  We review the (algebraic-)functional method devised by Galleas and further
developed by Galleas and the author. We first explain the method using the
simplest example: the computation of the partition function for the six-vertex
model with domain-wall boundary conditions. At the heart of the method lies a
linear functional equation for the partition function. After deriving this
equation we outline its analysis. The result is a closed expression in the form
of a symmetrized sum - or, equivalently, multiple-integral formula - that can
be rewritten to recover Izergin's determinant. Special attention is paid to the
relation with other approaches. In particular we show that the Korepin--Izergin
approach can be recovered within the functional method. We comment on the
functional method's range of applicability, and review how it is adapted to the
technically more involved example of the elliptic solid-on-solid model with
domain walls and a reflecting end. We present a new formula for the partition
function of the latter, which was expressed as a determinant by
Tsuchiya--Filali--Kitanine. Our result takes the form of a
'crossing-symmetrized' sum with $2^L$ terms featuring the elliptic domain-wall
partition function, which appears to be new also in the limiting case of the
six-vertex model. Further taking the rational limit we recover the expression
obtained by Frassek using the boundary perimeter Bethe ansatz.
