.. _chapter_matrices:

Matrices
--------

PETSc provides a variety of matrix implementations because no single
matrix format is appropriate for all problems. Currently, we support
dense storage and compressed sparse row storage (both sequential and
parallel versions) for CPU and GPU based matrices, as well as several specialized formats. Additional
specialized formats can be easily added.

This chapter describes the basics of using PETSc matrices in general
(regardless of the particular format chosen) and discusses tips for
efficient use of the several simple uniprocess and parallel matrix
types. The use of PETSc matrices involves the following actions: create
a particular type of matrix, insert values into it, process the matrix,
use the matrix for various computations, and finally destroy the matrix.
The application code does not need to know or care about the particular
storage formats of the matrices.

.. _sec_matcreate:

Creating matrices
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

As with vectors, PETSc has APIs that allow the user to specify the exact details of the matrix
creation process but also ``DM`` based creation routines that handle most of the details automatically
for specific families of applications. This is done with

.. code-block::

   DMCreateMatrix(DM dm,Mat *A)

The type of matrix created can be controlled with either

.. code-block::

   DMSetMatType(DM dm,MatType <MATAIJ or MATBAIJ or MATAIJCUSPARSE etc>)

or with

.. code-block::

   DMSetSetFromOptions(DM dm)

and the options database option ``-dm_mat_type <aij or baij or aijcusparse etc>`` Matrices can be created for CPU usage, for GPU usage and for usage on
both the CPUs and GPUs. 

The creation of ``DM`` objects is discussed in :any:`sec_struct`, :any:`sec_unstruct`, :any:`sec_network`.


Low-level matrix creation routines
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

When using a ``DM`` is not practical for a particular application on can create matrices directly
using

.. code-block::

   MatCreate(MPI_Comm comm,Mat *A)
   MatSetSizes(Mat A,PetscInt m,PetscInt n,PetscInt M,PetscInt N)

This routine generates a sequential matrix when running one process and
a parallel matrix for two or more processes; the particular matrix
format is set by the user via options database commands. The user
specifies either the global matrix dimensions, given by ``M`` and ``N``
or the local dimensions, given by ``m`` and ``n`` while PETSc completely
controls memory allocation. This routine facilitates switching among
various matrix types, for example, to determine the format that is most
efficient for a certain application. By default, ``MatCreate()`` employs
the sparse AIJ format, which is discussed in detail
:any:`sec_matsparse`. See the manual pages for further
information about available matrix formats.

Assembling (putting values into) matrices
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

To insert or add entries to a matrix on CPUs, one can call a variant of
``MatSetValues()``, either

.. code-block::

   MatSetValues(Mat A,PetscInt m,const PetscInt idxm[],PetscInt n,const PetscInt idxn[],const PetscScalar values[],INSERT_VALUES);

or

.. code-block::

     MatSetValues(Mat A,PetscInt m,const PetscInt idxm[],PetscInt n,const PetscInt idxn[],const PetscScalar values[],ADD_VALUES);

This routine inserts or adds a logically dense subblock of dimension
``m*n`` into the matrix. The integer indices ``idxm`` and ``idxn``,
respectively, indicate the global row and column numbers to be inserted.
``MatSetValues()`` uses the standard C convention, where the row and
column matrix indices begin with zero *regardless of the storage format
employed*. The array ``values`` is logically two-dimensional, containing
the values that are to be inserted. By default the values are given in
row major order, which is the opposite of the Fortran convention,
meaning that the value to be put in row ``idxm[i]`` and column
``idxn[j]`` is located in ``values[i*n+j]``. To allow the insertion of
values in column major order, one can call the command

.. code-block::

   MatSetOption(Mat A,MAT_ROW_ORIENTED,PETSC_FALSE);

*Warning*: Several of the sparse implementations do *not* currently
support the column-oriented option.

This notation should not be a mystery to anyone. For example, to insert
one matrix into another when using MATLAB, one uses the command
``A(im,in) = B;`` where ``im`` and ``in`` contain the indices for the
rows and columns. This action is identical to the calls above to
``MatSetValues()``.

When using the block compressed sparse row matrix format (``MATSEQBAIJ``
or ``MATMPIBAIJ``), one can insert elements more efficiently using the
block variant, ``MatSetValuesBlocked()`` or
``MatSetValuesBlockedLocal()``.

The function ``MatSetOption()`` accepts several other inputs; see the
manual page for details.

After the matrix elements have been inserted or added into the matrix,
they must be processed (also called “assembled”) before they can be
used. The routines for matrix processing are

.. code-block::

   MatAssemblyBegin(Mat A,MAT_FINAL_ASSEMBLY);
   MatAssemblyEnd(Mat A,MAT_FINAL_ASSEMBLY);

By placing other code between these two calls, the user can perform
computations while messages are in transit. Calls to ``MatSetValues()``
with the ``INSERT_VALUES`` and ``ADD_VALUES`` options *cannot* be mixed
without intervening calls to the assembly routines. For such
intermediate assembly calls the second routine argument typically should
be ``MAT_FLUSH_ASSEMBLY``, which omits some of the work of the full
assembly process. ``MAT_FINAL_ASSEMBLY`` is required only in the last
matrix assembly before a matrix is used.

Even though one may insert values into PETSc matrices without regard to
which process eventually stores them, for efficiency reasons we usually
recommend generating most entries on the process where they are destined
to be stored. To help the application programmer with this task for
matrices that are distributed across the processes by ranges, the
routine

.. code-block::

   MatGetOwnershipRange(Mat A,PetscInt *first_row,PetscInt *last_row);

informs the user that all rows from ``first_row`` to ``last_row-1``
(since the value returned in ``last_row`` is one more than the global
index of the last local row) will be stored on the local process.

In the sparse matrix implementations, once the assembly routines have
been called, the matrices are compressed and can be used for
matrix-vector multiplication, etc. Any space for preallocated nonzeros
that was not filled by a call to ``MatSetValues()`` or a related routine
is compressed out by assembling with ``MAT_FINAL_ASSEMBLY``. If you
intend to use that extra space later, be sure to insert explicit zeros
before assembling with ``MAT_FINAL_ASSEMBLY`` so the space will not be
compressed out. Once the matrix has been assembled, inserting new values
will be expensive since it will require copies and possible memory
allocation.

One may repeatedly assemble matrices that retain the same
nonzero pattern (such as within a nonlinear or time-dependent problem).
Where possible, data structures and communication
information will be reused (instead of regenerated) during successive
steps, thereby increasing efficiency. See
`KSP Tutorial ex5 <../../src/ksp/ksp/tutorials/ex5.c.html>`__
for a simple example of solving two linear systems that use the same
matrix data structure.

For matrices associated with ``DMDA`` there is a higher-level interface for providing
the numerical values based on the concept of stencils. See the manual page of ``MatSetValuesStencil()`` for usage.

For GPUs the routines ``MatSetPreallocationCOO()`` and ``MatSetValuesCOO()`` should be used for efficient matrix assembly
instead of ``MatSetValues()``.

We now introduce the various families of PETSc matrices. ``DMCreateMatrix()`` manages
the preallocation process (introduced below) automatically so many users do not need to
worry about the details of the preallocation process.

.. _sec_matsparse:

Sparse Matrices
^^^^^^^^^^^^^^^

The default matrix representation within PETSc is the general sparse AIJ
format (also called the compressed sparse
row format, CSR). This section discusses tips for *efficiently* using
this matrix format for large-scale applications. Additional formats
(such as block compressed row and block diagonal storage, which are
generally much more efficient for problems with multiple degrees of
freedom per node) are discussed below. Beginning users need not concern
themselves initially with such details and may wish to proceed directly
to :any:`sec_matoptions`. However, when an application code
progresses to the point of tuning for efficiency and/or generating
timing results, it is *crucial* to read this information.

Sequential AIJ Sparse Matrices
''''''''''''''''''''''''''''''

In the PETSc AIJ matrix formats, we store the nonzero elements by rows,
along with an array of corresponding column numbers and an array of
pointers to the beginning of each row. Note that the diagonal matrix
entries are stored with the rest of the nonzeros (not separately).

To create a sequential AIJ sparse matrix, ``A``, with ``m`` rows and
``n`` columns, one uses the command

.. code-block::

   MatCreateSeqAIJ(PETSC_COMM_SELF,PetscInt m,PetscInt n,PetscInt nz,PetscInt *nnz,Mat *A);

where ``nz`` or ``nnz`` can be used to preallocate matrix memory, as
discussed below. The user can set ``nz=0`` and ``nnz=NULL`` for PETSc to
control all matrix memory allocation.

The sequential and parallel AIJ matrix storage formats by default employ
*i-nodes* (identical nodes) when possible. We search for consecutive
rows with the same nonzero structure, thereby reusing matrix information
for increased efficiency. Related options database keys are
``-mat_no_inode`` (do not use inodes) and ``-mat_inode_limit <limit>``
(set inode limit (max limit=5)). Note that problems with a single degree
of freedom per grid node will automatically not use I-nodes.

The internal data representation for the AIJ formats employs zero-based
indexing.

Preallocation of Memory for Sequential AIJ Sparse Matrices
''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

The dynamic process of allocating new memory and copying from the old
storage to the new is *intrinsically very expensive*. Thus, to obtain
good performance when assembling an AIJ matrix, it is crucial to
preallocate the memory needed for the sparse matrix. The user has two
choices for preallocating matrix memory via ``MatCreateSeqAIJ()``.

One can use the scalar ``nz`` to specify the expected number of nonzeros
for each row. This is generally fine if the number of nonzeros per row
is roughly the same throughout the matrix (or as a quick and easy first
step for preallocation). If one underestimates the actual number of
nonzeros in a given row, then during the assembly process PETSc will
automatically allocate additional needed space. However, this extra
memory allocation can slow the computation,

If different rows have very different numbers of nonzeros, one should
attempt to indicate (nearly) the exact number of elements intended for
the various rows with the optional array, ``nnz`` of length ``m``, where
``m`` is the number of rows, for example

.. code-block::

   PetscInt nnz[m];
   nnz[0] = <nonzeros in row 0>
   nnz[1] = <nonzeros in row 1>
   ....
   nnz[m-1] = <nonzeros in row m-1>

In this case, the assembly process will require no additional memory
allocations if the ``nnz`` estimates are correct. If, however, the
``nnz`` estimates are incorrect, PETSc will automatically obtain the
additional needed space, at a slight loss of efficiency.

Using the array ``nnz`` to preallocate memory is especially important
for efficient matrix assembly if the number of nonzeros varies
considerably among the rows. One can generally set ``nnz`` either by
knowing in advance the problem structure (e.g., the stencil for finite
difference problems on a structured grid) or by precomputing the
information by using a segment of code similar to that for the regular
matrix assembly. The overhead of determining the ``nnz`` array will be
quite small compared with the overhead of the inherently expensive
``malloc``\ s and moves of data that are needed for dynamic allocation
during matrix assembly. Always guess high if an exact value is not known
(extra space is cheaper than too little).

Thus, when assembling a sparse matrix with very different numbers of
nonzeros in various rows, one could proceed as follows for finite
difference methods:

#. Allocate integer array ``nnz``.

#. Loop over grid, counting the expected number of nonzeros for the
   row(s) associated with the various grid points.

#. Create the sparse matrix via ``MatCreateSeqAIJ()`` or alternative.

#. Loop over the grid, generating matrix entries and inserting in matrix
   via ``MatSetValues()``.

For (vertex-based) finite element type calculations, an analogous
procedure is as follows:

#. Allocate integer array ``nnz``.

#. Loop over vertices, computing the number of neighbor vertices, which
   determines the number of nonzeros for the corresponding matrix
   row(s).

#. Create the sparse matrix via ``MatCreateSeqAIJ()`` or alternative.

#. Loop over elements, generating matrix entries and inserting in matrix
   via ``MatSetValues()``.

The ``-info`` option causes the routines ``MatAssemblyBegin()`` and
``MatAssemblyEnd()`` to print information about the success of the
preallocation. Consider the following example for the ``MATSEQAIJ``
matrix format:

.. code-block::

   MatAssemblyEnd_SeqAIJ:Matrix size 10 X 10; storage space:20 unneeded, 100 used
   MatAssemblyEnd_SeqAIJ:Number of mallocs during MatSetValues is 0

The first line indicates that the user preallocated 120 spaces but only
100 were used. The second line indicates that the user preallocated
enough space so that PETSc did not have to internally allocate
additional space (an expensive operation). In the next example the user
did not preallocate sufficient space, as indicated by the fact that the
number of mallocs is very large (bad for efficiency):

.. code-block::

   MatAssemblyEnd_SeqAIJ:Matrix size 10 X 10; storage space:47 unneeded, 1000 used
   MatAssemblyEnd_SeqAIJ:Number of mallocs during MatSetValues is 40000

Although at first glance such procedures for determining the matrix
structure in advance may seem unusual, they are actually very efficient
because they alleviate the need for dynamic construction of the matrix
data structure, which can be very expensive.

Parallel AIJ Sparse Matrices
''''''''''''''''''''''''''''

Parallel sparse matrices with the AIJ format can be created with the
command

.. code-block::

   MatCreateAIJ=(MPI_Comm comm,PetscInt m,PetscInt n,PetscInt M,PetscInt N,PetscInt d_nz,PetscInt *d_nnz, PetscInt o_nz,PetscInt *o_nnz,Mat *A);

``A`` is the newly created matrix, while the arguments ``m``, ``M``, and
``N``, indicate the number of local rows and the number of global rows
and columns, respectively. In the PETSc partitioning scheme, all the
matrix columns are local and ``n`` is the number of columns
corresponding to local part of a parallel vector. Either the local or
global parameters can be replaced with ``PETSC_DECIDE``, so that PETSc
will determine them. The matrix is stored with a fixed number of rows on
each process, given by ``m``, or determined by PETSc if ``m`` is
``PETSC_DECIDE``.

If ``PETSC_DECIDE`` is not used for the arguments ``m`` and ``n``, then
the user must ensure that they are chosen to be compatible with the
vectors. To do this, one first considers the matrix-vector product
:math:`y = A x`. The ``m`` that is used in the matrix creation routine
``MatCreateAIJ()`` must match the local size used in the vector creation
routine ``VecCreateMPI()`` for ``y``. Likewise, the ``n`` used must
match that used as the local size in ``VecCreateMPI()`` for ``x``.

The user must set ``d_nz=0``, ``o_nz=0``, ``d_nnz=``\ NULL, and
``o_nnz=NULL`` for PETSc to control dynamic allocation of matrix memory
space. Analogous to ``nz`` and ``nnz`` for the routine
``MatCreateSeqAIJ()``, these arguments optionally specify nonzero
information for the diagonal (``d_nz`` and ``d_nnz``) and off-diagonal
(``o_nz`` and ``o_nnz``) parts of the matrix. For a square global
matrix, we define each process’s diagonal portion to be its local rows
and the corresponding columns (a square submatrix); each process’s
off-diagonal portion encompasses the remainder of the local matrix (a
rectangular submatrix). The rank in the MPI communicator determines the
absolute ordering of the blocks. That is, the process with rank 0 in the
communicator given to ``MatCreateAIJ()`` contains the top rows of the
matrix; the i\ :math:`^{th}` process in that communicator contains the
i\ :math:`^{th}` block of the matrix.

Preallocation of Memory for Parallel AIJ Sparse Matrices
''''''''''''''''''''''''''''''''''''''''''''''''''''''''

As discussed above, preallocation of memory is critical for achieving
good performance during matrix assembly, as this reduces the number of
allocations and copies required. We present an example for three
processes to indicate how this may be done for the ``MATMPIAIJ`` matrix
format. Consider the 8 by 8 matrix, which is partitioned by default with
three rows on the first process, three on the second and two on the
third.

.. math::

   \left( \begin{array}{cccccccccc}
   1  & 2  & 0  & | & 0  & 3  & 0  & |  & 0  & 4  \\
   0  & 5  & 6  & | & 7  & 0  & 0  & |  & 8  & 0 \\
   9  & 0  & 10 & | & 11 & 0  & 0  & |  & 12 & 0  \\
   \hline \\
   13 & 0  & 14 & | & 15 & 16 & 17 & |  & 0  & 0  \\
   0  & 18 & 0  & | & 19 & 20 & 21 & |  & 0  & 0 \\
   0  & 0  & 0  & | & 22 & 23 & 0  & |  & 24 & 0 \\
   \hline \\
   25 & 26 & 27 & | & 0  & 0  & 28 & |  & 29 & 0 \\
   30 & 0  & 0  & | & 31 & 32 & 33 & |  & 0  &34
   \end{array} \right)

The “diagonal” submatrix, ``d``, on the first process is given by

.. math::

   \left( \begin{array}{ccc}
   1  & 2  & 0  \\
   0  & 5  & 6  \\
   9  & 0  & 10
   \end{array} \right),

while the “off-diagonal” submatrix, ``o``, matrix is given by

.. math::

   \left( \begin{array}{ccccc}
    0  & 3  & 0   & 0  & 4  \\
    7  & 0  & 0   & 8  & 0  \\
    11 & 0  & 0   & 12 & 0  \\
   \end{array} \right).

For the first process one could set ``d_nz`` to 2 (since each row has 2
nonzeros) or, alternatively, set ``d_nnz`` to :math:`\{2,2,2\}.` The
``o_nz`` could be set to 2 since each row of the ``o`` matrix has 2
nonzeros, or ``o_nnz`` could be set to :math:`\{2,2,2\}`.

For the second process the ``d`` submatrix is given by

.. math::

   \left( \begin{array}{cccccccccc}
    15 & 16 & 17 \\
    19 & 20 & 21 \\
    22 & 23 & 0
   \end{array} \right) .

Thus, one could set ``d_nz`` to 3, since the maximum number of nonzeros
in each row is 3, or alternatively one could set ``d_nnz`` to
:math:`\{3,3,2\}`, thereby indicating that the first two rows will have
3 nonzeros while the third has 2. The corresponding ``o`` submatrix for
the second process is

.. math::

   \left( \begin{array}{cccccccccc}
   13 & 0  & 14 &  0  & 0  \\
   0  & 18 & 0  &  0  & 0 \\
   0  & 0  & 0  &  24 & 0 \\
   \end{array} \right)

so that one could set ``o_nz`` to 2 or ``o_nnz`` to {2,1,1}.

Note that the user never directly works with the ``d`` and ``o``
submatrices, except when preallocating storage space as indicated above.
Also, the user need not preallocate exactly the correct amount of space;
as long as a sufficiently close estimate is given, the high efficiency
for matrix assembly will remain.

As described above, the option ``-info`` will print information about
the success of preallocation during matrix assembly. For the
``MATMPIAIJ`` and ``MATMPIBAIJ`` formats, PETSc will also list the
number of elements owned by on each process that were generated on a
different process. For example, the statements

.. code-block::

   MatAssemblyBegin_MPIAIJ:Stash has 10 entries, uses 0 mallocs
   MatAssemblyBegin_MPIAIJ:Stash has 3 entries, uses 0 mallocs
   MatAssemblyBegin_MPIAIJ:Stash has 5 entries, uses 0 mallocs

indicate that very few values have been generated on different
processes. On the other hand, the statements

.. code-block::

   MatAssemblyBegin_MPIAIJ:Stash has 100000 entries, uses 100 mallocs
   MatAssemblyBegin_MPIAIJ:Stash has 77777 entries, uses 70 mallocs

indicate that many values have been generated on the “wrong” processes.
This situation can be very inefficient, since the transfer of values to
the “correct” process is generally expensive. By using the command
``MatGetOwnershipRange()`` in application codes, the user should be able
to generate most entries on the owning process.

*Note*: It is fine to generate some entries on the “wrong” process.
Often this can lead to cleaner, simpler, less buggy codes. One should
never make code overly complicated in order to generate all values
locally. Rather, one should organize the code in such a way that *most*
values are generated locally.

The routine ``MatCreateAIJCusparse()`` allows one to create GPU based matrices for NVIDIA systems.
 ``MatCreateAIJKokkos()`` can create matrices for use with CPU, OpenMP, NVIDIA, AMD, or Intel based GPU systems.
 

Limited-Memory Variable Metric (LMVM) Matrices
''''''''''''''''''''''''''''''''''''''''''''''

Variable metric methods, also known as quasi-Newton methods, are
frequently used for root finding problems and approximate Jacobian
matrices or their inverses via sequential nonlinear updates based on the
secant condition. The limited-memory variants do not store the full
explicit Jacobian, and instead compute forward products and inverse
applications based on a fixed number of stored update vectors.

.. list-table:: PETSc LMVM matrix implementations.
  :name: tab_matlmvmimpl
  :header-rows: 1

  * - Method
    - PETSc Type
    - Name
    - Property
  * - "Good" Broyden   :cite:`KEYPREFIX-griewank2012broyden`
    - ``MATLMVMBrdn``
    - ``lmvmbrdn``
    - Square
  * - "Bad" Broyden :cite:`KEYPREFIX-griewank2012broyden`
    - ``MATLMVMBadBrdn``
    - ``lmvmbadbrdn``
    - Square
  * - Symmetric Rank-1 :cite:`KEYPREFIX-NW99`
    - ``MATLMVMSR1``
    - ``lmvmsr1``
    - Symmetric
  * - Davidon-Fletcher-Powell (DFP) :cite:`KEYPREFIX-NW99`
    - ``MATLMVMDFP``
    - ``lmvmdfp``
    - SPD
  * - Broyden-Fletcher-Goldfarb-Shanno (BFGS)
       :cite:`KEYPREFIX-NW99`
    - ``MATLMVMBFGS``
    - ``lmvmbfgs``
    - SPD
  * - Restricted Broyden Family :cite:`KEYPREFIX-erway2017solving`
    - ``MATLMVMSymBrdn``
    - ``lmvmsymbrdn``
    - SPD
  * - Restricted Broyden Family (full-memory diagonal)
    - ``MATLMVMDiagBrdn``
    - ``lmvmdiagbrdn``
    - SPD

PETSc implements seven different LMVM matrices listed in the
table above. They can be created using the
``MatCreate()`` and ``MatSetType()`` workflow, and share a number of
common interface functions. We will review the most important ones
below:

-  ``MatLMVMAllocate(Mat B, Vec X, Vec F)`` – Creates the internal data
   structures necessary to store nonlinear updates and compute
   forward/inverse applications. The ``X`` vector defines the solution
   space while the ``F`` defines the function space for the history of
   updates.

-  ``MatLMVMUpdate(MatB, Vec X, Vec F)`` – Applies a nonlinear update to
   the approximate Jacobian such that :math:`s_k = x_k - x_{k-1}` and
   :math:`y_k = f(x_k) - f(x_{k-1})`, where :math:`k` is the index for
   the update.

-  ``MatLMVMReset(Mat B, PetscBool destructive)`` – Flushes the
   accumulated nonlinear updates and resets the matrix to the initial
   state. If ``destructive = PETSC_TRUE``, the reset also destroys the
   internal data structures and necessitates another allocation call
   before the matrix can be updated and used for products and solves.

-  ``MatLMVMSetJ0(Mat B, Mat J0)`` – Defines the initial Jacobian to
   apply the updates to. If no initial Jacobian is provided, the updates
   are applied to an identity matrix.

LMVM matrices can be applied to vectors in forward mode via
``MatMult()`` or ``MatMultAdd()``, and in inverse mode via
``MatSolve()``. They also support ``MatGetVecs()``, ``MatDuplicate()``
and ``MatCopy()`` operations. The maximum number of :math:`s_k` and
:math:`y_k` update vectors stored can be changed via
``-mat_lmvm_num_vecs`` option.

Restricted Broyden Family, DFP and BFGS methods additionally implement
special Jacobian initialization and scaling options available via
``-mat_lmvm_scale_type <none,scalar,diagonal>``. We describe these
choices below:

-  ``none`` – Sets the initial Jacobian to be equal to the identity
   matrix. No extra computations are required when obtaining the search
   direction or updating the approximation. However, the number of
   function evaluations required to converge the Newton solution is
   typically much larger than what is required when using other
   initializations.

-  ``scalar`` – Defines the initial Jacobian as a scalar multiple of the
   identity matrix. The scalar value :math:`\sigma` is chosen by solving
   the one dimensional optimization problem

   .. math:: \min_\sigma \|\sigma^\alpha Y - \sigma^{\alpha - 1} S\|_F^2,

   where :math:`S` and :math:`Y` are the matrices whose columns contain
   a subset of update vectors :math:`s_k` and :math:`y_k`, and
   :math:`\alpha \in [0, 1]` is defined by the user via
   ``-mat_lmvm_alpha`` and has a different default value for each LMVM
   implementation (e.g.: default :math:`\alpha = 1` for BFGS produces
   the well-known :math:`y_k^T s_k / y_k^T y_k` scalar initialization).
   The number of updates to be used in the :math:`S` and :math:`Y`
   matrices is 1 by default (i.e.: the latest update only) and can be
   changed via ``-mat_lmvm_scalar_hist``. This technique is inspired by
   Gilbert and Lemarechal :cite:`KEYPREFIX-Gilbert-Lemarechal`.

-  ``diagonal`` – Uses a full-memory restricted Broyden update formula
   to construct a diagonal matrix for the Jacobian initialization.
   Although the full-memory formula is utilized, the actual memory
   footprint is restricted to only the vector representing the diagonal
   and some additional work vectors used in its construction. The
   diagonal terms are also re-scaled with every update as suggested in
   :cite:`KEYPREFIX-Gilbert-Lemarechal`. This initialization requires
   the most computational effort of the available choices but typically
   results in a significant reduction in the number of function
   evaluations taken to compute a solution.

Note that the user-provided initial Jacobian via ``MatLMVMSetJ0()``
overrides and disables all built-in initialization methods.

.. _sec_matdense:

Dense Matrices
^^^^^^^^^^^^^^

PETSc provides both sequential and parallel dense matrix formats, where
each process stores its entries in a column-major array in the usual
Fortran style. To create a sequential, dense PETSc matrix, ``A`` of
dimensions ``m`` by ``n``, the user should call

.. code-block::

   MatCreateSeqDense(PETSC_COMM_SELF,PetscInt m,PetscInt n,PetscScalar *data,Mat *A);

The variable ``data`` enables the user to optionally provide the
location of the data for matrix storage (intended for Fortran users who
wish to allocate their own storage space). Most users should merely set
``data`` to ``NULL`` for PETSc to control matrix memory allocation. To
create a parallel, dense matrix, ``A``, the user should call

.. code-block::

   MatCreateDense(MPI_Comm comm,PetscInt m,PetscInt n,PetscInt M,PetscInt N,PetscScalar *data,Mat *A)

The arguments ``m``, ``n``, ``M``, and ``N``, indicate the number of
local rows and columns and the number of global rows and columns,
respectively. Either the local or global parameters can be replaced with
``PETSC_DECIDE``, so that PETSc will determine them. The matrix is
stored with a fixed number of rows on each process, given by ``m``, or
determined by PETSc if ``m`` is ``PETSC_DECIDE``.

PETSc does not provide parallel dense direct solvers, instead
interfacing to external packages that provide these solvers. Our focus
is on sparse iterative solvers.

.. _sec_matnest:

Block Matrices
^^^^^^^^^^^^^^

Block matrices arise when coupling variables with different meaning,
especially when solving problems with constraints (e.g. incompressible
flow) and “multi-physics” problems. Usually the number of blocks is
small and each block is partitioned in parallel. We illustrate for a
:math:`3\times 3` system with components labeled :math:`a,b,c`. With
some numbering of unknowns, the matrix could be written as

.. math::

   \left( \begin{array}{ccc}
       A_{aa} & A_{ab} & A_{ac} \\
       A_{ba} & A_{bb} & A_{bc} \\
       A_{ca} & A_{cb} & A_{cc}
     \end{array} \right) .

There are two fundamentally different ways that this matrix could be
stored, as a single assembled sparse matrix where entries from all
blocks are merged together (“monolithic”), or as separate assembled
matrices for each block (“nested”). These formats have different
performance characteristics depending on the operation being performed.
In particular, many preconditioners require a monolithic format, but
some that are very effective for solving block systems (see
:any:`sec_block_matrices`) are more efficient when a nested
format is used. In order to stay flexible, we would like to be able to
use the same code to assemble block matrices in both monolithic and
nested formats. Additionally, for software maintainability and testing,
especially in a multi-physics context where different groups might be
responsible for assembling each of the blocks, it is desirable to be
able to use exactly the same code to assemble a single block
independently as to assemble it as part of a larger system. To do this,
we introduce the four spaces shown in :numref:`fig_localspaces`.

-  The monolithic global space is the space in which the Krylov and
   Newton solvers operate, with collective semantics across the entire
   block system.

-  The split global space splits the blocks apart, but each split still
   has collective semantics.

-  The split local space adds ghost points and separates the blocks.
   Operations in this space can be performed with no parallel
   communication. This is often the most natural, and certainly the most
   powerful, space for matrix assembly code.

-  The monolithic local space can be thought of as adding ghost points
   to the monolithic global space, but it is often more natural to use
   it simply as a concatenation of split local spaces on each process.
   It is not common to explicitly manipulate vectors or matrices in this
   space (at least not during assembly), but it is a useful for
   declaring which part of a matrix is being assembled.

.. figure:: /images/docs/manual/localspaces.svg
   :alt: The relationship between spaces used for coupled assembly.
   :name: fig_localspaces

   The relationship between spaces used for coupled assembly.

The key to format-independent assembly is the function

.. code-block::

   MatGetLocalSubMatrix(Mat A,IS isrow,IS iscol,Mat *submat);

which provides a “view” ``submat`` into a matrix ``A`` that operates in
the monolithic global space. The ``submat`` transforms from the split
local space defined by ``iscol`` to the split local space defined by
``isrow``. The index sets specify the parts of the monolithic local
space that ``submat`` should operate in. If a nested matrix format is
used, then ``MatGetLocalSubMatrix()`` finds the nested block and returns
it without making any copies. In this case, ``submat`` is fully
functional and has a parallel communicator. If a monolithic matrix
format is used, then ``MatGetLocalSubMatrix()`` returns a proxy matrix
on ``PETSC_COMM_SELF`` that does not provide values or implement
``MatMult()``, but does implement ``MatSetValuesLocal()`` and, if
``isrow,iscol`` have a constant block size,
``MatSetValuesBlockedLocal()``. Note that although ``submat`` may not be
a fully functional matrix and the caller does not even know a priori
which communicator it will reside on, it always implements the local
assembly functions (which are not collective). The index sets
``isrow,iscol`` can be obtained using ``DMCompositeGetLocalISs()`` if
``DMComposite`` is being used. DMComposite can also be used to create
matrices, in which case the MATNEST format can be specified using
``-prefix_dm_mat_type nest`` and MATAIJ can be specified using
``-prefix_dm_mat_type aij``. See
`SNES Tutorail ex28 <../../src/snes/tutorials/ex28.c.html>`__
for a simple example using this interface.

.. _sec_matoptions:

Basic Matrix Operations
~~~~~~~~~~~~~~~~~~~~~~~

Table :any:`2.2 <fig_matrixops>` summarizes basic PETSc matrix operations.
We briefly discuss a few of these routines in more detail below.

The parallel matrix can multiply a vector with ``n`` local entries,
returning a vector with ``m`` local entries. That is, to form the
product

.. code-block::

   MatMult(Mat A,Vec x,Vec y);

the vectors ``x`` and ``y`` should be generated with

.. code-block::

   VecCreateMPI(MPI_Comm comm,n,N,&x);
   VecCreateMPI(MPI_Comm comm,m,M,&y);

By default, if the user lets PETSc decide the number of components to be
stored locally (by passing in ``PETSC_DECIDE`` as the second argument to
``VecCreateMPI()`` or using ``VecCreate()``), vectors and matrices of
the same dimension are automatically compatible for parallel
matrix-vector operations.

Along with the matrix-vector multiplication routine, there is a version
for the transpose of the matrix,

.. code-block::

   MatMultTranspose(Mat A,Vec x,Vec y);

There are also versions that add the result to another vector:

.. code-block::

   MatMultAdd(Mat A,Vec x,Vec y,Vec w);
   MatMultTransposeAdd(Mat A,Vec x,Vec y,Vec w);

These routines, respectively, produce :math:`w = A*x + y` and
:math:`w = A^{T}*x + y` . In C it is legal for the vectors ``y`` and
``w`` to be identical. In Fortran, this situation is forbidden by the
language standard, but we allow it anyway.

One can print a matrix (sequential or parallel) to the screen with the
command

.. code-block::

   MatView(Mat mat,PETSC_VIEWER_STDOUT_WORLD);

Other viewers can be used as well. For instance, one can draw the
nonzero structure of the matrix into the default X-window with the
command

.. code-block::

   MatView(Mat mat,PETSC_VIEWER_DRAW_WORLD);

Also one can use

.. code-block::

   MatView(Mat mat,PetscViewer viewer);

where ``viewer`` was obtained with ``PetscViewerDrawOpen()``. Additional
viewers and options are given in the ``MatView()`` man page and
:any:`sec_viewers`.

.. list-table:: PETSc Matrix Operations
  :name: fig_matrixops
  :header-rows: 1

  * - Function Name
    - Operation
  * - ``MatAXPY(Mat Y, PetscScalar a, Mat X, MatStructure s);``
    - :math:`Y = Y + a*X`
  * - ``MatAYPX(Mat Y, PetscScalar a, Mat X, MatStructure s);``
    - :math:`Y = a*Y + X`
  * - ``MatMult(Mat A,Vec x, Vec y);``
    - :math:`y = A*x`
  * - ``MatMultAdd(Mat A,Vec x, Vec y,Vec z);``
    - :math:`z = y + A*x`
  * - ``MatMultTranspose(Mat A,Vec x, Vec y);``
    - :math:`y = A^{T}*x`
  * - ``MatMultTransposeAdd(Mat A, Vec x, Vec y, Vec z);``
    - :math:`z = y + A^{T}*x`
  * - ``MatNorm(Mat A,NormType type, PetscReal *r);``
    - :math:`r = A_{type}`
  * - ``MatDiagonalScale(Mat A,Vec l,Vec r);``
    - :math:`A = \text{diag}(l)*A*\text{diag}(r)`
  * - ``MatScale(Mat A,PetscScalar a);``
    - :math:`A = a*A`
  * - ``MatConvert(Mat A, MatType type, Mat *B);``
    - :math:`B = A`
  * - ``MatCopy(Mat A, Mat B, MatStructure s);``
    - :math:`B = A`
  * - ``MatGetDiagonal(Mat A, Vec x);``
    - :math:`x = \text{diag}(A)`
  * - ``MatTranspose(Mat A, MatReuse, Mat* B);``
    - :math:`B = A^{T}`
  * - ``MatZeroEntries(Mat A);``
    - :math:`A = 0`
  * - ``MatShift(Mat Y, PetscScalar a);``
    - :math:`Y =  Y + a*I`

.. list-table:: Values of MatStructure
  :name: fig_matstructure
  :header-rows: 1

  * - Name
    - Meaning
  * - ``SAME__NONZERO_PATTERN``
    - the matrices have an identical nonzero pattern
  * - ``DIFFERENT_NONZERO_PATTERN``
    - the matrices may have a different nonzero pattern
  * - ``SUBSET_NONZERO_PATTERN``
    - the second matrix has a subset of the nonzeros in the first matrix
  * - ``UNKNOWN_NONZERO_PATTERN``
    - there is nothing known about the relation between the nonzero patterns of the two matrices

The ``NormType`` argument to ``MatNorm()`` is one of ``NORM_1``,
``NORM_INFINITY``, and ``NORM_FROBENIUS``.

.. _sec_matrixfree:

Matrix-Free Matrices
~~~~~~~~~~~~~~~~~~~~

Some people like to use matrix-free methods, which do
not require explicit storage of the matrix, for the numerical solution
of partial differential equations. To support matrix-free methods in
PETSc, one can use the following command to create a ``Mat`` structure
without ever actually generating the matrix:

.. code-block::

   MatCreateShell(MPI_Comm comm,PetscInt m,PetscInt n,PetscInt M,PetscInt N,void *ctx,Mat *mat);

Here ``M`` and ``N`` are the global matrix dimensions (rows and
columns), ``m`` and ``n`` are the local matrix dimensions, and ``ctx``
is a pointer to data needed by any user-defined shell matrix operations;
the manual page has additional details about these parameters. Most
matrix-free algorithms require only the application of the linear
operator to a vector. To provide this action, the user must write a
routine with the calling sequence

.. code-block::

   UserMult(Mat mat,Vec x,Vec y);

and then associate it with the matrix, ``mat``, by using the command

.. code-block::

   MatShellSetOperation(Mat mat,MatOperation MATOP_MULT, (void(*)(void)) PetscErrorCode (*UserMult)(Mat,Vec,Vec));

Here ``MATOP_MULT`` is the name of the operation for matrix-vector
multiplication. Within each user-defined routine (such as
``UserMult()``), the user should call ``MatShellGetContext()`` to obtain
the user-defined context, ``ctx``, that was set by ``MatCreateShell()``.
This shell matrix can be used with the iterative linear equation solvers
discussed in the following chapters.

The routine ``MatShellSetOperation()`` can be used to set any other
matrix operations as well. The file
``$PETSC_DIR/include/petscmat.h`` (`source <../../include/petscmat.h.html>`__).
provides a complete list of matrix operations, which have the form
``MATOP_<OPERATION>``, where ``<OPERATION>`` is the name (in all capital
letters) of the user interface routine (for example, ``MatMult()``
:math:`\to` ``MATOP_MULT``). All user-provided functions have the same
calling sequence as the usual matrix interface routines, since the
user-defined functions are intended to be accessed through the same
interface, e.g., ``MatMult(Mat,Vec,Vec)`` :math:`\to`
``UserMult(Mat,Vec,Vec)``. The final argument for
``MatShellSetOperation()`` needs to be cast to a ``void *``, since the
final argument could (depending on the MatOperation) be a variety of
different functions.

Note that ``MatShellSetOperation()`` can also be used as a “backdoor”
means of introducing user-defined changes in matrix operations for other
storage formats (for example, to override the default LU factorization
routine supplied within PETSc for the ``MATSEQAIJ`` format). However, we
urge anyone who introduces such changes to use caution, since it would
be very easy to accidentally create a bug in the new routine that could
affect other routines as well.

See also :any:`sec_nlmatrixfree` for details on one set of
helpful utilities for using the matrix-free approach for nonlinear
solvers.

.. _sec_othermat:

Other Matrix Operations
~~~~~~~~~~~~~~~~~~~~~~~

In many iterative calculations (for instance, in a nonlinear equations
solver), it is important for efficiency purposes to reuse the nonzero
structure of a matrix, rather than determining it anew every time the
matrix is generated. To retain a given matrix but reinitialize its
contents, one can employ

.. code-block::

   MatZeroEntries(Mat A);

This routine will zero the matrix entries in the data structure but keep
all the data that indicates where the nonzeros are located. In this way
a new matrix assembly will be much less expensive, since no memory
allocations or copies will be needed. Of course, one can also explicitly
set selected matrix elements to zero by calling ``MatSetValues()``.

By default, if new entries are made in locations where no nonzeros
previously existed, space will be allocated for the new entries. To
prevent the allocation of additional memory and simply discard those new
entries, one can use the option

.. code-block::

   MatSetOption(Mat A,MAT_NEW_NONZERO_LOCATIONS,PETSC_FALSE);

Once the matrix has been assembled, one can factor it numerically
without repeating the ordering or the symbolic factorization. This
option can save some computational time, although it does require that
the factorization is not done in-place.

In the numerical solution of elliptic partial differential equations, it
can be cumbersome to deal with Dirichlet boundary conditions. In
particular, one would like to assemble the matrix without regard to
boundary conditions and then at the end apply the Dirichlet boundary
conditions. In numerical analysis classes this process is usually
presented as moving the known boundary conditions to the right-hand side
and then solving a smaller linear system for the interior unknowns.
Unfortunately, implementing this requires extracting a large submatrix
from the original matrix and creating its corresponding data structures.
This process can be expensive in terms of both time and memory.

One simple way to deal with this difficulty is to replace those rows in
the matrix associated with known boundary conditions, by rows of the
identity matrix (or some scaling of it). This action can be done with
the command

.. code-block::

   MatZeroRows(Mat A,PetscInt numRows,PetscInt rows[],PetscScalar diag_value,Vec x,Vec b),

or equivalently,

.. code-block::

   MatZeroRowsIS(Mat A,IS rows,PetscScalar diag_value,Vec x,Vec b);

For sparse matrices this removes the data structures for certain rows of
the matrix. If the pointer ``diag_value`` is ``NULL``, it even removes
the diagonal entry. If the pointer is not null, it uses that given value
at the pointer location in the diagonal entry of the eliminated rows.

One nice feature of this approach is that when solving a nonlinear
problem such that at each iteration the Dirichlet boundary conditions
are in the same positions and the matrix retains the same nonzero
structure, the user can call ``MatZeroRows()`` in the first iteration.
Then, before generating the matrix in the second iteration the user
should call

.. code-block::

   MatSetOption(Mat A,MAT_NEW_NONZERO_LOCATIONS,PETSC_FALSE);

From that point, no new values will be inserted into those (boundary)
rows of the matrix.

The functions ``MatZeroRowsLocal()`` and ``MatZeroRowsLocalIS()`` can
also be used if for each process one provides the Dirichlet locations in
the local numbering of the matrix. A drawback of ``MatZeroRows()`` is
that it destroys the symmetry of a matrix. Thus one can use

.. code-block::

   MatZeroRowsColumns(Mat A,PetscInt numRows,PetscInt rows[],PetscScalar diag_value,Vec x,Vec b),

or equivalently,

.. code-block::

   MatZeroRowsColumnsIS(Mat A,IS rows,PetscScalar diag_value,Vec x,Vec b);

Note that with all of these for a given assembled matrix it can be only
called once to update the x and b vector. It cannot be used if one
wishes to solve multiple right hand side problems for the same matrix
since the matrix entries needed for updating the b vector are removed in
its first use.

Once the zeroed rows are removed the new matrix has possibly many rows
with only a diagonal entry affecting the parallel load balancing. The
``PCREDISTRIBUTE`` preconditioner removes all the zeroed rows (and
associated columns and adjusts the right hand side based on the removed
columns) and then rebalances the resulting rows of smaller matrix across
the processes. Thus one can use ``MatZeroRows()`` to set the Dirichlet
points and then solve with the preconditioner ``PCREDISTRIBUTE``. Note
if the original matrix was symmetric the smaller solved matrix will also
be symmetric.

Another matrix routine of interest is

.. code-block::

   MatConvert(Mat mat,MatType newtype,Mat *M)

which converts the matrix ``mat`` to new matrix, ``M``, that has either
the same or different format. Set ``newtype`` to ``MATSAME`` to copy the
matrix, keeping the same matrix format. See
``$PETSC_DIR/include/petscmat.h`` (`source <../../include/petscmat.h.html>`__)
for other available matrix types; standard ones are ``MATSEQDENSE``,
``MATSEQAIJ``, ``MATMPIAIJ``, ``MATSEQBAIJ`` and ``MATMPIBAIJ``.

In certain applications it may be necessary for application codes to
directly access elements of a matrix. This may be done by using the the
command (for local rows only)

.. code-block::

   MatGetRow(Mat A,PetscInt row, PetscInt *ncols,const PetscInt (*cols)[],const PetscScalar (*vals)[]);

The argument ``ncols`` returns the number of nonzeros in that row, while
``cols`` and ``vals`` returns the column indices (with indices starting
at zero) and values in the row. If only the column indices are needed
(and not the corresponding matrix elements), one can use ``NULL`` for
the ``vals`` argument. Similarly, one can use ``NULL`` for the ``cols``
argument. The user can only examine the values extracted with
``MatGetRow()``; the values *cannot* be altered. To change the matrix
entries, one must use ``MatSetValues()``.

Once the user has finished using a row, he or she *must* call

.. code-block::

   MatRestoreRow(Mat A,PetscInt row,PetscInt *ncols,PetscInt **cols,PetscScalar **vals);

to free any space that was allocated during the call to ``MatGetRow()``.

.. _sec_partitioning:

Partitioning
~~~~~~~~~~~~

For almost all unstructured grid computation, the distribution of
portions of the grid across the process’s work load and memory can have
a very large impact on performance. In most PDE calculations the grid
partitioning and distribution across the processes can (and should) be
done in a “pre-processing” step before the numerical computations.
However, this does not mean it need be done in a separate, sequential
program; rather, it should be done before one sets up the parallel grid
data structures in the actual program. PETSc provides an interface to
the ParMETIS (developed by George Karypis; see
`the PETSc installation instructions <https://petsc.org/release/install/>`__.
for directions on installing PETSc to use ParMETIS) to allow the
partitioning to be done in parallel. PETSc does not currently provide
directly support for dynamic repartitioning, load balancing by migrating
matrix entries between processes, etc. For problems that require mesh
refinement, PETSc uses the “rebuild the data structure” approach, as
opposed to the “maintain dynamic data structures that support the
insertion/deletion of additional vector and matrix rows and columns
entries” approach.

Partitioning in PETSc is organized around the ``MatPartitioning``
object. One first creates a parallel matrix that contains the
connectivity information about the grid (or other graph-type object)
that is to be partitioned. This is done with the command

.. code-block::

   MatCreateMPIAdj(MPI_Comm comm,int mlocal,PetscInt n,const PetscInt ia[],const PetscInt ja[],PetscInt *weights,Mat *Adj);

The argument ``mlocal`` indicates the number of rows of the graph being
provided by the given process, ``n`` is the total number of columns;
equal to the sum of all the ``mlocal``. The arguments ``ia`` and ``ja``
are the row pointers and column pointers for the given rows; these are
the usual format for parallel compressed sparse row storage, using
indices starting at 0, *not* 1.

.. figure:: /images/docs/manual/usg.*
   :alt: Numbering on Simple Unstructured Grid
   :name: fig_usg

   Numbering on Simple Unstructured Grid

This, of course, assumes that one has already distributed the grid
(graph) information among the processes. The details of this initial
distribution is not important; it could be simply determined by
assigning to the first process the first :math:`n_0` nodes from a file,
the second process the next :math:`n_1` nodes, etc.

For example, we demonstrate the form of the ``ia`` and ``ja`` for a
triangular grid where we

(1) partition by element (triangle)

-  Process 0: ``mlocal = 2``, ``n = 4``, ``ja =``\ ``{2,3, 3}``,
   ``ia =`` ``{0,2,3}``

-  Process 1: ``mlocal = 2``, ``n = 4``, ``ja =``\ ``{0, 0,1}``,
   ``ia =``\ ``{0,1,3}``

Note that elements are not connected to themselves and we only indicate
edge connections (in some contexts single vertex connections between
elements may also be included). We use a space above to denote the
transition between rows in the matrix.

and (2) partition by vertex.

-  Process 0: ``mlocal = 3``, ``n = 6``,
   ``ja =``\ ``{3,4, 4,5, 3,4,5}``, ``ia =``\ ``{0, 2, 4, 7}``

-  Process 1: ``mlocal = 3``, ``n = 6``,
   ``ja =``\ ``{0,2, 4, 0,1,2,3,5, 1,2,4}``,
   ``ia =``\ ``{0, 3, 8, 11}``.

Once the connectivity matrix has been created the following code will
generate the renumbering required for the new partition

.. code-block::

   MatPartitioningCreate(MPI_Comm comm,MatPartitioning *part);
   MatPartitioningSetAdjacency(MatPartitioning part,Mat Adj);
   MatPartitioningSetFromOptions(MatPartitioning part);
   MatPartitioningApply(MatPartitioning part,IS *is);
   MatPartitioningDestroy(MatPartitioning *part);
   MatDestroy(Mat *Adj);
   ISPartitioningToNumbering(IS is,IS *isg);

The resulting ``isg`` contains for each local node the new global number
of that node. The resulting ``is`` contains the new process number that
each local node has been assigned to.

Now that a new numbering of the nodes has been determined, one must
renumber all the nodes and migrate the grid information to the correct
process. The command

.. code-block::

   AOCreateBasicIS(isg,NULL,&ao);

generates, see :any:`sec_ao`, an AO object that can be
used in conjunction with the ``is`` and ``isg`` to move the relevant
grid information to the correct process and renumber the nodes etc. In
this context, the new ordering is the “application” ordering so
``AOPetscToApplication()`` converts old global indices to new global
indices and ``AOApplicationToPetsc()`` converts new global indices back
to old global indices.

PETSc does not currently provide tools that completely manage the
migration and node renumbering, since it will be dependent on the
particular data structure you use to store the grid information and the
type of grid information that you need for your application. We do plan
to include more support for this in the future, but designing the
appropriate general user interface and providing a scalable
implementation that can be used for a wide variety of different grids
requires a great deal of time.

.. raw:: html

    <hr>

.. bibliography:: /petsc.bib
   :filter: docname in docnames
   :keyprefix: KEYPREFIX-
   :labelprefix: ref-
