Newton’s Method on TI83/84 or TI89
Copyright © 2002–2020 by Stan Brown
Copyright © 2002–2020 by Stan Brown
Summary: Newton’s Method is a fast way to home in on real solutions of an equation. Your TI83/84 or TI89 can do Newton’s Method for you, and this page shows two ways.
Newton’s Method is iterative, meaning that it uses a process or recipe to move from each guess x_{n} to the next guess x_{n+1}. The recipe for Newton’s Method is shown at right. This recipe takes a tangent line to the curve at x = x_{n}, finds the x value where that line crosses the x axis, and uses that x value as the next guess x_{n+1}. Any calculus textbook will have an illustration of the method.
What we are “guessing” is a zero of a realvalued function. If the guesses get progressively closer to the desired point, we say that the method converges. It turns out that if Newton’s Method converges at all to a particular zero, it converges rapidly, meaning that it takes relatively few steps. (See your textbook for conditions in which Newton’s Method converges.)
To start either method, put the equation you want to solve into f(x) = 0 form. Technically Newton’s Method finds zeroes of a function, not roots of an equation. Therefore you would rewrite something like x sin x = 2 as x sin x − 2 = 0. (Remember from algebra that a zero of function f is the same as a solution or root of the equation f(x) = 0 or an x intercept of the graph of f.)
To practice Newton’s Method, let’s find the square root of 2, since it will be easy to check the answer. √2 is a solution of x = √2 or x² = 2. (Yes, −√2 is a solution of this new equation but not of the original equation. But we’ll get the positive root because of our choice of initial guess.)
To start, rewrite it in the form f(x) = x² − 2 = 0. Then, differentiate the function: f′(x) = 2x.
Enter the function as Y1 and the derivative as Y2.  TI83/84: Press [Y= ].
In Y1, press [ x,T,θ,n ] [x² ] [− ] [2 ].
In Y2, press [ 2 ] [x,T,θ,n ].
Then cursor back to the = sign and press
[ENTER ]. This tells the TI83/84 not to graph the derivative
Y2.

TI89: Press [◆ ] [Y= ].
In Y1, press [ X ] [^ ] [2 ] [− ] [2 ] [ENTER ]
In Y2, press [ 2 ] [X ] [ENTER ] then [▲ ] [F4 ]
to tell the TI89 not to graph the derivative Y2.

(optional) Graph the function to plan your initial guess.  TI83/84: Press [ZOOM ] [6 ] for
ZStandard (standard zoom). 
TI89: Press [F2 ] [6 ] for ZoomStd . 
Return to the home screen.  TI83/84: Press [2nd MODE makes QUIT ]. 
TI89: Press [2nd ESC makes QUIT ]. 
Next you’ll set x to your first guess, then program the recipe for getting the next guess. Let’s guess 1 to start.
Store the initial guess, 1, in x.  TI83/84: Press [1 ] [STO→ ] [x,T,θ,n ]. 
TI89: Press [1 ] [. ] [STO→ ] [X ].
The decimal point is important: it tells the TI89 that you want decimal approximations rather than exact numbers. 
Since y1 is the lefthand side of the equation to be solved
and y2 is the derivative, you want to subtract y1/y2 from x and store
the result in x as the next guess.
After the first iteration, you see the result is 1.5. 
TI83/84: [x,T,θ,n ] [− ]
[ VARS ] [► ] [1 ] [1 ] pastes Y1 to screen.
[ ÷ ] [VARS ] [► ] [1 ] [2 ] pastes /Y2 to screen.
[ STO→ ] [x,T,θ,n ] [ENTER ]
computes the next guess and stores it in X.

TI89: You must supply the argument (x) to the two functions.
[ X ] [− ] [Y ] [1 ] [( ] [X ] [) ] [÷ ] [Y ] [2 ] [( ] [X ] [) ] [STO→ ] [X ]

Now iterate until the guesses converge.  Both: Each additional [ENTER ] keypress computes a further guess.
After just four more iterations, the process has converged. (If you want, you can press [ ^ ] [2 ] to verify that
this is indeed the square root of 2.)  
TI83/84:  TI89: 
The TI83/84 and TI89 are programmed somewhat differently. You can download programs for both calculator families in the file NEWTON.ZIP (31 KB, revised 28 Dec 2016), or you can key in the program following the directions below.
The TI83/84 program to run Newton’s Method should work for any function, even one you don’t know how to differentiate, because it computes the derivative numerically. The TI89 program will work for any differentiable function, because the TI89 finds the algebraic derivative for you.
Create a new program.  TI83/84: [PRGM ] [◄ ] [1 ] 
TI89: [APPs ] [7 ] [3 ] 
Name the program NEWTON.  TI83/84: The calculator has put you in alpha mode already. Look for the
green letters above each key and press
[N ] [E ] [W ] [T ] [O ] [N ]. Carefully check the screen, and
if necessary back up and make corrections. Press
[ENTER ]. 
TI89: Press [▼ ] [▼ ] to get to the
program name screen.
The calculator has put you in alpha mode already. Look for the
purple letters above each key and press
[N ] [E ] [W ] [T ] [O ] [N ]. Carefully check the screen, and
if necessary back up and make corrections. Press
[ENTER ]. 
Enter this program, adapted from LarsonHostetlerEdwards
Calculus 6/e.
(Check your calculator manual if you need help.)
Special notes for the TI89 program:

TI83/84: Disp "INITIAL GUESS" Input X 1→N Lbl 1 XY1/nDeriv(Y1,X,X)→R If abs(XR)<abs(X/1E10) Goto 2 R→X Disp {X,N} N+1→N Goto 1 Lbl 2 Disp "ROOT=",R Disp "ITER=",NAfter entering and carefully checking the program, press [ 2nd MODE makes QUIT ] to return to the home screen. 
TI89: newton() Prgm getMode("Exact/Approx")→str1 setMode("Exact/Approx","Approximate") ClrIO Disp "Initial guess=",t 1→z Loop ty1(t)/(d(y1(x),x))x=t→y If abs(ty)<abs(t/1E10) Exit z+1→z y→t EndLoop Disp "Root=",y Disp "Iterations=",z setMode("Exact/Approx",str1) DelVar str1 EndPrgmAfter entering and carefully checking the program, press [ 2nd ESC makes QUIT ]
to return to the home screen. 
Let’s test the program by finding a solution of x = cos(x). That means finding the positive zero of f(x) = x − cos(x). We’ll use an initial guess of 0.
Make sure the calculator is in radian mode.
Put the function into Y1.  TI83/84: [Y= ]
In Y1, press [ x,T,θ,n ] [− ] [COS ] [x,T,θ,n ] [) ] 
TI89: [◆ ] [Y= ]
In Y1, press [ X ] [− ] [2nd Z makes COS ]
[X ] [) ] 
Return to the home screen.  TI83/84: Press [2nd MODE makes QUIT ]. 
TI89: Press [2nd ESC makes QUIT ]. 
Run the program.  TI83/84: Press [PRGM ], then the number next to the NEWTON
program. (This number will vary depending on any other programs in
your calculator.) Press [ENTER ] to start the program.
You’re prompted for an initial guess. Enter 0. 
TI89:
Press [2nd − makes VARLINK ], scroll to newton , and press
[ENTER ]. This pastes newton( to your home screen.
Enter the initial guess, 0, and then [ ) ] [ENTER ]. 
The program finds the answer, about 0.7391, in six iterations.  TI83/84:  TI89: 
Explained why it’s okay to switch from x = √2 to x² = 2.
Converted from HTML 4.01 to HTML5 and italicized variables.
Updates and new info: https://BrownMath.com/ti83/