# Block Compressed Row Format (BSR)¶

• basically a CSR with dense sub-matrices of fixed shape instead of scalar items

• block size (R, C) must evenly divide the shape of the matrix (M, N)
• three NumPy arrays: indices, indptr, data
• indices is array of column indices for each block
• data is array of corresponding nonzero values of shape (nnz, R, C)
• subclass of `_cs_matrix` (common CSR/CSC functionality)
• subclass of `_data_matrix` (sparse matrix classes with .data attribute)
• fast matrix vector products and other arithmetics (sparsetools)

• constructor accepts:
• dense matrix (array)
• sparse matrix
• shape tuple (create empty matrix)
• (data, ij) tuple
• (data, indices, indptr) tuple
• many arithmetic operations considerably more efficient than CSR for sparse matrices with dense sub-matrices

• use:
• like CSR
• vector-valued finite element discretizations

## Examples¶

• create empty BSR matrix with (1, 1) block size (like CSR…):

```>>> mtx = sparse.bsr_matrix((3, 4), dtype=np.int8)
>>> mtx
<3x4 sparse matrix of type '<... 'numpy.int8'>'
with 0 stored elements (blocksize = 1x1) in Block Sparse Row format>
>>> mtx.todense()
matrix([[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0]], dtype=int8)
```
• create empty BSR matrix with (3, 2) block size:

```>>> mtx = sparse.bsr_matrix((3, 4), blocksize=(3, 2), dtype=np.int8)
>>> mtx
<3x4 sparse matrix of type '<... 'numpy.int8'>'
with 0 stored elements (blocksize = 3x2) in Block Sparse Row format>
>>> mtx.todense()
matrix([[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0]], dtype=int8)
```
• a bug?
• create using (data, ij) tuple with (1, 1) block size (like CSR…):

```>>> row = np.array([0, 0, 1, 2, 2, 2])
>>> col = np.array([0, 2, 2, 0, 1, 2])
>>> data = np.array([1, 2, 3, 4, 5, 6])
>>> mtx = sparse.bsr_matrix((data, (row, col)), shape=(3, 3))
>>> mtx
<3x3 sparse matrix of type '<... 'numpy.int64'>'
with 6 stored elements (blocksize = 1x1) in Block Sparse Row format>
>>> mtx.todense()
matrix([[1, 0, 2],
[0, 0, 3],
[4, 5, 6]]...)
>>> mtx.data
array([[[1]],

[[2]],

[[3]],

[[4]],

[[5]],

[[6]]]...)
>>> mtx.indices
array([0, 2, 2, 0, 1, 2], dtype=int32)
>>> mtx.indptr
array([0, 2, 3, 6], dtype=int32)
```
• create using (data, indices, indptr) tuple with (2, 2) block size:

```>>> indptr = np.array([0, 2, 3, 6])
>>> indices = np.array([0, 2, 2, 0, 1, 2])
>>> data = np.array([1, 2, 3, 4, 5, 6]).repeat(4).reshape(6, 2, 2)
>>> mtx = sparse.bsr_matrix((data, indices, indptr), shape=(6, 6))
>>> mtx.todense()
matrix([[1, 1, 0, 0, 2, 2],
[1, 1, 0, 0, 2, 2],
[0, 0, 0, 0, 3, 3],
[0, 0, 0, 0, 3, 3],
[4, 4, 5, 5, 6, 6],
[4, 4, 5, 5, 6, 6]])
>>> data
array([[[1, 1],
[1, 1]],

[[2, 2],
[2, 2]],

[[3, 3],
[3, 3]],

[[4, 4],
[4, 4]],

[[5, 5],
[5, 5]],

[[6, 6],
[6, 6]]])
```