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# Newton iteration and the Siegel linearisation theorem

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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