Create matrix class for FLINT's nmod_mat module

[roed]

This tickets adds a new class for dense matrices over Zmod(N) implemented using FLINT's nmod_mat_t type, along with some supporting ancillary changes:

• Changed echelon form over Zmod(N) for composite N to return the Howell form (see the introduction and Chapter 4 of ​Storjohann's thesis). Since this echelon form can have more rows than the input matrix, made some supporting changes including a new method _echelon_copy.
• Changed rank over Zmod(N) to return the number of leading 1s in Howell form (it raised a NotImplementedError before), and pivots to be the locations of these leading 1s. There is also a method _pivots for accessing the columns where the leading entry is not 1.
• Matrices modulo composite N have improved speed and functionality for inversion, charpoly, det, minpoly, echelon_form, solve_right and right_kernel_matrix. There is also a new method minpoly_ideal for the ideal vanishing on a matrix (the natural generalization of the minimal polynomial, which generates this ideal when it is principal).
• Changed stack and augment to return a matrix over a common base ring of the two inputs, rather than just using the top/left matrix to determine the base ring.
• A new method _change_implementation on matrices, together with rough heuristics for matrices on matrices mod N in determining when it's worth switching between FLINT and linbox (linbox is faster for large matrices, FLINT for smaller, and FLINT offers extra functionality for matrices modulo composite integers).
• Support multiplication of matrices with different implementations
• Add FLINT's ulong_extras for number theoretic functions on longs
• Add some iterators related to split primes in cyclotomic fields in support of changes to matrix_cyclo_dense (which uses a multimodular approach), and moved the _reduction_matrix method to the base field which will yield better caching behavior.
• Fix some doctests that relate to choosing different solutions in solve_right and getting different random matrices with a different implementation.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​763533d

Fix bug in cinit

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​8cdf9b1

Add some functions to Matrix_nmod_dense

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​9c6e9d3	level 1 and level 2
​518a32f	Updating todos

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​936668a

Add todos

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​aaf0680

Update todo

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​fb02349

Reorder todos

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​31d05e4	inverses, swap rows and columns
​d69baff	richmp
​ba439fc	Merge branch 'u/roed/nmod_mat' of git://trac.sagemath.org/sage into nmod_mat
​9285ed3	Merge branch 'u/roed/nmod_mat' of git://trac.sagemath.org/sage into nmod_mat
​6a3d32a	justifying flags
​a90423a	Merge branch 'u/roed/nmod_mat' of git://trac.sagemath.org/sage into nmod_mat
​c7c68cd	some progress
​61821e8	fixing various things
​cecbe30	Merge branch 'u/edgarcosta/nmod_mat' of git://trac.sagemath.org/sage into t/31548/nmod_mat

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​eb56a5e	Add all definitions to FLINT's ulong_extras, optimize matrix inversion, change default class for matrix mod n
​db33b05	Charpoly for Z/n

[mkoeppe]

Sage development has entered the release candidate phase for 9.3. Setting a new milestone for this ticket based on a cursory review of ticket status, priority, and last modification date.

[git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

​33dd395	comment
​b922d8e	Merge branch 'u/roed/nmod_mat' of git://trac.sagemath.org/sage into nmod_mat
​c1961a2	right kernel matrix
​b1d5396	Merge branch 'u/roed/nmod_mat' of git://trac.sagemath.org/sage into nmod_mat
​b9965be	remove comment
​1045207	touch up right kernel
​fb619b0	work around echenolize
​45de1c8	remove stack
​bf8aaab	Merge branch 'u/edgarcosta/nmod_mat' of git://trac.sagemath.org/sage into t/31548/nmod_mat
​3eb1cd4	Fix import

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​8c71a1c	31069: flint version upgrade
​a32be02	Merge branch 'public/31069' of git://trac.sagemath.org/sage into t/31548/nmod_mat
​aed2361	Working on solve_right

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​8b37863	Working on minpoly
​1541005	int64
​8a4b815	import
​1f40b95	int64
​2d157e1	Merge branch 'u/edgarcosta/nmod_mat' of git://trac.sagemath.org/sage into t/31548/nmod_mat
​5e9b8a7	Minpoly

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​fc48eca

Working on documentation

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​64e44fc

Fixing bugs and adding documentation to matrices mod n

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​e5c4c1e

Working on documentation

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​dad23f5	Working on picking appropriate matrix types and fixing some bugs
​1a7bb38	Fixing some bugs

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​6f41da9

Fixing doctest problems from switching default implementation to flint for small matrices

[roed]

Marking for patchbot to test....

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​d0507c4	Working on fixing reduction matrix bug
​9748ffc	Add deprecation warning to _reduction_matrix

[tscrim]

If you aren't yet ready for comments, I apologize. However, I made a quick skim over things before I realized you set this as needs review for the patchbot.

For def _change_implementation, wouldn't it make more sense to be cdef or cpdef since it should be called entirely from internal stuff? Also, in that implementation, I don't think we should use self.list() for spare matrices. I also generally find using sage.matrix.matrix_space.MatrixSpace unstable and think it is better to have a direct import.

I know you said this is for the patchbot, but it would be good to have doctests for _solve_right_modn. I would also add

cdef list X
cdef Py_ssize_t n

change

-            X = self.matrix_space(B.ncols(), self.ncols())(X)
-            return X.T
+            return self.matrix_space(B.ncols(), self.ncols())(X).T

(perhaps casting the result as a matrix too), and mark B as a Matrix.

I am not sure about the name of the files being matrix_nmod_dense and the subsequently the name of the class. It conflicts with the matrix_modn_* files and I feel it is too generic given that it is for a specific implementation.

Why is poly_crt in the matrix file?

can't -> cannot as we want to make the writing formal.

It would be good to implement a get_is_zero_unsafe method.

[roed]

No problem! I'm still working on stuff, so comments are easily incorporated. I also realized that the patchbot won't test this since it relies on spkgs until #31069 is merged.

Do you think matrix_modn_flint and Matrix_modn_flint would be better?

The poly_crt function is left over from an earlier implementation of charpoly or minpoly (I forget which). I will take care of it.

Thanks!

[tscrim]

Replying to roed:

No problem! I'm still working on stuff, so comments are easily incorporated. I also realized that the patchbot won't test this since it relies on spkgs until #31069 is merged.

Yea, I didn't quite realize that either. ;;

Do you think matrix_modn_flint and Matrix_modn_flint would be better?

The former for the name of the file, the latter for the name of the class considering the other matrix classes in Sage.

Let me know if there is anything I can do to help too.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​cd1b853	Merge branch 'u/roed/nmod_mat' of trac.sagemath.org:sage into t/31548/nmod_mat
​6e2b75a	Merge branch 'u/roed/nmod_mat' of trac.sagemath.org:sage into t/31548/nmod_mat
​ee10b06	Working on documentation and fixing tests
​465f51d	Working on documentation

[roed]

Replying to tscrim:

For def _change_implementation, wouldn't it make more sense to be cdef or cpdef since it should be called entirely from internal stuff? Also, in that implementation, I don't think we should use self.list() for spare matrices. I also generally find using sage.matrix.matrix_space.MatrixSpace unstable and think it is better to have a direct import.

Even for a 2 by 2 matrix where the calling overhead is most substantial, changing to cpdef makes a negligible difference:

sage: A = random_matrix(Zmod(120), 2)                                                                                                               
sage: A                                                                                                                                             
[ 32 108]
[  2 112]
sage: %time B = A._change_implementation("linbox") # using def                                                                                                
CPU times: user 339 µs, sys: 8 µs, total: 347 µs
Wall time: 350 µs

sage: A = random_matrix(Zmod(120), 2) # using cpdef                                                                                                                                                                                                                       
sage: %time B = A._change_implementation("linbox")                                                                                                                                                                                                            
CPU times: user 334 µs, sys: 5 µs, total: 339 µs
Wall time: 343 µs

I don't think it's worth the cost of recompiling everything depending on matrix0.pxd, and the risk of developers accidentally using def instead of cpdef on a matrix class in the future.

I agree about sparse matrices, but currently in sage.matrix.matrix_space.get_matrix_class we don't allow the implementation keyword for sparse matrices, so the issue is moot. I'd be happy to have more sparse matrices in Sage eventually, but for now there's no way to even test a version of _change_implementation in Matrix_sparse.

I'm not sure what you mean by using sage.matrix.matrix_space.MatrixSpace unstable. Do you mean switching to from sage.matrix.matrix_space import MatrixSpace? I was just copying what's done a couple dozen lines above in matrix2.pyx, but I'm happy to make this change if desired.

I know you said this is for the patchbot, but it would be good to have doctests for _solve_right_modn.

Agreed. I've added them.

I would also add

cdef list X
cdef Py_ssize_t n

change

-            X = self.matrix_space(B.ncols(), self.ncols())(X)
-            return X.T
+            return self.matrix_space(B.ncols(), self.ncols())(X).T

(perhaps casting the result as a matrix too), and mark B as a Matrix.

I've added the cdef Py_ssize_t n since that can speed up the loop, and made the change you suggested in the diff. I don't think it's necessary to mark X as a list or B as a matrix, since I'm not calling any functions that will benefit from knowing the types (and the runtime is surely dominated by the call to Pari's matsolvemod)

I am not sure about the name of the files being matrix_nmod_dense and the subsequently the name of the class. It conflicts with the matrix_modn_* files and I feel it is too generic given that it is for a specific implementation.

Agreed. I've changed it to matrix_modn_dense_flint.

Why is poly_crt in the matrix file?

I've created #31731 and removed it from this ticket.

can't -> cannot as we want to make the writing formal.

It would be good to implement a get_is_zero_unsafe method.

Both done.

[roed]

All tests pass in matrix modules rings modular crypto. I'm happy to run tests on all of Sage if people are happy with the changes in principle.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​760e0c9

Small fixes

[roed]

(I accidentally made a comment instead of editing the ticket description)

---

New commits:

​760e0c9

Small fixes

[roed]

This ticket is now actually ready for review, though the patchbot won't run currently (since it depends on #31069 which changes spkgs).

[tscrim]

Needs rebase.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​a3c8e38

Merge branch 'u/roed/nmod_mat' of git://trac.sagemath.org/sage into t/31548/nmod_mat

[mkoeppe]

Setting a new milestone for this ticket based on a cursory review.