  We consider the problem of explicitly computing Beilinson--Bloch heights of
homologically trivial cycles on varieties defined over number fields. Recent
results have established a congruence, up to the rational span of logarithms of
primes, between the height of certain limit mixed Hodge structures and certain
Beilinson--Bloch heights obtained from odd-dimensional hypersurfaces with a
node. This congruence suggests a new method to compute Beilinson--Bloch
heights. Here we explain how to compute the relevant limit mixed Hodge
structures in practice, then apply our computational method to a nodal quartic
curve and a nodal cubic threefold. In both cases, we explain the nature of the
primes occurring in the congruence.
