Experimentally determining computing complexity of matrix determinant

Question

I need help determining experimentally the computing complexity of the determinant of a matrix nxn

My code:

    import numpy as np
    import timeit
    t0 = time.time()
    for n in range(1, 10):
        A = np.random.rand(n, n)
        det = np.linalg.slogdet(A)
        t = timeit.timeit(lambda: det)
        print(t)

But I get the same time for every n, hence, computing complexity: O(N) which is not correct as it is meant to be O(N^3). Any help would be much appreciated.

Crashland

Answer 1

For what it's worth, any meaningful benchmarking typically requires sufficiently large N to give the computer something to chew on. A 10x10 matrix is not nearly large enough to start seeing complexity. Start throwing numbers like 100, 1000, 10000, etc, then you'll see your scaling.

For example if I slightly modify your code

for n in range(1, 14):
    t0 = time.time()
    p = 2**n
    A = np.random.rand(p,p)
    det = np.linalg.slogdet(A)
    print('N={:04d} : {:.2e}s'.format(p, time.time() - t0))

This results in

N=0002 : 4.35e-02s
N=0004 : 0.00e+00s
N=0008 : 0.00e+00s
N=0016 : 5.02e-04s
N=0032 : 0.00e+00s
N=0064 : 5.02e-04s
N=0128 : 5.01e-04s
N=0256 : 1.50e-03s
N=0512 : 8.00e-03s
N=1024 : 3.95e-02s
N=2048 : 2.05e-01s
N=4096 : 1.01e+00s
N=8192 : 7.14e+00s

You can see that for very small values of N, some small-value optimizations and tricks make it hard to see O() complexity, but as the values of N grow, you can start to see the scaling.

Washing

any idea why N = 2 is so 'slow'? – Succedaneum

Answer 2

There are some possible reasons:

1. The computer used to generate these numbers was busy doing something else when the and "slow" operations like n=2 or n=16 operations were done
2. particularly for n=2, maybe some caching was done after the first loop, speeding up subsequent runnings.

You'd also typically expect n=1 to have the worst ratio of running time to n, simply because of constant overhead like variable initialization.

Wakerobin