  The kernel polynomial method allows to sample overall spectral properties of
a quantum system, while sparse diagonalization provides accurate information
about a few important states. We present a method combining these two
approaches without loss of performance or accuracy. We apply this hybrid kernel
polynomial method to improve the computation of thermodynamic quantities and
the construction of perturbative effective models, in a regime where neither of
the methods is sufficient on its own. To achieve this we develop a perturbative
kernel polynomial method to compute arbitrary order series expansions of
expectation values. We demonstrate the efficiency of our approach on three
examples: the calculation of supercurrent and inductance in a Josephson
junction, the interaction of spin qubits defined in a two dimensional electron
gas, and the calculation of the effective band structure in a realistic model
of a semiconductor nanowire.
