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Try Newton's method on f(x) = √[3]x (use np.cbrt), which has the root x = 0…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
Try Newton's method on (use np.cbrt), which has the root . Its derivative is . Start at and print the first six guesses. What happens, and why?
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Plan. Write the step: . For this function, , so the step is . Every step doubles the distance from the root and changes the side.
Check. The guesses are 1, , 4, , 16 and . They jump from side to side and go farther from the root at every step. This is the opposite of convergence.
Answer: the method diverges. The curve is so steep at the root that the tangent always points beyond the root. Newton's method needs a function that is smooth enough near the root. A bracket method such as bisection or brentq would find this root without trouble.
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