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Consider the function, $\rm g(x) = e^{-x} (x-1)^m$ which has a single root $\rm m$ at $\rm x_*$ at $\rm 1$.

Now, if we consider the following modified iteration: $$\rm x_{k+1} = x_k + \frac{g(x_k)}{g`(x_k)\lambda} $$ how do we find the values for which the constant $\lambda$ provides the fastest convergence?

So I have the newton method for the above eqn. $$\rm g`(x) = (m-1) e^{-x} (x-1)^m, $$ and the Newton iterate simplifies to this: $\rm x_k - \frac{1}{m-1}$ for the unmodified version, while the modified version gives an amplification by scale factor of $\lambda$. Are we supposed to equate the latter term to the root $\rm x_*$?? Are there any intuitive explanations to this solution? I should also say that I am utterly new and confused to numerical analysis.

Thank you.

Lutz Lehmann
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skidjoe
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    As the derivative of this example is easy to compute: What did you get, what were your conclusions or suspicions? – Lutz Lehmann Oct 05 '19 at 12:51
  • What have you attempted? Where are you stuck? Frankly, I am surprised that you were actually upvoted without showing any effort of your own here. Usually that behavior receives downvotes and votes to close. – Paul Sinclair Oct 05 '19 at 14:46
  • Sorry about that, post has been updated. – skidjoe Oct 05 '19 at 15:04
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    Your derivative is wrong, you have to apply the product rule correctly, $$g'(x)=e^{-x}[m(x-1)^{m-1}-(x-1)^m]=e^{-x}m+1-x^{m-1}.$$ – Lutz Lehmann Oct 05 '19 at 17:04
  • Ok thanks for that, I adjusted that. I am still not quite clear on how do determine the value of $\lambda$ for which convergence is fastest... can you provide insight into that? – skidjoe Oct 05 '19 at 18:57
  • There should remain only terms that are quadratic in $(x-1)$. For general $λ$ you also have terms that are linear in $(x-1)$. – Lutz Lehmann Oct 06 '19 at 05:15
  • On the general topic of Newton's method close to multiple roots, see also https://math.stackexchange.com/q/805490/115115, https://math.stackexchange.com/q/2614354/115115, https://math.stackexchange.com/q/312819/115115, https://math.stackexchange.com/q/3089817/115115 – Lutz Lehmann Oct 06 '19 at 06:00

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