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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] |
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| - | === An example of a fixed point iteration not converging to a fixed point in one countable sequence === | ||
| - | |||
| - | \begin{equation*} | ||
| - | \begin{array}{l} | ||
| - | f : [0,2] \to [0,2] \\[1ex] | ||
| - | f(x) = \left\{\begin{array}{rl} | ||
| - | \displaystyle\frac{1+x}{2}, & x < 1 \\[2ex] | ||
| - | \displaystyle \frac{3+2x}{4}, & x \geq 1 \end{array}\right. | ||
| - | \end{array} | ||
| - | \end{equation*} | ||
| - | |||
| - | (draw figure) | ||
| - | |||
| - | 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. | ||
| - | |||
| - | 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). | ||