  We present a numerical method for the evaluation of the mass gap, and the
low-lying energy gaps, of a large family of free-fermionic and
free-parafermionic quantum chains. The method is suitable for some
generalizations of the quantum Ising and XY models with multispin interactions.
We illustrate the method by considering the Ising quantum chains with uniform
and random coupling constants. The mass gaps of these quantum chains are
obtained from the largest root of a characteristic polynomial. We also show
that the Laguerre bound, for the largest root of a polynomial, used as an
initial guess for the largest root in the method, gives us estimates for the
mass gaps sharing the same leading finite-size behavior as the exact results.
This opens an interesting possibility of obtaining precise critical properties
very efficiently which we explore by studying the critical point and the
paramagnetic Griffiths phase of the quantum Ising chain with random couplings.
In this last phase, we obtain the effective dynamical critical exponent as a
function of the distance-to-criticality. Finally, we compare the mass gap
estimates derived from the Laguerre bound and the strong-disorder
renormalization-group method. Both estimates require comparable computational
efforts, with the former having the advantage of being more accurate and also
being applicable away from infinite-randomness fixed points. We believe this
method is a relevant tool for tackling critical quantum chains with and without
quenched disorder.
