Some on sparse matrix solve speed.

The sparse solvers in scipy don't necessarily seem faster but these are tiny matrices.

In [1]: from scipy import sparse

In [6]: a = sparse.lil_matrix([[1., 0.], [0., 4.]])

In [7]: a
Out[7]:
<2x2 sparse matrix of type '<type 'numpy.float64'>'
   with 2 stored elements in LInked List format>

In [8]: b = sparse.lil_matrix([[5.], [6.0]])

In [9]: b
Out[9]:
<2x1 sparse matrix of type '<type 'numpy.float64'>'
   with 2 stored elements in LInked List format>

In [10]: from scipy.sparse.linalg import spsolve

In [12]: spsolve(a.tocsr(), b)
Out[12]: array([ 5. ,  1.5])

In [13]: import numpy

In [14]: A = numpy.array([[1.0, 0.0], [0.0, 4.0]])

In [15]: B = numpy.array([[5.], [6.0]])

In [16]: %timeit numpy.linalg.solve(A, B)
10000 loops, best of 3: 19.6 µs per loop

In [17]: %timeit spsolve(a, b)
1000 loops, best of 3: 370 µs per loop

In [18]: from scipy.sparse.linalg import factorized

In [20]: a0 = a.tocsr()

In [21]: %timeit spsolve(a, b)
1000 loops, best of 3: 370 µs per loop

In [22]: a1 = a.tocsc()

In [23]: %timeit spsolve(a1, b)
1000 loops, best of 3: 196 µs per loop

In [24]: solve = factorized(a1)

In [29]: b = numpy.array([5., 6.0])

In [30]: solve(b)
Out[30]: array([ 5. ,  1.5])

In [31]: %timeit solve(b)
100000 loops, best of 3: 11 µs per loop

In [32]: %timeit solve = factorized(a1); solve(b)
10000 loops, best of 3: 158 µs per loop