Linear index upper triangular matrix

Question

If I have the upper triangular portion of a matrix, offset above the diagonal, stored as a linear array, how can the (i,j) indices of a matrix element be extrac

Answer 1

The equations going from linear index to (i,j) index are

i = n - 2 - floor(sqrt(-8*k + 4*n*(n-1)-7)/2.0 - 0.5)
j = k + i + 1 - n*(n-1)/2 + (n-i)*((n-i)-1)/2

The inverse operation, from (i,j) index to linear index is

k = (n*(n-1)/2) - (n-i)*((n-i)-1)/2 + j - i - 1

Verify in Python with:

from numpy import triu_indices, sqrt
n = 10
for k in range(n*(n-1)/2):
    i = n - 2 - int(sqrt(-8*k + 4*n*(n-1)-7)/2.0 - 0.5)
    j = k + i + 1 - n*(n-1)/2 + (n-i)*((n-i)-1)/2
    assert np.triu_indices(n, k=1)[0][k] == i
    assert np.triu_indices(n, k=1)[1][k] == j

for i in range(n):
    for j in range(i+1, n):
        k = (n*(n-1)/2) - (n-i)*((n-i)-1)/2 + j - i - 1
        assert triu_indices(n, k=1)[0][k] == i
        assert triu_indices(n, k=1)[1][k] == j