# MatCreateAIJ
Creates a sparse parallel matrix in `MATAIJ` format (the default parallel PETSc format).  For good matrix assembly performance the user should preallocate the matrix storage by setting the parameters d_nz (or d_nnz) and o_nz (or o_nnz).  By setting these parameters accurately, performance can be increased by more than a factor of 50. 
## Synopsis
```
#include "petscmat.h" 
PetscErrorCode MatCreateAIJ(MPI_Comm comm, PetscInt m, PetscInt n, PetscInt M, PetscInt N, PetscInt d_nz, const PetscInt d_nnz[], PetscInt o_nz, const PetscInt o_nnz[], Mat *A)
```
Collective


## Input Parameters

- ***comm -*** MPI communicator
- ***m -*** number of local rows (or `PETSC_DECIDE` to have calculated if M is given)
This value should be the same as the local size used in creating the
y vector for the matrix-vector product y = Ax.
- ***n -*** This value should be the same as the local size used in creating the
x vector for the matrix-vector product y = Ax. (or PETSC_DECIDE to have
calculated if N is given) For square matrices n is almost always m.
- ***M -*** number of global rows (or `PETSC_DETERMINE` to have calculated if m is given)
- ***N -*** number of global columns (or `PETSC_DETERMINE` to have calculated if n is given)
- ***d_nz  -*** number of nonzeros per row in DIAGONAL portion of local submatrix
(same value is used for all local rows)
- ***d_nnz -*** array containing the number of nonzeros in the various rows of the
DIAGONAL portion of the local submatrix (possibly different for each row)
or NULL, if d_nz is used to specify the nonzero structure.
The size of this array is equal to the number of local rows, i.e 'm'.
- ***o_nz  -*** number of nonzeros per row in the OFF-DIAGONAL portion of local
submatrix (same value is used for all local rows).
- ***o_nnz -*** array containing the number of nonzeros in the various rows of the
OFF-DIAGONAL portion of the local submatrix (possibly different for
each row) or NULL, if o_nz is used to specify the nonzero
structure. The size of this array is equal to the number
of local rows, i.e 'm'.



## Output Parameter

- ***A -*** the matrix


It is recommended that one use the `MatCreate()`, `MatSetType()` and/or `MatSetFromOptions()`,
MatXXXXSetPreallocation() paradigm instead of this routine directly.
[MatXXXXSetPreallocation() is, for example, `MatSeqAIJSetPreallocation()`]


## Notes
If the *_nnz parameter is given then the *_nz parameter is ignored

m,n,M,N parameters specify the size of the matrix, and its partitioning across
processors, while d_nz,d_nnz,o_nz,o_nnz parameters specify the approximate
storage requirements for this matrix.

If `PETSC_DECIDE` or  `PETSC_DETERMINE` is used for a particular argument on one
processor than it must be used on all processors that share the object for
that argument.

The user MUST specify either the local or global matrix dimensions
(possibly both).

The parallel matrix is partitioned across processors such that the
first m0 rows belong to process 0, the next m1 rows belong to
process 1, the next m2 rows belong to process 2 etc.. where
m0,m1,m2,.. are the input parameter 'm'. i.e each processor stores
values corresponding to [m x N] submatrix.

The columns are logically partitioned with the n0 columns belonging
to 0th partition, the next n1 columns belonging to the next
partition etc.. where n0,n1,n2... are the input parameter 'n'.

The DIAGONAL portion of the local submatrix on any given processor
is the submatrix corresponding to the rows and columns m,n
corresponding to the given processor. i.e diagonal matrix on
process 0 is [m0 x n0], diagonal matrix on process 1 is [m1 x n1]
etc. The remaining portion of the local submatrix [m x (N-n)]
constitute the OFF-DIAGONAL portion. The example below better
illustrates this concept.

For a square global matrix we define each processor's diagonal portion
to be its local rows and the corresponding columns (a square submatrix);
each processor's off-diagonal portion encompasses the remainder of the
local matrix (a rectangular submatrix).

If o_nnz, d_nnz are specified, then o_nz, and d_nz are ignored.

When calling this routine with a single process communicator, a matrix of
type SEQAIJ is returned.  If a matrix of type MPIAIJ is desired for this
type of communicator, use the construction mechanism
```none
     MatCreate(...,&A); MatSetType(A,MATMPIAIJ); MatSetSizes(A, m,n,M,N); MatMPIAIJSetPreallocation(A,...);
```


```none
MatCreate(...,&A);
```
```none
MatSetType(A,MATMPIAIJ);
```
```none
MatSetSizes(A, m,n,M,N);
```
```none
MatMPIAIJSetPreallocation(A,...);
```

By default, this format uses inodes (identical nodes) when possible.
We search for consecutive rows with the same nonzero structure, thereby
reusing matrix information to achieve increased efficiency.


## Options Database Keys

- ***-mat_no_inode  -*** Do not use inodes
- ***-mat_inode_limit <limit> -*** Sets inode limit (max limit=5)
- ***-matmult_vecscatter_view <viewer> -*** View the vecscatter (i.e., communication pattern) used in `MatMult()` of sparse parallel matrices.
See viewer types in manual of `MatView()`. Of them, ascii_matlab, draw or binary cause the vecscatter be viewed as a matrix.
Entry (i,j) is the size of message (in bytes) rank i sends to rank j in one `MatMult()` call.



## Example usage

Consider the following 8x8 matrix with 34 non-zero values, that is
assembled across 3 processors. Lets assume that proc0 owns 3 rows,
proc1 owns 3 rows, proc2 owns 2 rows. This division can be shown
as follows

```none
            1  2  0  |  0  3  0  |  0  4
    Proc0   0  5  6  |  7  0  0  |  8  0
            9  0 10  | 11  0  0  | 12  0
    -------------------------------------
           13  0 14  | 15 16 17  |  0  0
    Proc1   0 18  0  | 19 20 21  |  0  0
            0  0  0  | 22 23  0  | 24  0
    -------------------------------------
    Proc2  25 26 27  |  0  0 28  | 29  0
           30  0  0  | 31 32 33  |  0 34
```


This can be represented as a collection of submatrices as

```none
      A B C
      D E F
      G H I
```


Where the submatrices A,B,C are owned by proc0, D,E,F are
owned by proc1, G,H,I are owned by proc2.

The 'm' parameters for proc0,proc1,proc2 are 3,3,2 respectively.
The 'n' parameters for proc0,proc1,proc2 are 3,3,2 respectively.
The 'M','N' parameters are 8,8, and have the same values on all procs.

The DIAGONAL submatrices corresponding to proc0,proc1,proc2 are
submatrices [A], [E], [I] respectively. The OFF-DIAGONAL submatrices
corresponding to proc0,proc1,proc2 are [BC], [DF], [GH] respectively.
Internally, each processor stores the DIAGONAL part, and the OFF-DIAGONAL
part as SeqAIJ matrices. for eg: proc1 will store [E] as a SeqAIJ
matrix, ans [DF] as another SeqAIJ matrix.

When d_nz, o_nz parameters are specified, d_nz storage elements are
allocated for every row of the local diagonal submatrix, and o_nz
storage locations are allocated for every row of the OFF-DIAGONAL submat.
One way to choose d_nz and o_nz is to use the max nonzerors per local
rows for each of the local DIAGONAL, and the OFF-DIAGONAL submatrices.
In this case, the values of d_nz,o_nz are
```none
     proc0 : dnz = 2, o_nz = 2
     proc1 : dnz = 3, o_nz = 2
     proc2 : dnz = 1, o_nz = 4
```

We are allocating m*(d_nz+o_nz) storage locations for every proc. This
translates to 3*(2+2)=12 for proc0, 3*(3+2)=15 for proc1, 2*(1+4)=10
for proc3. i.e we are using 12+15+10=37 storage locations to store
34 values.

When d_nnz, o_nnz parameters are specified, the storage is specified
for every row, corresponding to both DIAGONAL and OFF-DIAGONAL submatrices.
In the above case the values for d_nnz,o_nnz are
```none
     proc0: d_nnz = [2,2,2] and o_nnz = [2,2,2]
     proc1: d_nnz = [3,3,2] and o_nnz = [2,1,1]
     proc2: d_nnz = [1,1]   and o_nnz = [4,4]
```

Here the space allocated is sum of all the above values i.e 34, and
hence pre-allocation is perfect.




## See Also
 [Sparse Matrix Creation](sec_matsparse), `MatCreate()`, `MatCreateSeqAIJ()`, `MatSetValues()`, `MatMPIAIJSetPreallocation()`, `MatMPIAIJSetPreallocationCSR()`,
`MATMPIAIJ`, `MatCreateMPIAIJWithArrays()`

## Level
intermediate

## Location
<A HREF="PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/mat/impls/aij/mpi/mpiaij.c.html#MatCreateAIJ">src/mat/impls/aij/mpi/mpiaij.c</A>

## Examples
<A HREF="PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/mat/tutorials/ex4.c.html">src/mat/tutorials/ex4.c.html</A><BR>
<A HREF="PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/mat/tutorials/ex4f.F90.html">src/mat/tutorials/ex4f.F90.html</A><BR>
<A HREF="PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/mat/tutorials/ex9.c.html">src/mat/tutorials/ex9.c.html</A><BR>
<A HREF="PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/ksp/ksp/tutorials/ex14f.F90.html">src/ksp/ksp/tutorials/ex14f.F90.html</A><BR>
<A HREF="PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/ksp/ksp/tutorials/ex27.c.html">src/ksp/ksp/tutorials/ex27.c.html</A><BR>
<A HREF="PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/ksp/ksp/tutorials/ex79.c.html">src/ksp/ksp/tutorials/ex79.c.html</A><BR>
<A HREF="PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/ksp/ksp/tutorials/ex81.c.html">src/ksp/ksp/tutorials/ex81.c.html</A><BR>
<A HREF="PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/ts/tutorials/ex44.c.html">src/ts/tutorials/ex44.c.html</A><BR>
<A HREF="PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/tao/bound/tutorials/plate2.c.html">src/tao/bound/tutorials/plate2.c.html</A><BR>
<A HREF="PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/tao/bound/tutorials/plate2f.F90.html">src/tao/bound/tutorials/plate2f.F90.html</A><BR>


---
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