How to compute scipy sparse matrix determinant without turning it to dense?

Question

I am trying to figure out the fastest method to find the determinant of sparse symmetric and real matrices in python. using scipy sparse module but really surprised

Answer 1

You can use scipy.sparse.linalg.splu to obtain sparse matrices for the lower (L) and upper (U) triangular matrices of an M=LU decomposition:

from scipy.sparse.linalg import splu

lu = splu(M)

The determinant det(M) can be then represented as:

det(M) = det(LU) = det(L)det(U)

The determinant of triangular matrices is just the product of the diagonal terms:

diagL = lu.L.diagonal()
diagU = lu.U.diagonal()
d = diagL.prod()*diagU.prod()

However, for large matrices underflow or overflow commonly occurs, which can be avoided by working with the logarithms.

diagL = diagL.astype(np.complex128)
diagU = diagU.astype(np.complex128)
logdet = np.log(diagL).sum() + np.log(diagU).sum()

Note that I invoke complex arithmetic to account for negative numbers that might appear in the diagonals. Now, from logdet you can recover the determinant:

det = np.exp(logdet) # usually underflows/overflows for large matrices

whereas the sign of the determinant can be calculated directly from diagL and diagU (important for example when implementing Crisfield's arc-length method):

sign = swap_sign*np.sign(diagL).prod()*np.sign(diagU).prod()

where swap_sign is a term to consider the number of permutations in the LU decomposition. Thanks to @Luiz Felippe Rodrigues, it can be calculated:

swap_sign = minimumSwaps(lu.perm_r)

def minimumSwaps(arr): 
    """
    Minimum number of swaps needed to order a
    permutation array
    """
    # from https://www.thepoorcoder.com/hackerrank-minimum-swaps-2-solution/
    a = dict(enumerate(arr))
    b = {v:k for k,v in a.items()}
    count = 0
    for i in a:
        x = a[i]
        if x!=i:
            y = b[i]
            a[y] = x
            b[x] = y
            count+=1
    return count