  The kernel polynomial method (KPM) is a powerful numerical method for
approximating spectral densities. Typical implementations of the KPM require an
a prior estimate for an interval containing the support of the target spectral
density, and while such estimates can be obtained by classical techniques, this
incurs addition computational costs. We propose an spectrum adaptive KPM based
on the Lanczos algorithm without reorthogonalization which allows the selection
of KPM parameters to be deferred to after the expensive computation is
finished. Theoretical results from numerical analysis are given to justify the
suitability of the Lanczos algorithm for our approach, even in finite precision
arithmetic. While conceptually simple, the paradigm of decoupling computation
from approximation has a number of practical and pedagogical benefits which we
highlight with numerical examples.
