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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) Both sides previous revision Previous revision 2007/04/19 09:48 vkuncak 2007/04/15 19:10 vkuncak 2007/04/15 19:10 vkuncak 2007/04/15 19:06 vkuncak 2007/04/15 19:06 vkuncak 2007/04/15 19:05 vkuncak 2007/04/15 19:04 vkuncak 2007/04/15 18:55 vkuncak 2007/04/15 18:45 vkuncak 2007/04/14 18:45 vkuncak 2007/04/14 18:45 vkuncak 2007/04/14 18:44 vkuncak 2007/04/14 18:44 vkuncak 2007/04/14 18:44 vkuncak created Next revision Previous revision 2007/04/19 09:48 vkuncak 2007/04/15 19:10 vkuncak 2007/04/15 19:10 vkuncak 2007/04/15 19:06 vkuncak 2007/04/15 19:06 vkuncak 2007/04/15 19:05 vkuncak 2007/04/15 19:04 vkuncak 2007/04/15 18:55 vkuncak 2007/04/15 18:45 vkuncak 2007/04/14 18:45 vkuncak 2007/04/14 18:45 vkuncak 2007/04/14 18:44 vkuncak 2007/04/14 18:44 vkuncak 2007/04/14 18:44 vkuncak created Line 3: Line 3: \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} Line 14: Line 14: 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).