  The eigenvalues of a matrix polynomial can be determined classically by
solving a generalized eigenproblem for a linearized matrix pencil, for instance
by writing the matrix polynomial in companion form. We introduce a general
scaling technique, based on tropical algebra, which applies in particular to
this companion form. This scaling, which is inspired by an earlier work of
Akian, Bapat, and Gaubert, relies on the computation of "tropical roots". We
give explicit bounds, in a typical case, indicating that these roots provide
accurate estimates of the order of magnitude of the different eigenvalues, and
we show by experiments that this scaling improves the accuracy (measured by
normwise backward error) of the computations, particularly in situations in
which the data have various orders of magnitude. In the case of quadratic
polynomial matrices, we recover in this way a scaling due to Fan, Lin, and Van
Dooren, which coincides with the tropical scaling when the two tropical roots
are equal. If not, the eigenvalues generally split in two groups, and the
tropical method leads to making one specific scaling for each of the groups.
