Let $x$ be a real number. Then the irrationality measure $\mu(x)$ can be defined as the smallest positive real number $\mu(x)$ such that the inequality $ \left|x-\dfrac{p}{q}\right|>\dfrac{1}{q^{\mu+\epsilon}}$ holds for any $\epsilon>0$ and all integers $p,q$ with $q$ sufficiently large. If no such $\mu(x)$ exists, then $\mu(x)=\infty$ and $x$ is said to be a Liouville number
Per the Mathworld page, it is known that $\mu(x)=1$ if $x$ is rational, $\mu(x)=2$ if $x$ is algebraic of degree $>1$, and $\mu(x)\geq 2$ if $x$ is transcendental. (Hence all Liouville numbers are transcendent, but not vice versa). It goes on to cite various upper bounds on $\mu(x)$ for various common constants.
Are there effective algorithms for computing such upper bounds numerically?