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I have this question I am trying to solve:

Consider the case $F (x) = x^3 − x^2$

a.) Find the exact solutions of $F (x^∗) = 0$ by polynomial factorisation.

b.) For which of these solutions $x^∗$ is the convergence of Newton’s method quadratic(with order 2) and for which is it only linear (with order 1)? Justify your answer bystudying the properties of $F (x)$ at $x^∗$.

I was wondering what the $^*$ in $x^*$ is supposed to mean. I have tried looking online but haven't had any luck.

Cheers.

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  • $\begingroup$ If $F(x) =x^2-1$ then $x^*$ could be either $-1$ or $1$. $\endgroup$
    – Randall
    Jan 10, 2023 at 20:09

1 Answer 1

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In this context, $x^*$ is just a composite symbol for a variable, similar to $x_0$. This particular symbol is commonly used for the solution to an equation, optimization problem, or similar such thing. By contrast something like $x_0$ is usually some sort of "starting point" that is given rather than needing to be determined. But this is just convention.

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  • $\begingroup$ Thank you so much! It all makes sense now :) $\endgroup$
    – bigbong2
    Jan 10, 2023 at 20:58

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