  We present fully polynomial-time (deterministic or randomised) approximation
schemes for Holant problems, defined by a non-negative constraint function
satisfying a generalised second order recurrence modulo a couple of exceptional
cases. As a consequence, any non-negative Holant problem on cubic graphs has an
efficient approximation algorithm unless the problem is equivalent to
approximately counting perfect matchings, a central open problem in the area.
This is in sharp contrast to the computational phase transition shown by
2-state spin systems on cubic graphs. Our main technique is the recently
established connection between zeros of graph polynomials and approximate
counting. We also use the "winding" technique to deduce the second result on
cubic graphs.
