  This paper proposes a new and efficient numerical algorithm for recovering
the damping coefficient from the spectrum of a damped wave operator, which is a
classical Borg-Levinson inverse spectral problem. The algorithm is based on
inverting a sequence of trace formulas, which are deduced by a recursive
formula, bridging geometrical and spectrum information explicitly in terms of
Fredholm integral equations. Numerical examples are presented to illustrate the
efficiency of the proposed algorithm.
