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]]])