  This article presents a fast direct solver, termed Algebraic Inverse Fast
Multipole Method (from now on abbreviated as AIFMM), for linear systems arising
out of $N$-body problems. AIFMM relies on the following three main ideas: (i)
Certain sub-blocks in the matrix corresponding to $N$-body problems can be
efficiently represented as low-rank matrices; (ii) The low-rank sub-blocks in
the above matrix are leveraged to construct an extended sparse linear system;
(iii) While solving the extended sparse linear system, certain fill-ins that
arise in the elimination phase are represented as low-rank matrices and are
"redirected" though other variables maintaining zero fill-in sparsity. The main
highlights of this article are the following: (i) Our method is completely
algebraic (as opposed to the existing Inverse Fast Multipole Method~\cite{
arXiv:1407.1572,doi:10.1137/15M1034477,TAKAHASHI2017406}, from now on
abbreviated as IFMM). We rely on our new Nested Cross
Approximation~\cite{arXiv:2203.14832} (from now on abbreviated as NNCA) to
represent the matrix arising out of $N$-body problems. (ii) A significant
contribution is that the algorithm presented in this article is more efficient
than the existing IFMMs. In the existing IFMMs, the fill-ins are compressed and
redirected as and when they are created. Whereas in this article, we update the
fill-ins first without affecting the computational complexity. We then compress
and redirect them only once. (iii) Another noteworthy contribution of this
article is that we provide a comparison of AIFMM with Hierarchical Off-Diagonal
Low-Rank (from now on abbreviated as HODLR) based fast direct solver and NNCA
powered GMRES based fast iterative solver. (iv) Additionally, AIFMM is also
demonstrated as a preconditioner.
