  We propose herein an extension of truncated spectrum methodologies (TSMs), a
non-perturbative numerical approach able to elucidate the low energy properties
of quantum field theories. TSMs, in their various flavors, involve a division
of a computational Hilbert space, $\mathcal{H}$, into two parts, one part,
$\mathcal{H}_1$ that is `kept' for the numerical computations, and one part,
$\mathcal{H}_2$, that is discarded or `truncated'. Even though $\mathcal{H}_2$
is discarded, TSMs will often try to incorporate the effects of $\mathcal{H}_2$
in some effective way. In these terms, we propose to keep the dimension of
$\mathcal{H}_1$ small. We pair this choice of $\mathcal{H}_1$ with a Krylov
subspace iterative approach able to take into account the effects of
$\mathcal{H}_2$. This iterative approach can be taken to arbitrarily high order
and so offers the ability to compute quantities to arbitrary precision. In many
cases it also offers the advantage of not needing an explicit UV cutoff. To
compute the matrix elements that arise in the Krylov iterations, we employ a
Feynman diagrammatic representation that is then evaluated with Monte Carlo
techniques. Each order of the Krylov iteration is variational and is guaranteed
to improve upon the previous iteration. The first Krylov iteration is akin to
the NLO approach of Elias-Mir\'o et al. \cite{Elias-Miro:2017tup}. To
demonstrate this approach, we focus on the $1+1d$ dimensional $\phi^4$ model
and compute the bulk energy and mass gaps in both the $\mathbb{Z}_2$-broken and
unbroken sectors. We estimate the critical $\phi^4$ coupling in the broken
phase to be $g_c=0.2645\pm0.002$.
