Efficiently computing a matrix's induced p-norm
On the negative side, there is a result by myself and Julien Hendrickx that the matrix $p$-norm is NP-hard to approximate whenever $p$ is not $1,2,$ or $\infty$.
On the positive side, the M.S. thesis of Daureen Steinberg has an efficient algorithm for computing the $p$-norm of a nonnegative matrix (see Remark 3.4 on page 48).
S.W. Drury derives a method to find the operator norm of a general real matrix $$ A : \ell^p \longrightarrow \ell^q $$ in a recent paper in Lin. Alg. Appl (and using it, refutes a long-standing conjecture of Matsaev).
In keeping with the answer of Alex Olshevsky, the algorithm seems have a complexity exponential in the number of columns of the matrix (but linear in the number of rows).
Drury's implementation for Visual C++ and Maple can be found here, and a C version targeted at Unix and with bindings for Matlab, Octave and Python can be found here.
Nicholas Higham gives an algorithm for estimating the Hölder $p$-norm of a matrix with the estimate being within a factor of $n^{1-1/p} \|\mathbf{A}\|_p$ ; maybe you can somehow adapt this approach to your needs?
(added 5/13/2011)
I posted a Mathematica translation of Higham's original MATLAB code here.