Condition Number Example - Exponent
Thomas J. Kennedy
Suppose we are interested in the condition number of the exponential function.
$$ f(x) = \alpha e^{bx} $$
We will use the general form of the condition number.
$$ (cond\phantom{1}f)(x) = \left|\frac{xf^\prime(x)}{f(x)} \right| $$
1 Two Methods
1.1 Compute f’(x)
We can compute the derivative of
$$ f(x) = \alpha e^{bx} $$
directly using the exponent rule.
$$ f^\prime(x) = \alpha b e^{bx} $$
1.2 Natural Log Trick
Sometimes… the natural log trick can simplify the algebra. The natural log trick used the derivative of the natural log, i.e.,
$$ \frac{d}{dx}\ln(f(x)) = \frac{f^\prime(x)}{f(x)} $$
Let use use the natural log trick for $f(x) = \alpha e^{bx}$:
$$ \begin{eqnarray} \frac{d}{dx}\ln(\alpha e^{bx}) &=& \frac{d}{dx}\left(\ln(\alpha) + \ln(e^{bx})\right) \\ &=& \frac{d}{dx}\left(\ln(\alpha)\right) + \frac{d}{dx}\left(\ln(e^{bx})\right) \\ &=& \frac{d}{dx}\left(\ln(e^{bx})\right) \\ &=& \frac{d}{dx}\left(bx\right) \\ &=& b \\ \end{eqnarray} $$
The natural log trick often allows a problem to be simplified algebraically (through properties of the natural log) before computing any deviates.
2 Computing the Condition Number
Now… we can plug the result into the condition number formula.
$$ \begin{eqnarray} (cond\phantom{1}f)(x) &=& \left|\frac{xf^\prime(x)}{f(x)} \right| \\ &=& \left|x b \right| \\ &=& |x|*|b|\\ \end{eqnarray} $$
3 Examining Behavior
Suppose we are interested in three cases:
- $x \to 0$
- $x \to \infty$
- $x \to -\infty$
As $x$ approaches 0.. the condition number also approaches zero.
$$ \lim_{x \to 0} = \lim_{x \to 0} |xb| \to 0 \therefore \text{Well Conditioned} $$
As $x$ approaches infinity.. the condition number also approaches infinity.
$$ \lim_{x \to \infty} = \lim_{x \to \infty} |xb| \to \infty \therefore \text{Ill Conditioned} $$
The third case (i.e., $x \to -\infty$) is simliar to $x \to \infty$.