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How do I solve for $\delta$ in $[r−\delta,r+\delta]$ where Newton's method will surely converge? For example in:

Explain Newton’s method for $$f(x) = x^3+x−2 = 0.$$ Show that Newton’s method converges if $$x_0 \in [1−1/30 , 1+1/30 ]$$ to a limit $L$. Find an error estimate for the error $$e_n = |x_n−L|.$$ (Hint: $x^3 −3x^2 +2 = (x−1)(x^2 −2x−2)$ and $|x^2 − 2x − 2| ≤ 10$ if $0 ≤ x ≤ 2$.)

How was the $1/30$ obtained?

Lutz Lehmann
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Guppy_00
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Following the theory explained in https://math.stackexchange.com/a/1653829/115115, determine over $[0,2]$ $$ m_1=\min_{x\in[0,2]} |f'(x)|=\min_{x\in[0,2]} 3x^2+1=1 $$ and $$ M_2=\max_{x\in[0,2]} |f''(x)|=\max_{x\in[0,2]}6x=12 $$ and determine the "contraction" constant $$ C=\frac{M_2}{2m_1}=6. $$ From $$ |x_{n+1}-L|\le C·|x_n-L|^2=(C·|x_n-L|)·|x_n-L|\\ \implies |x_n-L|\le C^{-1}· (C·|x_0-L|)^{2^n} $$ one sees that the method is contractive and quadratically convergent for $$ |x_0-L|<\frac16. $$

Starting with the smaller interval $[\frac12,\frac32]$ these estimates give $m_1=\frac74$, $M_2=9$, $C=18/7<3$ leading to the greater radius $$|x_0-L|<\frac13$$ for the initial interval of good starting points.

Lutz Lehmann
  • 126,666
  • So [1-1/6, 1+1/6] is like the maximum interval for convergence to the root=1 and since [1-1/30,1+1/30] is within the maximum interval, any guesses in [1-1/30,1+1/30] will lead to convergence? is that it? – Guppy_00 Feb 20 '16 at 13:13
  • Not the maximum interval, since $[1-1/3, 1+1/3]$ is larger, and even that is smaller than what $C=18/7$ leading to a radius of $δ=\min(1/2, 7/18)=7/18=1/3+1/18$ actually gives. But yes, $[1-1/30,1+1/30]$ is contained in these larger intervals. – Lutz Lehmann Feb 20 '16 at 13:45
  • so how do I find the maximum interval? Also I've come across this problem: f(x)=x^3-2x^2-11x+12=(x-4)(x-1)(x+3) for which x_o will newton's method converge for which root??? https://en.wikipedia.org/wiki/Newton%27s_method#Zero_derivative – Guppy_00 Feb 20 '16 at 13:50
  • Take a look at the Newton fractal wp-page to see that this is a non-trivial problem for polynomials of degree 3 or higher. – Lutz Lehmann Feb 20 '16 at 14:54
  • See for instance http://usefuljs.net/fractals/index.html?{"current":"newton","colourOptions":8,"TERMS":[1,0,-2,0,-11,0,12,0]} (all one address). – Lutz Lehmann Feb 20 '16 at 15:14