PETSc version 3.15.5
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# MatCreateAIJ

Creates a sparse parallel matrix in AIJ 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

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

        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.

### Options Database Keys

| -mat_no_inode | - Do not use inodes
| -mat_inode_limit <limit> | - Sets inode limit (max limit=5)

### 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 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

         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, coresponding 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

MatCreate(), MatCreateSeqAIJ(), MatSetValues(), MatMPIAIJSetPreallocation(), MatMPIAIJSetPreallocationCSR(),
MATMPIAIJ, MatCreateMPIAIJWithArrays()

### Level

intermediate

### Location

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

### Examples

src/mat/tutorials/ex4.c.html
src/mat/tutorials/ex9.c.html
src/mat/tutorials/ex4f.F90.html
src/ksp/ksp/tutorials/ex79.c.html
src/ksp/ksp/tutorials/ex81.c.html
src/ksp/ksp/tutorials/ex14f.F90.html
src/tao/bound/tutorials/plate2.c.html
src/tao/bound/tutorials/plate2f.F90.html
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