On Tue, Feb 17, 2009 at 7:56 PM, Jason Grout
<[email protected]> wrote:
>
> mabshoff wrote:
>>
>>
>> On Feb 17, 10:36 am, Johan Oudinet <[email protected]> wrote:
>>> Hi Michael,
>>
>> Hi Johan,
>>
>> <SNIP>
>>
>>> I've just downloaded the Linux binaries from Sage website (the
>>> ubuntu-64bit-intel-xeon version)
>>>
>>> Sage Version 3.2.3, Release Date: 2009-01-05
>>>
>>
>> Ok.
>>
>>>> The matrix is sparse and pre Sage 3.3 rank was computed using pari
>>> How can I get Sage 3.3? From the Sage website, I can only find the
>>> 3.2.3 version.
>>
>> Sage 3.3 isn't released yet, but there is another release candidate
>> tonight, i.e. 3.3.rc2. Sage 3.3 it self should be out in the next 2, 3
>> days, but it has been slower than we had planned.
>>
>>>> which tends to blow up since it uses *a lot* of memory. We now switch
>>>> back to computing the rank of such a matrix by using the much faster
>>>> dense representation, but John Palmieri as the author of that code
>>>> should fill you in on the details there.
>>> Actually, I plan to using this code for larger matrices (up to 10^4),
>>> so I don't think I could use a dense representation, do you?
>>
>> Well, you do multiplies AFAIK and that should destroy the sparse
>> structure rather quickly assuming your matrix doesn't have a special
>> structure. So, any ideas if the sparse structure is preserved?
>>
>>>
>>>>> I'd appreciate any help.
>>>> I am running the computation right now on a box with 128 GB, so we
>>>> will see how far I get :) Right now we are in the 30th iteration of
>>> Wow! 128GB, it's very nice for doing such computations!
>>
>> Well, I didn't pay for it :)
>>
>>> How can you know the number of iterations in a Sage script? Is there a
>>> debug mode, or something like that?
>>
>> I just added a print statement in the loops. If this all ended I
>> didn't really want to see "True" or "False" at the end :)
>>
>> After about 60 minutes and maybe 80 iterations in rt I killed it
>> consuming about 40GB of RAM. So something is going wrong. Two
>> thoughts:
>>
>> * you are hitting a yet diagnosed memory leak - I will check that
>> * since your matrix coefficients are rational they just explode - not
>> much we can do about that. Using a ring with finite precision, i.e.
>> RealField() might avoid that.
>> * your sparse structure gets destroyed and you end up with dense
>> matrices anyway, ergo bye bye free RAM
>>
>> Obviously it can be all three :)
Since I'm just adding sparse vectors, I don't think the sparse
structure is destroyed here.
But, as the computation works with a ring with finite precision
(GF(997)), the problem should be on the coefficient explosion (when
computing the rank) and/or a memory leak in the rank function applied
to a sparse matrix over Rational field.
>>
>>>> while rt != d:
>>>> while rt == rank(T.augment(matrix(d,1,{(i,0):1}))):
>>>> i+=1
>>>> T=T.augment(matrix(d,1,{(i,0):1}))
>>>> rt+=1
>>>> and we are already consuming about 2.5GB RAM. There are some known
>>>> problem with LA in Sage that are leaky, but I suspect those are
>>>> reference count issues in Cython. Cython 0.11 out soon should help
>>>> there with the new reference count nanny.
>>> Well, my goal is to find a matrix B and a number n such that for every
>>> number k>n, B*A^(k+1) == A^k with respect to A is a dxd sparse matrix
>>> over Rational field (actually A is an adjacency matrix of a finite
>>> directed graph).
>>> The algorithm I implemented works (at least for small matrix and where
>>> rank(A^k) > 0), but it's not really optimized (I'm far from being an
>>> expert in linear algebra)!
>>> I'm interesting in a faster algorithm for both numerical and exact
>>> solutions. So, if someone knows a better solution (for example, a
>>> classical algorithm in linear algebra that solves this problem), I'll
>>> be glad to have some references ;-)
>>
>> Ok, I am not the guy who knows the literature well, so someone else
>> needs to answer this. But there are plenty of people from graph theory
>> around here. :)
>
>
> If you're willing, could you rephrase the problem in terms of directed
> graphs, if that is a natural way to look at it?
Well, in terms of directed graphs, I have a graph where its vertices
and edges are related to a recurrence relation as follows:
V(0) = 1 (or could be 0 in a generalized version of my problem)
and for each n > 0:
V(n+1) = \sum_{over the successors, V^\prime, of V} V^\prime(n)
And I want to build a graph that its edges are labeled by rational
numbers and that represents the inverse of the previous recurrence
relation:
V(N) = CN (a finite number)
V(0) = 1, V(1) = C1, ..., V(k) = Ck
and for each n st k < n < N:
V(n) = \sum_{V - e -> V^\prime} e * V^\prime(n+1)
But I have no idea how to build this graph directly from the first
graph, that's why I prefer formulating the problem as a linear algebra
problem.
--
Johan
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