Howell form

[joschmitt]

I copied over the generic Howell form from Hecke.
This requires the operations 'ann', 'gcdex' and 'quo' as Storjohann calls them; I implemented these under the names annihilator, gcdex and _div for ResElem.jl. Better names are welcome.
I also added some lines that normalize the pivot elements with the canonical_unit to make the Howell form unique; is that correct?

[codecov]

Codecov Report

Attention: Patch coverage is 98.02632% with 3 lines in your changes missing coverage. Please review.

Project coverage is 87.39%. Comparing base (762eab1) to head (d39d3fc).

Files	Patch %	Lines
src/MatrixNormalForms.jl	98.57%	2 Missing ⚠️
src/Residue.jl	91.66%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1735      +/-   ##
==========================================
+ Coverage   87.33%   87.39%   +0.06%     
==========================================
  Files         117      117              
  Lines       29777    29929     +152     
==========================================
+ Hits        26006    26157     +151     
- Misses       3771     3772       +1     

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[fieker]

elsewhere this is done using the euclidean algo

[fieker]

On Tue, Jun 25, 2024 at 05:42:44AM -0700, Johannes Schmitt wrote: I copied over the generic Howell form from Hecke. This requires the operations 'ann', 'gcdex' and 'quo' as Storjohann calls them; I implemented these under the names `annihilator`, `gcdex` and `_div` for `ResElem.jl`. Better names are welcome.

gcdex = xxgcd in Hecke? Giving the 2x2 matrix of cofactors? bad name - but then, Hecke's is also bad.... why _div? Is this an exact or an euc division? In which case, div should do? or divrem?

I also added some lines that normalize the pivot elements with the `canonical_unit` to make the Howell form unique; is that correct?

Yep

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You can view, comment on, or merge this pull request online at: #1735 -- Commit Summary -- * Copy Howell form from Hecke * Normalize the pivots to make it unique * Tests -- File Changes -- M src/MatrixNormalForms.jl (252) M src/Residue.jl (23) M src/exports.jl (2) M test/MatrixNormalForms-test.jl (34) -- Patch Links -- https://github.com/Nemocas/AbstractAlgebra.jl/pull/1735.patch https://github.com/Nemocas/AbstractAlgebra.jl/pull/1735.diff -- Reply to this email directly or view it on GitHub: #1735 You are receiving this because you are subscribed to this thread. Message ID: ***@***.***>

[joschmitt]

gcdex = xxgcd in Hecke? Giving the 2x2 matrix of cofactors? bad name - but then, Hecke's is also bad.... why _div? Is this an exact or an euc division? In which case, div should do? or divrem?

Yes, gcdex is what is called xxgcd in Hecke and gives the 2x2 matrix.

_div does not do an exact division. And I'm not sure I would call it "euclidean" in general? In the PhD thesis, Storjohann defines it for a principal ideal ring $R$ as

$\mathrm{Quo}(a, b)$: return a $q\in R$ such $a − qb \in \mathcal{R}(R, b)$

where $\mathcal{R}(R, b)$ is a "prescribed complete set of residues with respect to $b$".

[fieker]

On Tue, Jun 25, 2024 at 06:14:43AM -0700, Johannes Schmitt wrote: > gcdex = xxgcd in Hecke? Giving the 2x2 matrix of cofactors? bad name - but then, Hecke's is also bad.... why _div? Is this an exact or an euc division? In which case, div should do? or divrem? Yes, `gcdex` is what is called `xxgcd` in Hecke and gives the 2x2 matrix. `_div` does not do an exact division. And I'm not sure I would call it "euclidean" in general? In the PhD thesis, Storjohann defines it for a principal ideal ring $R$ as > $\mathrm{Quo}(a, b)$: return a $q\in R$ such $a − qb \in \mathcal{R}(R, b)$ where $\mathcal{R}(R, b)$ is a "prescribed complete set of residues with respect to $b$".

Do we have any non-euc ring where this is used/ useful? Z/nZ is euc as is O_K/I. However, already in O_K/I I don't think we have a unique residue system

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