Question

We are going to compute the cube root of 9 to 6 significant digits using Newton's method.

Choose a function of the form

𝑓(𝑥)=(𝑥^𝑚)−𝐶

Give the formula for 𝑥_(𝑛+1) in terms of the previous value 𝑥_(𝑛) .

Answer 1

you want y = x^3 - 9
Now just apply Newton's method, which I'm sure you can find
(a) in your text
(b) online.
It doesn't even really matter how close your first guess is, since it converges quite rapidly (if at all, but x^3 is a nice well-behaved function).

Answer 2

Here is my simplistic GWBASIC program (from the early 80's)

10 x = 2
20 xnew = x - (x^3 - 9)/(3*x^2)
30 if ABS(xnew - x) < .0000001 THEN END
40 PRINT XNEW
50 X = XNEW
60 GOTO 20

2.083333
2.080089
2.080084