  We formulate the quadratic eigenvalue problem underlying the mathematical
model of a linear vibrational system as an eigenvalue problem of a
diagonal-plus-low-rank matrix $A$. The eigenvector matrix of $A$ has a
Cauchy-like structure. Optimal viscosities are those for which $trace(X)$ is
minimal, where $X$ is the solution of the Lyapunov equation $AX+XA^{*}=GG^{*}$.
Here $G$ is a low-rank matrix which depends on the eigenfrequencies that need
to be damped. After initial eigenvalue decomposition of linearized problem
which requires $O(n^3)$ operations, our algorithm computes optimal viscosities
for each choice of external dampers in $O(n^2)$ operations, provided that the
number of dampers is small. Hence, the subsequent optimization is order of
magnitude faster than in the standard approach which solves Lyapunov equation
in each step, thus requiring $O(n^3)$ operations. Our algorithm is based on
$O(n^2)$ eigensolver for complex symmetric diagonal-plus-rank-one matrices and
fast $O(n^2)$ multiplication of linked Cauchy-like matrices.
