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Fragment from http://tori.ils.uec.ac.jp/TORI/index.php/File:DSC01693bangkok.JPG

Presentation of Regular iteration method at the conference 2012.09.27.AMSE (see AMSE 2012 report ), photo by Devchonka:

.. In order to get $a_3$, we equalize coefficients with $\varepsilon^3$, and so on, getting the asymptotic series. The series gives the good approximation at small values of $\varepsilon$. Small $\varepsilon$ corresponds to large negative values of $x$. For these values we have good primary approximation.

For moderate values of $x$, apply to the primary approximation the iteration shown at the bottom of the screen. ..