## Solve a multitude of linear least square system efficiently

Question

I have to find the best solution for >10^7 equation systems with 5 equations in 2 variables each (5 measurements to find 2 parameters with the least amount of error in a long series). The following code (normally used to do curve fitting) does what I want:

``````#Create_example_Data
n = 100
T_Arm = np.arange(10*n).reshape(-1, 5, 2)
Erg = np.arange(5*n).reshape(-1, 5)
m = np.zeros(n)
c = np.zeros(n)
#Run
for counter in xrange(n):
m[counter], c[counter] = np.linalg.lstsq(T_Arm[counter, :, :],
Erg[counter, :])[0]
``````

Unfortunately it is too slow. Is there any way how to speed this code up significantly? I tried to vectorise it, but I did not succeed. Using the last solution as a initial guess might be a good idea as well. Using `scipy.optimize.leastsq` did not speed it up as well.

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## Answers to Solve a multitude of linear least square system efficiently ( 1 )

1. You could use a sparse block matrix A which stores the (5, 2) entries of T_Arm on its diagonal, and solve AX = b where b is the vector composed of stacked entries of `Erg`. Then solve the system with scipy.sparse.linalg.lsqr(A, b).

To construct A and b I use n=3 for visualisation purposes:

``````import numpy as np
import scipy
from scipy.sparse import bsr_matrix
n = 3
col = np.hstack(5 * [np.arange(10 * n / 5).reshape(n, 2)]).flatten()
array([ 0.,  1.,  0.,  1.,  0.,  1.,  0.,  1.,  0.,  1.,  2.,  3.,  2.,
3.,  2.,  3.,  2.,  3.,  2.,  3.,  4.,  5.,  4.,  5.,  4.,  5.,
4.,  5.,  4.,  5.])

row = np.tile(np.arange(10 * n / 2), (2, 1)).T.flatten()
array([  0.,   0.,   1.,   1.,   2.,   2.,   3.,   3.,   4.,   4.,   5.,
5.,   6.,   6.,   7.,   7.,   8.,   8.,   9.,   9.,  10.,  10.,
11.,  11.,  12.,  12.,  13.,  13.,  14.,  14.])

A = bsr_matrix((T_Arm[:n].flatten(), (row, col)), shape=(5 * n, 2 * n))
A.toarray()
array([[ 0,  1,  0,  0,  0,  0],
[ 2,  3,  0,  0,  0,  0],
[ 4,  5,  0,  0,  0,  0],
[ 6,  7,  0,  0,  0,  0],
[ 8,  9,  0,  0,  0,  0],
[ 0,  0, 10, 11,  0,  0],
[ 0,  0, 12, 13,  0,  0],
[ 0,  0, 14, 15,  0,  0],
[ 0,  0, 16, 17,  0,  0],
[ 0,  0, 18, 19,  0,  0],
[ 0,  0,  0,  0, 20, 21],
[ 0,  0,  0,  0, 22, 23],
[ 0,  0,  0,  0, 24, 25],
[ 0,  0,  0,  0, 26, 27],
[ 0,  0,  0,  0, 28, 29]], dtype=int64)

b = Erg[:n].flatten()
``````

And then

``````scipy.sparse.linalg.lsqr(A, b)[0]
array([  5.00000000e-01,  -1.39548109e-14,   5.00000000e-01,
8.71088538e-16,   5.00000000e-01,   2.35398726e-15])
``````

EDIT: A is not as huge in memory as it seems: more on block sparse matrices here.