Question

Suppose $A$ is an $m\times n$ real matrix and we need to find $||A||_p$ for $p \notin {1,2,\infty }$. What is the most efficient way to compute $||A_p||$?

Here's one naive approach I can think of. Sample random points ||x|| on the unit hypersphere , computing $||Ax||_p$ for each such and take the maximum. What I would like to know is the runtime of this approach for the "average" A,and how we can optimize this for special classes of matrices( like Diagonal, Orthonormal, etc.)?

Answer 1

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).

Answer 2

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?

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(added 5/13/2011)

I posted a Mathematica translation of Higham's original MATLAB code here.

Answer 3

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.