2.5.1. Introduction

(dense) matrix is:

  • mathematical object
  • data structure for storing a 2D array of values

important features:

  • memory allocated once for all items
    • usually a contiguous chunk, think NumPy ndarray
  • fast access to individual items (*) Why Sparse Matrices?

  • the memory, that grows like n**2

  • small example (double precision matrix):

    >>> import numpy as np
    >>> import matplotlib.pyplot as plt
    >>> x = np.linspace(0, 1e6, 10)
    >>> plt.plot(x, 8.0 * (x**2) / 1e6, lw=5)
    [<matplotlib.lines.Line2D object at ...>]
    >>> plt.xlabel('size n')
    <matplotlib.text.Text object at ...>
    >>> plt.ylabel('memory [MB]')
    <matplotlib.text.Text object at ...> Sparse Matrices vs. Sparse Matrix Storage Schemes

  • sparse matrix is a matrix, which is almost empty
  • storing all the zeros is wasteful -> store only nonzero items
  • think compression
  • pros: huge memory savings
  • cons: depends on actual storage scheme, (*) usually does not hold Typical Applications

  • solution of partial differential equations (PDEs)
    • the finite element method
    • mechanical engineering, electrotechnics, physics, …
  • graph theory
    • nonzero at (i, j) means that node i is connected to node j Sparsity Structure Visualization

  • spy() from matplotlib
  • example plots:
../../_images/graph.png ../../_images/graph_g.png ../../_images/graph_rcm.png