In the previous article we have seen how to calculate Padé approximations on the example of
The study (ii)
We are now going to see how efficient Padé’s ansatz is and compare it to the well known series representation . We also focus on the “diagonal” Padé’ , which seems to be one of the two popular choices (the other is ).
Example 1 (inside of Taylor r.o.c): convergence at . Here the value is
Since three quarters lies inside the radius of convergence for the series, we can directly compare how fast the individual approaches to converge to the exact value:
The numbers on the horizontal axis (0, … ,10) denote the order of the approximation . Here and below, the black dots represent the exact value, green squares stand for the diagonal Padé approximation , and blue diamonds correspond to the -th Taylor polynomial evaluated at the point of interest. At this resolution, already the third Padé does not differ visually from the exact value, whereas the series still deviates significantly!
Example 2 (at the edge of Taylor r.o.c): consider . The series is rather famous
and known to be slowly convergent. Have a look how the Padés outperform the Taylors:
Example 3 (Padé converges where Taylor explodes): consider . At this point the Taylor series does not work anymore, however the Padé approximants do converge to the correct value