  We present explicit algorithms for computing structured matrix-vector
products that are optimal in the sense of Strassen, i.e., using a provably
minimum number of multiplications. These structures include
Toeplitz/Hankel/circulant, symmetric, Toeplitz-plus-Hankel, sparse, and
multilevel structures. The last category include \textsc{bttb}, \textsc{bhhb},
\textsc{bccb} but also any arbitrarily complicated nested structures built out
of other structures.
