Fast way to convert upper triangular matrix into symmetric matrix

Question

I have an upper-triangular matrix of np.float64 values, like this:

array([[ 1.,  2.,  3.,  4.],
       [ 0.,  5.,  6.,  7.],
       [ 0.,  0.,  8.,  9.],
       [ 0.,  0.,  0., 10.]])

I would like to convert this into the corresponding symmetric matrix, like this:

array([[ 1.,  2.,  3.,  4.],
       [ 2.,  5.,  6.,  7.],
       [ 3.,  6.,  8.,  9.],
       [ 4.,  7.,  9., 10.]])

The conversion can be done in place, or as a new matrix. I would like it to be as fast as possible. How can I do this quickly?

Answer 1

This is the fastest routine I've found so far that doesn't use Cython or a JIT like Numba. I takes about 1.6 μs on my machine to process a 4x4 array (average time over a list of 100K 4x4 arrays):

inds_cache = {}

def upper_triangular_to_symmetric(ut):
    n = ut.shape[0]
    try:
        inds = inds_cache[n]
    except KeyError:
        inds = np.tri(n, k=-1, dtype=np.bool)
        inds_cache[n] = inds
    ut[inds] = ut.T[inds]

Here are some other things I've tried that are not as fast:

The above code, but without the cache. Takes about 8.3 μs per 4x4 array:

def upper_triangular_to_symmetric(ut):
    n = ut.shape[0]
    inds = np.tri(n, k=-1, dtype=np.bool)
    ut[inds] = ut.T[inds]

A plain Python nested loop. Takes about 2.5 μs per 4x4 array:

def upper_triangular_to_symmetric(ut):
    n = ut.shape[0]
    for r in range(1, n):
        for c in range(r):
            ut[r, c] = ut[c, r]

Floating point addition using np.triu. Takes about 11.9 μs per 4x4 array:

def upper_triangular_to_symmetric(ut):
    ut += np.triu(ut, k=1).T

Numba version of Python nested loop. This was the fastest thing I found (about 0.4 μs per 4x4 array), and was what I ended up using in production, at least until I started running into issues with Numba and had to revert back to a pure Python version:

import numba

@numba.njit()
def upper_triangular_to_symmetric(ut):
    n = ut.shape[0]
    for r in range(1, n):
        for c in range(r):
            ut[r, c] = ut[c, r]

Cython version of Python nested loop. I'm new to Cython so this may not be fully optimized. Since Cython adds operational overhead, I'm interested in hearing both Cython and pure-Numpy answers. Takes about 0.6 μs per 4x4 array:

cimport numpy as np
cimport cython

@cython.boundscheck(False)
@cython.wraparound(False)
def upper_triangular_to_symmetric(np.ndarray[np.float64_t, ndim=2] ut):
    cdef int n, r, c
    n = ut.shape[0]
    for r in range(1, n):
        for c in range(r):
            ut[r, c] = ut[c, r]