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GNU Emacs Calc 2.02 Manual

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### 3.7.66 Programming Tutorial Exercise 8

The first step is to compute the derivative f'(x) and thus the formula x - f(x)/f'(x).

(Because this definition is long, it will be repeated in concise form below. You can use M-# m to load it from there. While you are entering a Z ` Z ' body in a macro, Calc simply collects keystrokes without executing them. In the following diagrams we'll pretend Calc actually executed the keystrokes as you typed them, just for purposes of illustration.)

 ```2: sin(cos(x)) - 0.5 3: 4.5 1: 4.5 2: sin(cos(x)) - 0.5 . 1: -(sin(x) cos(cos(x))) . ' sin(cos(x))-0.5 RET 4.5 m r C-x ( Z ` TAB RET a d x RET ```
 ```2: 4.5 1: x + (sin(cos(x)) - 0.5) / sin(x) cos(cos(x)) . / ' x RET TAB - t 1 ```

Now, we enter the loop. We'll use a repeat loop with a 20-repetition limit just in case the method fails to converge for some reason. (Normally, the Z / command will stop the loop before all 20 repetitions are done.)

 ```1: 4.5 3: 4.5 2: 4.5 . 2: x + (sin(cos(x)) ... 1: 5.24196456928 1: 4.5 . . 20 Z < RET r 1 TAB s l x RET ```

This is the new guess for x. Now we compare it with the old one to see if we've converged.

 ```3: 5.24196 2: 5.24196 1: 5.24196 1: 5.26345856348 2: 5.24196 1: 0 . . 1: 4.5 . . RET M-TAB a = Z / Z > Z ' C-x ) ```

The loop converges in just a few steps to this value. To check the result, we can simply substitute it back into the equation.

 ```2: 5.26345856348 1: 0.499999999997 . RET ' sin(cos(\$)) RET ```

Let's test the new definition again:

 ```2: x^2 - 9 1: 3. 1: 1 . . ' x^2-9 RET 1 X ```

Once again, here's the full Newton's Method definition:

 ```C-x ( Z ` TAB RET a d x RET / ' x RET TAB - t 1 20 Z < RET r 1 TAB s l x RET RET M-TAB a = Z / Z > Z ' C-x ) ```

It turns out that Calc has a built-in command for applying a formula repeatedly until it converges to a number. See section 10.8.4 Nesting and Fixed Points, to see how to use it.

Also, of course, a R is a built-in command that uses Newton's method (among others) to look for numerical solutions to any equation. See section 11.7.1 Root Finding.

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