This example illustrates the use of the rocSPARSE
incomplete LU factorization preconditioner using the BSR storage format.
Given an arbitrary matrix
The incomplete LU decomposition is a sparse approximation of the above-mentioned LU decomposition. Thus, rocSPARSE
allows us to compute a sparse lower triangular matrix
- Setup input data.
- Allocate device memory and offload input data to the device.
- Initialize rocSPARSE by creating a handle.
- Prepare utility variables for rocSPARSE bsrilu0 invocation.
- Perform the analysis step.
- Call dbsrilu0 to compute the incomplete LU decomposition.
- Check zero-pivots.
- Convert the resulting BSR sparse matrix to a dense matrix. Check and print the resulting matrix.
- Free rocSPARSE resources and device memory.
- Print validation result.
The Block Compressed Sparse Row (BSR) storage format describes a sparse matrix using three arrays. The idea behind this storage format is to split the given sparse matrix into equal sized blocks of dimension bsr_dim
and store those using the CSR format. Because the CSR format only stores non-zero elements, the BSR format introduces the concept of non-zero block: a block that contains at least one non-zero element. Note that all elements of non-zero blocks are stored, even if some of them are equal to zero.
Therefore, defining
mb
: number of rows of blocksnb
: number of columns of blocksnnzb
: number of non-zero blocksbsr_dim
: dimension of each block
we can describe a sparse matrix using the following arrays:
-
bsr_val
: contains the elements of the non-zero blocks of the sparse matrix. The elements are stored block by block in column- or row-major order. That is, it is an array of sizennzb
$\cdot$ bsr_dim
$\cdot$ bsr_dim
. -
bsr_row_ptr
: given$i \in [0, mb]$ - if
$0 \leq i < mb$ ,bsr_row_ptr[i]
stores the index of the first non-zero block in row$i$ of the block matrix - if
$i = mb$ ,bsr_row_ptr[i]
storesnnzb
.
This way, row
$j \in [0, mb)$ contains the non-zero blocks of indices frombsr_row_ptr[j]
tobsr_row_ptr[j+1]-1
. The corresponding values inbsr_val
can be accessed frombsr_row_ptr[j] * bsr_dim * bsr_dim
to(bsr_row_ptr[j+1]-1) * bsr_dim * bsr_dim
. - if
-
bsr_col_ind
: given$i \in [0, nnzb-1]$ ,bsr_col_ind[i]
stores the column of the$i^{th}$ non-zero block in the block matrix.
Note that, for a given
For instance, consider a sparse matrix as
Taking
with the following non-zero blocks:
and the zero matrix:
Therefore, the BSR representation of
bsr_val = { 8, 0, 7, 2, 0, 3, 0, 5, 2, 0, 1, 0, 0, 0, 0, 0 // A_{00}
4, 7, 0, 0, 0, 7, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0 // A_{10}
0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 // A_{12}
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 5, 0 // A_{20}
0, 9, 6, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0 } // A_{21}
bsr_row_ptr = { 0, 1, 3, 4 }
bsr_col_ind = { 0, 0, 2, 0, 1 }
- rocSPARSE is initialized by calling
rocsparse_create_handle(rocsparse_handle*)
and is terminated by callingrocsparse_destroy_handle(rocsparse_handle)
. -
rocsparse_direction dir
: matrix storage of BSR blocks. The following values are accepted:-
rocsparse_direction_row
: parse blocks by rows. -
rocsparse_direction_column
: parse blocks by columns.
-
-
rocsparse_mat_descr descr
: holds all properties of a matrix. The properties set in this example are the following:-
rocsparse_fill_mode
: indicates whether a (triangular) matrix is lower (rocsparse_fill_mode_lower
) or upper (rocsparse_fill_mode_upper
) triangular.
-
-
rocsparse_solve_policy policy
: specifies the policy to follow for triangular solvers and factorizations. The only value accepted isrocsparse_solve_policy_auto
. -
rocsparse_analysis_policy analysis
: specifies the policy to follow for analysis data. The following values are accepted:-
rocsparse_analysis_policy_reuse
: the analysis data gathered is re-used. -
rocsparse_analysis_policy_force
: the analysis data will be re-built.
-
-
rocsparse_[sdcz]bsrilu0
computes the incomplete LU factorization of a sparse BSR matrix$A$ , such that$A \approx L \cdot U$ . The correct function signature should be chosen based on the datatype of the input matrix:-
s
single-precision real (float
) -
d
double-precision real (double
) -
c
single-precision complex (rocsparse_float_complex
) -
z
double-precision complex (rocsparse_double_complex
)
-
-
rocsparse_[sdcz]bsrilu0_analysis
performs the analysis step forrocsparse_[sdcz]bsrilu0
. The character matched in[sdcz]
coincides with the one matched inrocsparse_[sdcz]bsrilu0
. -
rocsparse_[sdcz]bsrilu0_buffer_size
allows to obtain the size (in bytes) of the temporary storage buffer required for therocsparse_[sdcz]bsrilu0_analysis
androcsparse_[sdcz]bsrilu0
functions. The character matched in[sdcz]
coincides with the one matched in any of the mentioned functions. -
rocsparse_bsrilu0_zero_pivot(rocsparse_handle, rocsparse_mat_info, rocsparse_int *position)
returnsrocsparse_status_zero_pivot
if either a structural or numerical zero has been found during the execution ofrocsparse_[sbcz]bsrilu0(....)
and stores inposition
the index$i$ of the first zero pivot$A_{ii}$ found. If no zero pivot is found it returnsrocsparse_status_success
.
rocsparse_analysis_policy
rocsparse_analysis_policy_reuse
rocsparse_bsrilu0_zero_pivot
rocsparse_create_handle
rocsparse_create_mat_descr
rocsparse_create_mat_info
rocsparse_dbsr2csr
rocsparse_dbsrilu0
rocsparse_dbsrilu0_analysis
rocsparse_dbsrilu0_buffer_size
rocsparse_dcsr2dense
rocsparse_destroy_handle
rocsparse_destroy_mat_descr
rocsparse_destroy_mat_info
rocsparse_direction
rocsparse_direction_column
rocsparse_fill_mode_lower
rocsparse_handle
rocsparse_int
rocsparse_mat_descr
rocsparse_mat_info
rocsparse_set_mat_fill_mode
rocsparse_solve_policy
rocsparse_solve_policy_auto
rocsparse_status
rocsparse_status_zero_pivot
hipFree
hipMalloc
hipMemcpy
hipMemcpyDeviceToHost
hipMemcpyHostToDevice