How do I ensure that my function below is well conditioned as $s$ approaches $\infty$? The problem I get is that for large $s$ the function returns an indeterminate form $\frac{0}{0}$. I would otherwise have expected the function to increase monotonically from 0 to 1 as $s$ approaches $\infty$.

\begin{align} f(s) = \frac{x_s^2+\sqrt{1-x_s^2}-1}{\sqrt{x_s^2(x_s(\sqrt{1-x_s^2}-1))^2+(x_s^2+\sqrt{1-x_s^2}-1)^2}} \end{align}

\begin{align} x_s = e^{-s} \end{align}

import numpy as np

def func(s):
    xs = np.exp(-1*s)
    num = xs**2+np.sqrt(1-xs**2)-1
    den = np.sqrt(xs**2*(xs*(np.sqrt(1-xs**2)-1))**2+(xs**2+np.sqrt(1-xs**2)-1)**2) 
    return num/den

s = np.linspace(0,20, 1000)
plot(s, func(s))

enter image description here

  • $\begingroup$ In code you have (xs*(np.sqrt(1-xs**2)-1))**2, but in the definition you instead have $(x\sqrt{1-x^2}-1)^2$. Which of these is supposed to be correct? $\endgroup$
    – Kirill
    Sep 28 '14 at 23:16
  • $\begingroup$ Edited. The code was the correct version. Nice spot! $\endgroup$
    – Dipole
    Sep 29 '14 at 8:08

The function is really smooth as $s \rightarrow \infty$. This allows you to do the following strategy:

  • For $s\le s_0$, do the usual evaluation, i.e., return $f(s)$.

  • For $s>s_0$, define the function $g(h)=f(1/h)$, do a Taylor expansion around $h=0$ (i.e., around $s=\infty$) and return $\bar g(h)|_{h=1/s}$ where $\bar g$ contains the first few Taylor terms of the expansion of $g(h)$. In the evaluation of these Taylor terms, you need to compute the $k$th derivatives $g^{(k)}(0)$ which will involve factors $\frac 00$ for which you will have to apply l'Hopital's rule.

  • $\begingroup$ Thanks, I will try this. How would I pick $s_0$ ? Also I don't look forward to expanding the taylor series for this function, but I guess doing it by hand is the only way? $\endgroup$
    – Dipole
    Sep 28 '14 at 21:43
  • $\begingroup$ @Jack Use a CAS. I get this for the Taylor series at $x=0$: $$\textstyle \frac{1}{2} x+\frac{1}{2} x^2+\frac{5}{16} x^3-\frac{1}{16}x^4-\frac{129}{256}x^5-\frac{197}{256}x^6-\frac{1459}{2048}x^7 -\frac{437}{2048}x^8 +\frac{35451}{65536} x^9+\frac{81235}{65536}x^{10} + O(x^{11})$$ $\endgroup$
    – Kirill
    Sep 28 '14 at 22:58
  • $\begingroup$ @Kirill Would Sympy be able to do that? $\endgroup$
    – Dipole
    Sep 29 '14 at 8:10
  • 1
    $\begingroup$ @Jack: Yes, Sympy can do that. $\endgroup$ Sep 29 '14 at 9:07
  • $\begingroup$ You'd choose $s_0$ so that for $s<s_0$ no cancellation happens, or not to a degree that is detrimental. In the graph you show, you could choose $s_0=10$. $\endgroup$ Sep 29 '14 at 12:18

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