds
• Encircled with flowers
• That have leaves with buds
• And so on ad infinitum

If you have MATLAB installed and would like to generate these Newton basins yourself, then download (with a Windows 95 right-click or a Macintosh option-click) the following two files:

basins12.m

newtbasn.m

You can edit the script basins12.m (with calls the function newtbasn.m where Newton's method is implemented in complex matrix form) to change the polynomial to use and to set the window you want to see. Each of these files is commented to tell what's happening. And newtbasn.m can be edited to construct the Newton basins for any polynomial you choose.

To carry out 50 iterations for each point in a 400 by 300 grid requires about 1.25 BILLION floating point operations (flops). A 266-MHz Pentium II executes about 6 million flops per second in this computation, and hence requires a bit over 3 minutes.

You might therefore like to start with a simpler situation. In order to construct the Newton basins of the cubic equation

(as illustrated in Section 3.9 of E & P), download the files

basins3.m

newtbas3.m

whose code is perhaps easier to understand. These files are easily edited to construct the Newton basins of any cubic of your choice.