Question

use bisection in five stages to find a real root of the equation x^3-x-1=0
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Answer 1

Answer:

Bisection Method:

This is a very simple method. Identify two points x = a and x = b such that f (a) and f (b) are having opposite signs. Let f (a) be negative and f (b) be positive. Then there will be a root of f (x) = 0 in between a and b. Let the first approximation be the mid point of the interval (a, b). i.e.

x1 = (a + b)/2

If f (x1) = 0, then x1 is a root, other wise root lies between a and x1 or x1 and b according as f (x1) is positive or negative. Then again we bisect the interval and continue the process until the root is found to desired accuracy.

Solution:

let a = 0 and b = 1

x1 = 0.5, f(x1) = -ve

between x1 = 0.5 and b = 1

x2 = 0.75 (+ve)

between x1 and x2

x3 = 0.625

x4 = 0.6875

x5 = … (you get the gist)

I think you should continue this about 10 times till you get x(n) = x(n+1) approximately to 3 decimal places.

x(n) will be the answer.