.. Copyright (c) 1998 Lawrence Livermore National Security, LLC and other
   HYPRE Project Developers. See the top-level COPYRIGHT file for details.

   SPDX-License-Identifier: (Apache-2.0 OR MIT)


.. _PILUT:

PILUT: Parallel Incomplete Factorization
==============================================================================

.. warning::
   PILUT is not actively supported by the hypre development team. We recommend using
   :ref:`ilu` for parallel ILU algorithms. This new ILU implementation includes
   64-bit integers support (for linear systems with more than 2,147,483,647 global
   unknowns) through both *mixedint* and *bigint* builds of hypre and NVIDIA/AMD GPUs
   support through the CUDA/HIP backends.

PILUT is a parallel preconditioner based on Saad's dual-threshold incomplete
factorization algorithm. The original version of PILUT was done by Karypis and
Kumar [KaKu1998]_ in terms of the Cray SHMEM library. The code was subsequently
modified by the hypre team: SHMEM was replaced by MPI; some algorithmic changes
were made; and it was software engineered to be interoperable with several
matrix implementations, including hypre's ParCSR format, PETSc's matrices, and
ISIS++ RowMatrix. The algorithm produces an approximate factorization :math:`L U`,
with the preconditioner :math:`M` defined by :math:`M = L U`.

.. note::
   PILUT produces a nonsymmetric preconditioner even when the original matrix is
   symmetric. Thus, it is generally inappropriate for preconditioning symmetric methods
   such as Conjugate Gradient.

Parameters:
------------------------------------------------------------------------------

* ``SetMaxNonzerosPerRow( int LFIL ); (Default: 20)`` Set the maximum number of
  nonzeros to be retained in each row of :math:`L` and :math:`U`.  This
  parameter can be used to control the amount of memory that :math:`L` and
  :math:`U` occupy. Generally, the larger the value of ``LFIL``, the longer it
  takes to calculate the preconditioner and to apply the preconditioner and the
  larger the storage requirements, but this trades off versus a higher quality
  preconditioner that reduces the number of iterations.

* ``SetDropTolerance( double tol ); (Default: 0.0001)`` Set the tolerance
  (relative to the 2-norm of the row) below which entries in L and U are
  automatically dropped. PILUT first drops entries based on the drop tolerance,
  and then retains the largest LFIL elements in each row that remain.  Smaller
  values of ``tol`` lead to more accurate preconditioners, but can also lead to
  increases in the time to calculate the preconditioner.

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