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# Differences

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non-converging_iteration_in_reals [2007/04/15 19:10]
vkuncak
non-converging_iteration_in_reals [2015/04/21 17:50] (current)
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\begin{equation*} \begin{equation*}
\begin{array}{l} ​ \begin{array}{l} ​
-  f : [0,2] \to [0,2] \\[1ex]+  f : [0,2] \to [0,2] \\begin{equation*}1ex]
f(x) = \left\{\begin{array}{rl} ​   f(x) = \left\{\begin{array}{rl} ​
-      \displaystyle\frac{1+x}{2},​ & x < 1 \\[2ex]+      \displaystyle\frac{1+x}{2},​ & x < 1 \\begin{equation*}2ex]
\displaystyle \frac{3+2x}{4},​ & x \geq 1 \end{array}\right.       \displaystyle \frac{3+2x}{4},​ & x \geq 1 \end{array}\right.
\end{array} \end{array}
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Start from e.g. x=1/​2. ​ Obtain a series converging to 1, but 1 is not a fixed point because $f(1)=5/4$. Start from e.g. x=1/​2. ​ Obtain a series converging to 1, but 1 is not a fixed point because $f(1)=5/4$.

-This is because $f$ is not continuous.+This is because $f$ is not continuous.  If it was continous (at least, continuous from the left), we would have the desired property.

Whenever we converge to some $x^*$, we take $f(x^*)$ and continue iterating and taking limits (the "​number of times" to iterate, even if infinite, depends on the size of the lattice, which can be formalized [[wk>​ordinal number]]s). Whenever we converge to some $x^*$, we take $f(x^*)$ and continue iterating and taking limits (the "​number of times" to iterate, even if infinite, depends on the size of the lattice, which can be formalized [[wk>​ordinal number]]s).