Utility Functions for Matrices¶
The actual computation of homology groups ends up being linear algebra with the differentials thought of as matrices. This module contains some utility functions for this purpose.
Preprocess a matrix using the “Elimination algorithm” described by Dumas et al. [DHSW2003], and then call
elementary_divisorson the resulting (smaller) matrix.
‘snf’ stands for ‘Smith Normal Form’.
mat– an integer matrix, either sparse or dense.
(They use the transpose of the matrix considered here, so they use rows instead of columns.)
matone column at a time. For each column, add multiples of previous columns to it until either
it’s zero, in which case it should be deleted.
its first nonzero entry is 1 or -1, in which case it should be kept.
its first nonzero entry is something else, in which case it is deferred until the second pass.
Then do a second pass on the deferred columns.
At this point, the columns with 1 or -1 in the first entry contribute to the rank of the matrix, and these can be counted and then deleted (after using the 1 or -1 entry to clear out its row). Suppose that there were \(N\) of these.
The resulting matrix should be much smaller; we then feed it to Sage’s
elementary_divisorsfunction, and prepend \(N\) 1’s to account for the rows deleted in the previous step.
sage: from sage.homology.matrix_utils import dhsw_snf sage: mat = matrix(ZZ, 3, 4, range(12)) sage: dhsw_snf(mat) [1, 4, 0] sage: mat = random_matrix(ZZ, 20, 20, x=-1, y=2) sage: mat.elementary_divisors() == dhsw_snf(mat) True