Condition Number Example - A Simple Monomial
Thomas J. Kennedy
Suppose we are interested in the condition number of a monomial, not just one monomial, but all monomials…
$$ f(x) = b x^n $$
where
- $b \in \mathbb{R}$
- $x \in \mathbb{R}$
- $n \ne 0$
We will assume the general form of the condition number, i.e., we will not address $x = 0$ or $f(x) = 0$.
Computing the condition number is fairly quick… once we recall that
$$ \begin{eqnarray} f^\prime(x) &=& \frac{d}{dx}(b x^n)\\ &=& bnx^{n-1}\\ \end{eqnarray} $$
Recall the various properties of exponents
Do not forget that $x^n * x^m = x^{n+m}$.
The condition number can be computed fairly quickly…
$$ \begin{eqnarray} (cond\phantom{1}f)(x) &=& \left| \frac{xf^\prime(x)}{f(x)} \right|\\ &=& \left| \frac{x * bnx^{n-1}}{bx^n}\right|\\ &=& \left| \frac{x * x^{n-1}}{x^n}\right|\\ &=& \left| \frac{nx^{n}}{x^n}\right|\\ (cond\phantom{1}f)(x) &=& \left| n \right|\\ \end{eqnarray} $$
Interesting… does the result make sense? This result indicates that the power (exponent) of a monomial determines its conditioning. This is a useful result to keep in mind.