Let f = y = x3 − 3x + 3 = 0
Let the root be a
If x = a = −2, y = +1
If x = a = −3, y = −15
Thus x = a **** somewhere between −2 and −3.
For x = a − 2.1, y = 0.039, which is close to zero.
Assume as a first approximation, the root to be a = v + h
Put v = −2.1
To a first approximation the root is −2.1 − 0.0038123 or −2.1038123. As a second approximation, assume the root to be
a = −2.1038123 + h,
Put v1 = −2.1038123
h1 = − f (v1)/ f (v1) = −0.000814/6.967 = −0.0001168
The second approximation, therefore, gives a = −2.1039291.
The third and higher approximations can be made in this fashion. The first approximation will be usually good enough in practice.
