Sparse Matrices

Dense Matrices

A \(n \times n\) matrix is called dense if it has \(O(n^2)\) non-zero entries. For example:

\[\mathbf{A} = \begin{bmatrix} 1.0 & 2.0 & 3.0 \\ 4.0 & 5.0 & 6.0 \\ 7.0 & 8.0 & 9.0 \end{bmatrix}.\]

To store the matrix, all components are saved in row-major order. For \(\mathbf{A}\) given above, we would store:

\[AA = \begin{bmatrix} 1.0 & 2.0 & 3.0 & 4.0 & 5.0 & 6.0 & 7.0 & 8.0 & 9.0 \end{bmatrix}.\]

The dimensions of the matrix are stored separately.

Sparse Matrices

A \(n \times n\) matrix is called sparse if it has \(O(n)\) non-zero entries. For example:

\[A = \begin{bmatrix} 1.0 & 0 & 0 & 2.0 & 0 \\ 3.0 & 4.0 & 0 & 5.0 & 0 \\ 6.0 & 0 & 7.0 & 8.0 & 9.0 \\ 0 & 0 & 10.0 & 11.0 & 0 \\ 0 & 0 & 0 & 0 & 12.0 \end{bmatrix}.\]

COO (Coordinate Format) stores arrays of row indices, column indices and the corresponding non-zero data values in any order. This format provides fast methods to construct sparse matrices and convert to different sparse formats. For \({\bf A}\) the COO format is:

\[\textrm{data} = \begin{bmatrix} 12.0 & 9.0 & 7.0 & 5.0 & 1.0 & 2.0 & 11.0 & 3.0 & 6.0 & 4.0 & 8.0 & 10.0\end{bmatrix}\] \[\textrm{row} = \begin{bmatrix} 4 & 2 & 2 & 1 & 0 & 0 & 3 & 1 & 2 & 1 & 2 & 3 \end{bmatrix}, \\ \textrm{col} = \begin{bmatrix} 4 & 4 & 2 & 3 & 0 & 3 & 3 & 0 & 0 & 1 & 3 & 2 \end{bmatrix}\]

How to interpret: The first entries of \(\textrm{data}\), \(\textrm{row}\), \(\textrm{col}\) are 12.0, 4, 4, respectively, meaning there is a 12.0 at position (4, 4) of the matrix; second entries are 9.0, 2, 4, so there is a 9.0 at (2, 4).

CSR (Compressed Sparse Row) encodes rows offsets, column indices and the corresponding non-zero data values. This format provides fast arithmetic operations between sparse matrices, and fast matrix vector product. The row offsets are defined by the followign recursive relationship (starting with \(\textrm{rowptr}[0] = 0\)):

\[ \textrm{rowptr}[j] = \textrm{rowptr}[j-1] + \mathrm{nnz}(\textrm{row}_{j-1}), \\ \]

where \(\mathrm{nnz}(\textrm{row}_k)\) is the number of non-zero elements in the \(k^{th}\) row. Note that the length of \(\textrm{rowptr}\) is \(n_{rows} + 1\), where the last element in \(\textrm{rowptr}\) is the number of nonzeros in \(A\). For \({\bf A}\) the CSR format is:

\[\textrm{data} = \begin{bmatrix} 1.0 & 2.0 & 3.0 & 4.0 & 5.0 & 6.0 & 7.0 & 8.0 & 9.0 & 10.0 & 11.0 & 12.0 \end{bmatrix}\] \[\textrm{col} = \begin{bmatrix} 0 & 3 & 0 & 1 & 3 & 0 & 2 & 3 & 4 & 2 & 3 & 4\end{bmatrix}\] \[\textrm{rowptr} = \begin{bmatrix} 0 & 2 & 5 & 9 & 11 & 12 \end{bmatrix}\]

How to interpret: The first two entries of \(\textrm{rowptr}\) gives us the elements in the first row. Interval [0, 2) of \(\textrm{data}\) and \(\textrm{col}\), corresponding to two (data, column) pairs: (1.0, 0) and (2.0, 3), means the first row has 1.0 at column 0 and 2.0 at column 3. The second and third entries of \(\textrm{rowptr}\) tells us [2, 5) of \(\textrm{data}\) and \(\textrm{col}\) corresponds to the second row. The three pairs (3.0, 0), (4.0, 1), (5.0, 3) means in the second row, there is a 3.0 at column 0, a 4.0 at column 1, and a 5.0 at column 3.

CSR Matrix Vector Product Algorithm

The following code snippet performs CSR matrix vector product for square matrices:

import numpy as np
def csr_mat_vec(A, x):
  Ax = np.zeros_like(x)
  for i in range(x.shape[0]):
    for k in range(A.rowptr[i], A.rowptr[i+1]):
      Ax[i] += A.data[k]*x[A.col[k]]
  return Ax

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