Complex Damped Newton's Method for TI-85/6

This program graphs the complex Newton's method so you can see how initial points converge to zeros. A damping factor of about .3 gives a nicer, but perhaps not accurate, picture.

Enter the following as a new program in your TI-85/6. Do not enter the comments starting with #

PROGRAM:NEWTG
:ClDrw
:Lbl L1
:Input x
:For(J,1,15)    #15 is the number of iterations
:real x ->U
:imag x ->V
:PtOn(U,V)
:PtOn(U,-V)     #optional - good for real polynomials
:x-H*F/D->x
:End
:DispG
:Pause          #optional, gives time to look at graph
:Goto L1

Note that -> is STO, real, imag come from the CPLX menu and PtOn and ClDrw comes from the GRAPH-DRAW menu, Lbl,For,End,Pause from CTL menu and DispG and Input from the I/O menu in program mode.

Before you run the program you must set the following variables:

F will be the polynomial -- Use ALPHA-= (on STO key) to define the polynomial. Use the variable x-var
D will be the derivative of F, you should calculate this yourself. Enter as above.
H is the damping factor between 0 and 1. A good value is .3 Use STO to set this variable.

Also you need to go to the GRAPH menu, unselect any graphs and set the range, remembering that x is the real axis and y is the imaginary axis. You may also want to go to the GRAPH-FORMAT menu and select or de-select axes, grid, etc.

Then go to the PRGM-NAMES menu and select NEWTG

At the ? prompt enter the complex initial point a + bi in the form (a,b), for example:

?(.3,2)

You should generally pick the initial point on the boundary of your range for best results.

After plotting some points the program will pause so you can see it, usually you can tell as the screen gets darker. Press ENTER and put a new initial point. Occasionally Newton's method does not do enough iterations to get to a root. You can continue from the same spot by entering x-VAR.

This program has no formal end, to stop it when you are done press ON, then QUIT