Diagonal Format (DIA)

  • very simple scheme
  • diagonals in dense NumPy array of shape (n_diag, length)
    • fixed length -> waste space a bit when far from main diagonal
    • subclass of _data_matrix (sparse matrix classes with .data attribute)
  • offset for each diagonal
    • 0 is the main diagonal
    • negative offset = below
    • positive offset = above
  • fast matrix * vector (sparsetools)
  • fast and easy item-wise operations
    • manipulate data array directly (fast NumPy machinery)
  • constructor accepts:
    • dense matrix (array)
    • sparse matrix
    • shape tuple (create empty matrix)
    • (data, offsets) tuple
  • no slicing, no individual item access
  • use:
    • rather specialized
    • solving PDEs by finite differences
    • with an iterative solver

Examples

  • create some DIA matrices:

    >>> data = np.array([[1, 2, 3, 4]]).repeat(3, axis=0)

    >>> data

    array([[1, 2, 3, 4],

           [1, 2, 3, 4],

           [1, 2, 3, 4]])

    >>> offsets = np.array([0, -1, 2])

    >>> mtx = sparse.dia_matrix((data, offsets), shape=(4, 4))

    >>> mtx   

    <4x4 sparse matrix of type '<... 'numpy.int64'>'

            with 9 stored elements (3 diagonals) in DIAgonal format>

    >>> mtx.todense()

    matrix([[1, 0, 3, 0],

            [1, 2, 0, 4],

            [0, 2, 3, 0],

            [0, 0, 3, 4]])

    

    >>> data = np.arange(12).reshape((3, 4)) + 1

    >>> data

    array([[ 1,  2,  3,  4],

           [ 5,  6,  7,  8],

           [ 9, 10, 11, 12]])

    >>> mtx = sparse.dia_matrix((data, offsets), shape=(4, 4))

    >>> mtx.data        

    array([[ 1,  2,  3,  4],

           [ 5,  6,  7,  8],

           [ 9, 10, 11, 12]])

    >>> mtx.offsets

    array([ 0, -1,  2], dtype=int32)

    >>> print(mtx)   

      (0, 0)        1

      (1, 1)        2

      (2, 2)        3

      (3, 3)        4

      (1, 0)        5

      (2, 1)        6

      (3, 2)        7

      (0, 2)        11

      (1, 3)        12

    >>> mtx.todense()

    matrix([[ 1,  0, 11,  0],

            [ 5,  2,  0, 12],

            [ 0,  6,  3,  0],

            [ 0,  0,  7,  4]])

  • explanation with a scheme:

    offset: row

    

         2:  9

         1:  --10------

         0:  1  . 11  .

        -1:  5  2  . 12

        -2:  .  6  3  .

        -3:  .  .  7  4

             ---------8

  • matrix-vector multiplication

    >>> vec = np.ones((4, ))

    >>> vec

    array([1.,  1.,  1.,  1.])

    >>> mtx * vec

    array([12.,  19.,   9.,  11.])

    >>> mtx.toarray() * vec

    array([[ 1.,   0.,  11.,   0.],

           [ 5.,   2.,   0.,  12.],

           [ 0.,   6.,   3.,   0.],

           [ 0.,   0.,   7.,   4.]])