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In the fixed point iteration method and Newton's method, how could I choose the first approximate solution $P_0$?

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    Take a good educated guess if possible. Newton's method is a locally convergent method, no guarantees are given for global convergence. Take a look at Newton fractals for mostly "nice" chaotic global behavior. With higher dimensions it gets worse, divergence becomes a real possibility. – 2017-02-27
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    What does go wrong if I choose any arbitrary $P_0$ from the interval? – 2017-02-27
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    Is there any value in the interval $[a,b]$ that we can't use it as an initial point?? – 2017-02-27
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    You can combine Newton and bisection methods. Have a look at http://math.stackexchange.com/questions/1644377/how-to-find-the-root-of-a-polynomial-function-closest-to-the-initial-guess/1644396#1644396. The rsik with Newton is that the iterates can go outside the interval. – 2017-02-27
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    Please update your question to the actual task description. Esp. if the interval end points have a sign variation in the values. -- In general you can formulate the Newton method for any differentiable function $F:\Bbb R^n\to\Bbb R^n$, and in that generality globalizing the Newton method is still an active research topic. – 2017-02-27

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