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Speed ​​up random matrix computation

I create random Toeplitz matrices to estimate the probability of their reversibility. My current code

import random
from scipy.linalg import toeplitz
import numpy as np
for n in xrange(1,25):
    rankzero = 0
    for repeats in xrange(50000):
        column = [random.choice([0,1]) for x in xrange(n)]
        row = [column[0]]+[random.choice([0,1]) for x in xrange(n-1)]
        matrix = toeplitz(column, row)
        if  (np.linalg.matrix_rank(matrix) < n):
            rankzero += 1
    print n, (rankzero*1.0)/50000

Can this be accelerated?

I would like to increase the value of 50,000 to get more accuracy, but now it is too slow.

Profiling using only for n in xrange(10,14)shows

  400000    9.482    0.000    9.482    0.000 {numpy.linalg.lapack_lite.dgesdd}
  4400000    7.591    0.000   11.089    0.000 random.py:272(choice)
   200000    6.836    0.000   10.903    0.000 index_tricks.py:144(__getitem__)
        1    5.473    5.473   62.668   62.668 toeplitz.py:3(<module>)
   800065    4.333    0.000    4.333    0.000 {numpy.core.multiarray.array}
   200000    3.513    0.000   19.949    0.000 special_matrices.py:128(toeplitz)
   200000    3.484    0.000   20.250    0.000 linalg.py:1194(svd)
6401273/6401237    2.421    0.000    2.421    0.000 {len}
   200000    2.252    0.000   26.047    0.000 linalg.py:1417(matrix_rank)
  4400000    1.863    0.000    1.863    0.000 {method 'random' of '_random.Random' objects}
  2201015    1.240    0.000    1.240    0.000 {isinstance}
[...]
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3 answers

One way is to save some work from re-invoking the toeplitz () function by caching indexes into which values ​​are placed. The following code is 30% faster than the source code. The rest of the performance is in rank calculation ... And I don't know if there is a faster rank calculation for greenhouse matrices with 0s and 1s.

(). ~ 4 , matrix_rank scipy.linalg.det() == 0 ( , )

import random
from scipy.linalg import toeplitz, det
import numpy as np,numpy.random

class si:
    #cache of info for toeplitz matrix construction
    indx = None
    l = None

def xtoeplitz(c,r):
    vals = np.concatenate((r[-1:0:-1], c))
    if si.indx is None or si.l != len(c):
        a, b = np.ogrid[0:len(c), len(r) - 1:-1:-1]
        si.indx = a + b
        si.l = len(c)
    # `indx` is a 2D array of indices into the 1D array `vals`, arranged so
    # that `vals[indx]` is the Toeplitz matrix.
    return vals[si.indx]

def doit():
    for n in xrange(1,25):
        rankzero = 0
        si.indx=None

        for repeats in xrange(5000):

            column = np.random.randint(0,2,n)
            #column=[random.choice([0,1]) for x in xrange(n)] # original code

            row = np.r_[column[0], np.random.randint(0,2,n-1)]
            #row=[column[0]]+[random.choice([0,1]) for x in xrange(n-1)] #origi

            matrix = xtoeplitz(column, row)
            #matrix=toeplitz(column,row) # original code

            #if  (np.linalg.matrix_rank(matrix) < n): # original code
            if  np.abs(det(matrix))<1e-4: # should be faster for small matrices
                rankzero += 1
        print n, (rankzero*1.0)/50000
+3

, 0s 1s:

column = [random.choice([0,1]) for x in xrange(n)]
row = [column[0]]+[random.choice([0,1]) for x in xrange(n-1)]

. , , random.choice() , . 500% :

column = [0 for i in xrange(n)]
row = [0 for i in xrange(n)]

# NOTE: n must be less than 32 here, or remove int() and lose some speed
cbits = int(random.getrandbits(n))
rbits = int(random.getrandbits(n))

for i in xrange(n):
    column[i] = cbits & 1
    cbits >>= 1
    row[i] = rbits & 1
    rbits >>= 1

row[0] = column[0]
+2

, lapack dgesdd , LU- .

matrix_rank det , lapack dgetrf, LU- (http://docs.scipy.org/doc/numpy/reference/generated/numpy.linalg.det.html).

Thus, the asymptotic complexity of both calls matrix_rankand detis O (n ^ 3), the complexity of the decomposition of LU.

However, Topelitsa systems can be solved in O (n ^ 2) (according to Wikipedia). So, if you want to run your code on large matrices, it makes sense to write a python extension to call a specialized library.
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