Find the largest real root α of by f ( x) = x 6 − x − 1 = 0 lying between 1 and 2. Perform three iterations by
i) bisection method
ii) secant method ( x0 = 2, x1 = 1) .
Sure, I can help you find the largest real root α of the equation f(x) = x^6 - x - 1 = 0 lying between 1 and 2 using the bisection method and the secant method with the given initial values.
i) Bisection Method:
The bisection method involves iteratively narrowing down the interval that contains the root by halving it. Here's how it works for the given equation:
Initial interval [a, b] = [1, 2]
Iteration 1:
- Midpoint c = (a + b) / 2 = (1 + 2) / 2 = 1.5
- Calculate f(c) = f(1.5) = (1.5)^6 - 1.5 - 1 ≈ -1.328125
- Since f(c) is negative, the root lies in the left subinterval. Update the interval to [a, c] = [1, 1.5].
Iteration 2:
- Midpoint c = (a + b) / 2 = (1 + 1.5) / 2 = 1.25
- Calculate f(c) = f(1.25) = (1.25)^6
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