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# 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`).

## 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

## 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` to 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.

## Notes

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

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

The `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.

If `m` and `n` are not `PETSC_DECIDE`, then the values determine the `PetscLayout` of the matrix and the ranges returned by `MatGetOwnershipRange()`, `MatGetOwnershipRanges()`, `MatGetOwnershipRangeColumn()`, and `MatGetOwnershipRangesColumn()`.

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` on each MPI process. I.e., each MPI process 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. The two matrices, the DIAGONAL portion and the OFF-DIAGONAL portion are each stored as `MATSEQAIJ` matrices.

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 `MATSEQAIJ` is returned. If a matrix of type `MATMPIAIJ` is desired for this type of communicator, use the construction mechanism

      MatCreate(..., &A);
      MatSetType(A, MATMPIAIJ);
      MatSetSizes(A, m, n, M, N);
      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.

## 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

                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

          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 `MATSEQAIJ` matrices. For example, proc1 will store [E] as a `MATSEQAIJ` matrix, and [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 submatrix. One way to choose `d_nz` and `o_nz` is to use the maximum number of nonzeros over the local rows for each of the local DIAGONAL, and the OFF-DIAGONAL submatrices. In this case, the values of `d_nz`,`o_nz` are

         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

         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

Matrices, `Mat`, Sparse Matrix Creation, `MatCreate()`, `MatCreateSeqAIJ()`, `MatSetValues()`, `MatMPIAIJSetPreallocation()`, `MatMPIAIJSetPreallocationCSR()`, `MATMPIAIJ`, `MatCreateMPIAIJWithArrays()`, `MatGetOwnershipRange()`, `MatGetOwnershipRanges()`, `MatGetOwnershipRangeColumn()`, `MatGetOwnershipRangesColumn()`, `PetscLayout`

## Level

intermediate

## Location

src/mat/impls/aij/mpi/mpiaij.c

## Examples

src/ksp/ksp/tutorials/ex81a.c
src/tao/bound/tutorials/plate2f.F90
src/mat/tutorials/ex4.c
src/ksp/ksp/tutorials/ex81.c
src/ksp/ksp/tutorials/ex79.c
src/mat/tutorials/ex4f.F90
src/mat/tutorials/ex9.c
src/ts/tutorials/ex44.c

Index of all Mat routines
Table of Contents for all manual pages
Index of all manual pages