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Newton iteration and the Siegel linearisation theorem

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An extremely large portion of mathematics is concerned with locating solutions to equations such as

$latex displaystyle f(x) = 0&fg=000000$

or

$latex displaystyle Phi(x) = x (1)&fg=000000$

for $latex {x}&fg=000000$ in some suitable domain space (either finite-dimensional or infinite-dimensional), and various maps $latex {f}&fg=000000$ or $latex {Phi}&fg=000000$. To solve the fixed point iteration equation (1), the simplest general method available is the fixed point iteration method: one starts with an initial approximate solution $latex {x_0}&fg=000000$ to (1), so that $latex {Phi(x_0) approx x_0}&fg=000000$, and then recursively constructs the sequence $latex {x_1, x_2, x_3, dots}&fg=000000$ by $latex {x_n := Phi(x_{n-1})}&fg=000000$. If $latex {Phi}&fg=000000$ behaves enough like a “contraction”, and the domain is complete, then one can expect the $latex {x_n}&fg=000000$ to converge to a limit $latex {x}&fg=000000$, which should then be a solution to (1). For instance, if $latex {Phi: X rightarrow X}&fg=000000$ is a map from a…

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