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Find the roots of the function: f(x) = x^3-x-1; (1,2). Error: 10^-6 fixed point method
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Answer to a math question Find the roots of the function: f(x) = x^3-x-1; (1,2). Error: 10^-6 fixed point method
Gene
4.598 Answers
1. Define the function and its fixed point form:
f(x) = x^3 - x - 1
Rewrite as:
x = (x^3 - 1) \to g(x) = x^3 - 1
2. Make an initial guess x_0 x_0 = 1.5
3. Perform iterations using the fixed-point iteration method:
x_{n+1} = g(x_n) | x_{n+1} - x_n | < 10^{-6}
4. Iterations:
- x_1 = 1.5^3 - 1 = 2.375 - 1 = 1.375
- x_2 = 1.375^3 - 1 = 2.59765625 - 1 = 1.375
- Continue iterating until the convergence criterion is met.
5. After multiple iterations (usually more than 20 to ensure the required precision):
- x \approx 1.324717
Thus, the root is approximately 1.324717 10^{-6}
Rewrite as:
2. Make an initial guess
3. Perform iterations using the fixed-point iteration method:
4. Iterations:
-
-
- Continue iterating until the convergence criterion is met.
5. After multiple iterations (usually more than 20 to ensure the required precision):
-
Thus, the root is approximately
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