#12103: Use MeatAxe as an optional back end for dense matrices over `GF(p^n)`, p
odd, n>1, `p^n<255`
------------------------------+---------------------------------------------
Reporter: SimonKing | Owner: jason, was
Type: defect | Status: new
Priority: major | Milestone: sage-4.8
Component: linear algebra | Keywords: linear algebra, MeatAxe
Work_issues: | Upstream: None of the above - read trac
for reasoning.
Reviewer: | Author: Simon King
Merged: | Dependencies: #9562 #4260
------------------------------+---------------------------------------------
Sage has (or will soon have) fairly good implementations of dense matrices
over `GF(2)`, over `GF(2^e)` (#9562) and over `GF(p)` (p prime, #4260).
However, it uses generic code for dense matrices over `GF(p^n)`, p odd,
n>1, `p^n<255`.
I suggest to use a major modification of `MeatAxe Release 2.2.4` instead
of the basic implementation. More on the modifications and the reason for
choosing an old release is explained in the comments.
This is awfully slow:
{{{
sage: MS = MatrixSpace(GF(5^3,'y'),2000)
sage: %time A = MS.random_element()
CPU times: user 6.36 s, sys: 0.02 s, total: 6.39 s
Wall time: 6.41 s
sage: type(A)
<type 'sage.matrix.matrix_generic_dense.Matrix_generic_dense'>
sage: B = MS.random_element()
sage: %time A*B # using 6.3% of my computer's memory
CPU times: user 744.20 s, sys: 1.18 s, total: 745.38 s
Wall time: 747.69 s
2000 x 2000 dense matrix over Finite Field in y of size 5^3
sage: %time ~A # using 10.4% of my computer's memory
CPU times: user 1096.74 s, sys: 1.30 s, total: 1098.05 s
Wall time: 1101.24 s
2000 x 2000 dense matrix over Finite Field in y of size 5^3
sage: %time A.echelon_form() # using 10.4% of my computer's memory
CPU times: user 378.62 s, sys: 0.33 s, total: 378.95 s
Wall time: 380.06 s
2000 x 2000 dense matrix over Finite Field in y of size 5^3
}}}
With the optional spkg and the patch, one gets a clear improvement.
{{{
sage: MS = MatrixSpace(GF(5^3,'y'),2000)
sage: %time A = MS.random_element()
CPU times: user 0.32 s, sys: 0.00 s, total: 0.32 s
Wall time: 0.33 s
sage: type(A)
<type 'sage.matrix.matrix_modpn_dense.Matrix_modpn_dense'>
sage: B = MS.random_element()
# The following uses Strassen-Winograd multiplication
sage: %time A*B # using 3.5% of my computer's memory
CPU times: user 7.68 s, sys: 0.01 s, total: 7.69 s
Wall time: 7.72 s
2000 x 2000 dense matrix over Finite Field in y of size 5^3
# The following is school book multiplication;
# that's more or less the original meataxe speed:
sage: %time A._multiply_classical(B) # using 3.6% of my computer's
memory
CPU times: user 11.68 s, sys: 0.02 s, total: 11.70 s
Wall time: 11.73 s
2000 x 2000 dense matrix over Finite Field in y of size 5^3
# Strassen is not implemented for inversion and echelon form.
sage: %time ~A # using 3.8% of my computer's memory
CPU times: user 23.55 s, sys: 0.00 s, total: 23.55 s
Wall time: 23.62 s
2000 x 2000 dense matrix over Finite Field in y of size 5^3
sage: %time A.echelon_form() #using 3.9% of my computer's memory
CPU times: user 11.73 s, sys: 0.01 s, total: 11.74 s
Wall time: 11.78 s
2000 x 2000 dense matrix over Finite Field in y of size 5^3
}}}
I think the component is "linear algebra", even though it is about an
optional package.
'''__How to install stuff__'''
* Apply the dependencies
* Install
[http://sage.math.washington.edu/home/SimonKing/LibMeatAxe/libmeataxe-1.0.spkg
the optional libmeataxe spkg]
* Apply either [attachment:] or [attachment:], depending on whether or
not you have #11900 applied.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/12103>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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