Find the largest real root α of f (x) = x6 − x − 1 = 0 lying between 1 and 2. Perform three iterations by
i) bisection method
ii) secant method ( x0 = 2, x1 = 1) .
To find the largest real root α of the equation \(f(x) = x^6 - x - 1 = 0\) lying between 1 and 2, you can use the bisection method and the secant method. Let's perform three iterations for each method:
i) Bisection Method:
The bisection method is an iterative method for finding the root of an equation within a given interval. In this method, you start with an interval [a, b] where \(f(a)\) and \(f(b)\) have opposite signs, which ensures that the root lies within the interval. Then, you repeatedly bisect the interval and update the endpoints based on the signs of \(f\) at the midpoint.
Given the interval [1, 2], let's perform three iterations:
Iteration 1:
- Initial interval: [1, 2]
- Midpoint: \(c = \frac{1 + 2}{2} = 1.5\)
- Evaluate \(f(c)\): \(f(1.5) = (1.5)^6 - 1.5 - 1 \approx 12.984375\)
- Since \(f(1.5)\) is positive, update the interval to [1, 1.5] because
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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) .