  This paper proposes a verification method for sparse linear systems $Ax=b$
with general and nonsingular coefficients. A verification method produces the
error bound for a given approximate solution. Conventional methods use one of
two approaches. One approach is to verify the computed solution of the normal
equation $A^TAx=A^Tb$ by exploiting symmetric and positive definiteness;
however, the condition number of $A^TA$ is the square of that for $A$. The
other approach uses an approximate inverse matrix of the coefficient; however,
the approximate inverse may be dense even if $A$ is sparse. Here, we propose a
method for the verification of solutions of sparse linear systems based on
$LDL^T$ decomposition. The proposed method can reduce the fill-in and is
applicable to many problems. Moreover, an efficient iterative refinement method
is proposed for obtaining accurate solutions.
