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#### 2.7.65 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 `C-x * 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 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 Root Finding.

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