CODEWITHSUNDEEP
pythonperformancenumpymatrixmatrix-inverse
Babak
Babak

Reputation: 487

efficiency of inverting a matrix in numpy with Cholesky decomposition

I have a symmetric positive-definite matrix (e.g. Covariance matrix), and I want to calculate its inverse. In math, I know that it is more efficient to use Cholesky decomposition to invert the matrix, especially if your matrix is big. But I was not sure how does "numpy.lianlg.inv()" works. Say I have the following code:

import numpy as np

X = np.arange(10000).reshape(100,100)
X = X + X.T - np.diag(X.diagonal()) #  symmetry 
X = np.dot(X,X.T) # positive-definite

# simple inversion:
inverse1 = np.linalg.inv(X) 

# Cholesky decomposition inversion:
c = np.linalg.inv(np.linalg.cholesky(X))
inverse2 = np.dot(c.T,c)

Which one is more efficient (inverse1 or inverse2)? If the second one is more efficient, why is numpy.linalg.inv() not using this instead?

Upvotes: 6

Views: 9478

Answers (1)

Eric
Eric

Reputation: 97571

With the following setup:

import numpy as np

N = 100
X = np.linspace(0, 1, N*N).reshape(N, N)
X = 0.5*(X + X.T) + np.eye(N) * N

I get the following timings with IPython's %timeit:

In [28]: %timeit np.linalg.inv(X)
255 µs ± 30.9 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)

In [29]: %timeit c = np.linalg.inv(np.linalg.cholesky(X)); np.dot(c.T,c)
414 µs ± 15.4 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)

Upvotes: 2

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