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Topic: prgmNEWT
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Alvar J. Garcia

Posts: 304
Registered: 12/6/04
prgmNEWT
Posted: Nov 1, 1997 12:34 PM
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Greetings to ALL members of the ListServs:

Thank you all for you kind words and suggestions wrt Newton's
Method to find Roots. Due to popular demand, what follows are the
programs my students and I came up with in class.

Method 1: On the TI 82/83 one can do recursive functions right on the home
screen.
STEP 1: let y1=f(x) on the y= screen
STEP 2: enter the following store commands on the home screen
1 -> G
G - y1(G)/nDeriv(y1,x,G) -> G
where the '1' is your initial guess (good for the
y1 = x^3 + x - 1 example)
STEP 3: Hit <Enter> to get the first recursion
STEP 4: Continue hitting <Enter> until you get the desired
convergence or you've demonstrated divergence making
for a pathological f(x).

Method 2: The method above will not work on the TI 85/86 as y1(G) does not
mean f(G) but f(x)*G for the last value of x stored in RAM! Hence, the
following program.

prgmNEWT
:Disp "Guess?"
:Input G
:Disp "Number of Tangents?"
:Input N
:ClrHome
:For(I,1,N,1)
: Disp G
: G - y1(G)/nDeriv(y1,x,G) -> G
:End

This program requires that you define y1=f(x), the function whose root you
are estimating, in the y= screen before you run it. The program will
generate a table of values converging to the root in question (closest to
your initial guess) or divergent values for pathological f(x).

Method 3: This is the final program (finessing prgmNEWT with graphics).
After you define y1 in the y= screen and get an appropriate viewing window
showing the root in question, this program displays the convergence table,
as prgmNEWT does, then displays a graph of f(x) with all the tangent lines
also demonstrating convergence or divergence of the tangent line roots to
the root of f(x). We acomplish this by storing all the xn's in L1 as an
array of values to plot the tangent lines after the table loop has
completed.

prgmNEWTG
:{0} -> L1
:Disp "Guess?"
:Input G
:Disp "Nunber of Tangents?"
:Input N
:ClrHome
:For(I,1,N,1)
: Disp G
: G -> L1(I)
: G - y1(G)/nDeriv(y1,x,G) -> G
: If int (I/7) = I/7
: Then
: Pause
: ClrHome
: End
:End
:Pause
:"nDeriv(y1,x,L1)(x-L1)+y1(L1)" -> y2
:DispGraph

For these programs to work on the TI 85/86, simply change
G - y1(G)/nDeriv(y1,x,G) -> G
to G - y1/nDeriv(y1,x,G) -> G
and add G -> x right before the same line of code.

Enjoy!

Regards,
A. Jorge Garcia
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Teacher/Professor Applied-Mathematics/Computer-Science Baldwin-SHS/Nassau-CC
http://freenet.buffalo.edu/~aj317
mailto:aj317@freenet.buffalo.edu
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