Newton's method on a the TI graphing calculators MATH 101FALL 2003

Newton's method on the TI graphing calculators
Here is an example showing how to implement Newton's approximation on a TI-83, TI-85, or TI-89 graphing calculator. We will approximate a root of the function f(x) = x2 - 5 with an initial guess of x = 2.

The iteration rule for Newton's method is

For this problem, we have x0 = 2 and xn+1 = xn - (xn2 - 5)/(2xn). Here's how we can find the iterates quickly using a [TI-83], [TI-85], or [TI-89].

On the TI-83

We first store the iteration rule as the function Y1. To do this we press , clear whatever function happened to be stored there, and enter the iteration rule above.

Next we press to return to the Home screen. We store the value 2 (our initial guess) into the variable X by pressing . We want to evaluate Y1 (the next iterate) and store the value back into X. To do this, we press , then use the arrow keys to highlight Y-VARS, then press twice to get the variable name Y1 on the home screen. Next we press . When we press , the calculator evaluates Y1 and stores the result in X.

We now press again and again. Each time, the calculator evaluates the next iterate and stores the result into X

When the values of the iterates stop changing, we stop and report the answer, correct to nine decimal places.

On the TI-85

We first store the iteration rule as the function, which we may as well call I. We can do this on the home screen. We press followed by the iteration rule itself and the key.

We store the value 2 (our initial guess) into the variable x by pressing .

Now to evaluate I and store the result back into x, we press .

From now on, each time we press , the calculator finds the next iterate and stores the result into the variable x. We continue pressing until the numbers stop changing, and report the result, correct to ten decimal places.

On the TI-89 To define the iteration rule, we press , select "Define", and type in the rule, which we'll call i(x). Here's what we get:

Now we press , and enter our initial guess by pressing .

To get the iteration started, we enter . From now on, each time we press , the calculator will use the most recent entry as the input to the iterative step, and replace it with the next iterate. To get the iterates to display as decimal approximations (rather than as fractions), we press the green diamond key. Then we press until the numbers stop changing.