  The dense matrix resulting from an integral equation (IE) based solution of
Maxwell's equations can be compactly represented by an ${\cal H}^2$-matrix.
Given a general dense ${\cal H}^2$-matrix, prevailing fast direct solutions
involve approximations whose accuracy can only be indirectly controlled. In
this work, we propose new accuracy-controlled direct solution algorithms,
including both factorization and inversion, for solving general ${\cal
H}^2$-matrices, which does not exist prior to this work. Different from
existing direct solutions, where the cluster bases are kept unchanged in the
solution procedure thus lacking explicit accuracy control, the proposed new
algorithms update the cluster bases and their rank level by level based on
prescribed accuracy, without increasing computational complexity. Zeros are
also introduced level by level such that the size of the matrix blocks computed
at each tree level is the rank at that level, and hence being small. The
proposed new direct solution has been applied to solve electrically large
volume IEs whose rank linearly grows with electric size. A complexity of
$O(NlogN)$ in factorization and inversion time, and a complexity of $O(N)$ in
storage and solution time are both theoretically proven and numerically
demonstrated. For constant-rank cases, the proposed direct solution has a
strict $O(N)$ complexity in both time and memory. Rapid direct solutions of
millions of unknowns can be obtained on a single CPU core with directly
controlled accuracy.
