On Wed, Feb 18, 2009 at 1:21 AM, William Stein <wst...@gmail.com> wrote:
>
> On Tue, Feb 17, 2009 at 4:13 PM, Johan Oudinet <johan.oudi...@gmail.com>
> wrote:
>>
>> On Tue, Feb 17, 2009 at 7:56 PM, Jason Grout
>> <jason-s...@creativetrax.com> wrote:
>>>
>>> mabshoff wrote:
>>>>
>>>>
>>>> On Feb 17, 10:36 am, Johan Oudinet <johan.oudi...@gmail.com> 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.
>
> You do this:
>
> A = Ap * A
>
> which is matrix multiplication over QQ. That quickly leads to dense
> matrices with huge entries, which could simply use a lot of RAM.
Yes, but for this matrix, the memory problem only occurs later, when I
have a dxr matrix (T0) and try to get a square invertible matrix (T).
And in the loop, there is only addition of sparse vectors, no matrix
multiplication.
>
>> 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
--~--~---------~--~----~------------~-------~--~----~
To post to this group, send email to sage-support@googlegroups.com
To unsubscribe from this group, send email to
sage-support-unsubscr...@googlegroups.com
For more options, visit this group at
http://groups.google.com/group/sage-support
URLs: http://www.sagemath.org
-~----------~----~----~----~------~----~------~--~---
