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non-converging_iteration_in_reals [2007/04/15 19:10] vkuncak |
non-converging_iteration_in_reals [2007/04/19 09:48] vkuncak |
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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). | ||