  The last two decades witnessed the increasing of the interests on the
absolute value equations (AVE) of finding $x\in\mathbb{R}^n$ such that
$Ax-|x|-b=0$, where $A\in \mathbb{R}^{n\times n}$ and $b\in \mathbb{R}^n$. In
this paper, we pay our attention on designing efficient algorithms. To this
end, we reformulate AVE to a generalized linear complementarity problem (GLCP),
which, among the equivalent forms, is the most economical one in the sense that
it does not increase the dimension of the variables. For solving the GLCP, we
propose an inexact Douglas-Rachford splitting method which can adopt a relative
error tolerance. As a consequence, in the inner iteration processes, we can
employ the LSQR method ([C.C. Paige and M.A. Saunders, ACM Trans. Mathe. Softw.
(TOMS), 8 (1982), pp. 43--71]) to find a qualified approximate solution for
each subproblem, which makes the cost per iteration very low. We prove the
convergence of the algorithm and establish its global linear rate of
convergence. Comparing results with the popular algorithms such as the exact
generalized Newton method [O.L. Mangasarian, Optim. Lett., 1 (2007), pp. 3--8],
the inexact semi-smooth Newton method [J.Y.B. Cruz, O.P. Ferreira and L.F.
Prudente, Comput. Optim. Appl., 65 (2016), pp. 93--108] and the exact SOR-like
method [Y.-F. Ke and C.-F. Ma, Appl. Math. Comput., 311 (2017), pp. 195--202]
are reported, which indicate that the proposed algorithm is very promising.
Moreover, our method also extends the range of numerically solvable of the AVE;
that is, it can deal with not only the case that $\|A^{-1}\|<1$, the commonly
used in those existing literature, but also the case where $\|A^{-1}\|=1$.
