On Wed, Jan 25, 2012 at 10:15 AM, javier <vengor...@gmail.com> wrote: > Hi all! > > I noticed that doing > A.is_singular() > is the same thing as > A.determinant() == 0 > > I am not sure this is optimal, as we can know that A.det() != 0 > without having to compute it. > I was wondering if it would be quicker to make A.is_singular() > equivalent to > > if A.nrows() == A.ncols(): > return A.rank() == A.ncols() > else: > return False
I think is_singular should raise a ValueError if the input matrix is not square. To quote wikipedia: "A square matrix that is not invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is 0. Singular matrices are rare in the sense that if you pick a random square matrix, it will almost surely not be singular." > > The rationale is that we only need to know if the determinant is > different from 0, not its precise value. > I have done some tests and the timings suggest that computing the rank > is a bit faster than computing the determinant: > > L = range(3000^2) > > A = Matrix(ZZ, 3000, L) > time A.det() > 0 > Time: CPU 4.16 s, Wall: 4.07 s > > A = Matrix(ZZ, 3000, L) > time A.rank() > 2 > Time: CPU 4.00 s, Wall: 3.92 s That is probably a good idea over ZZ or QQ and probably many other rings, especially since for a random matrix over ZZ or QQ, rank will be full and Sage will quickly figure this out by reducing modulo one small random prime. So, even if the coefficients of A are enormous, the rank approach will usually be very fast, whereas computing the determinant will be slowed down by the size of the entries. > Small improvement, but an improvement. Also, I observed that some > matrix functions use hand-made caches. Should we change those to use > @cahced_method instead, or is this done by some design reason? There was no cached_method when I wrote most of the matrix code. Also, I think decorators weren't supported in Cython then (?). The matrix caching code is definitely more sophisticated than what cached_method does though. It has to behave differently for mutable and immutable matrices. > > Cheers > Javier > > -- > To post to this group, send an email to sage-devel@googlegroups.com > To unsubscribe from this group, send an email to > sage-devel+unsubscr...@googlegroups.com > For more options, visit this group at > http://groups.google.com/group/sage-devel > URL: http://www.sagemath.org -- William Stein Professor of Mathematics University of Washington http://wstein.org -- To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
