extend Cholesky decomposition method for finite fields #39200

jacksonwalters · 2024-12-24T21:23:41Z

Given a Hermitian matrix $U$ over $GF(q^2)$, returns a matrix $A$ such that $U = AA^\ast$, where $^\ast$ denotes conjugate-transpose.

Note that the Cholesky decomposition is meant for characteristic zero, and also only returns a lower/upper triangular matrix.

Corresponding: Issue #39202
Discussion: Issue #38456
Required by: PR #38455

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The title is concise and informative.
The description explains in detail what this PR is about.
I have linked a relevant issue or discussion.
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github-actions · 2024-12-26T19:36:00Z

Documentation preview for this PR (built with commit 704ae31; changes) is ready! 🎉
This preview will update shortly after each push to this PR.

src/sage/matrix/matrix2.pyx

tscrim · 2024-12-30T00:38:25Z

I really dislike these apply-suggestion-commits with a completely useless commit message. Could you please squash and rebase? It makes it easier to see which changes have been done and looking through the logs much later.

add hermitian_decomposition method given a Hermitian matrix U, returns a matrix A such that U = AA* use translated GAP source code for BaseChangeCanonical Co-Authored-By: Travis Scrimshaw <clfrngrown@aol.com>

jacksonwalters · 2024-12-30T00:56:13Z

Sure. I agree, the commit messages are pretty useless. I've squashed the commits here. Which branch should I rebase on? I rebased PR#38455 on this one (PR#39200) already.

tscrim · 2024-12-30T01:03:43Z

Thanks. Sorry, I misspoke slightly, I meant force-push.You will need to rebase #38455 again though…

jacksonwalters · 2024-12-30T01:22:05Z

@tscrim Okay, I have rebased #38455 on this again and attempted to resolve merge conflicts.

jacksonwalters · 2024-12-30T02:32:54Z

I don't think this handles the 1x1 case, so I'll add a fix.

Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

…alters/sage into hermitian_decomposition

jacksonwalters · 2025-01-02T20:00:03Z

@tscrim How do I squash these commits given there's a merge commit? We may've done this before, but I don't see how.

tscrim · 2025-01-04T06:47:14Z

I would just rebase the entire branch over the latest develop branch and squash commits as reasonable.

tscrim

Just a few more details, modulo making row 0-based.

src/sage/matrix/matrix2.pyx

tscrim · 2025-01-04T06:49:19Z

src/sage/matrix/matrix2.pyx

+            row += 1
+
+            # Look for a non-zero element on the main diagonal, starting from `row`
+            i = row - 1  # Adjust for zero-based indexing in Sage


I don't think this is possible as we mutate i compared to row.

change \ast to * for consistency Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

add word "package" to GAP ``forms`` Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

jacksonwalters · 2025-01-15T16:02:00Z

@tscrim I've refactored it so that row is initialized to -1, so that many row - 1 statements are just row. I agree, I don't think we can decouple it from i in any way.

tscrim

As I look into the Cholesky decomposition and its doc, I am thinking we should actually call this method cholesky_extended() for "extended Cholesky decomposition". Or possibly accessing this via cholesky(extended=True) (with the default being False), in which case this method would be called something like _cholesky_extended_ff. I am actually leaning towards the latter right now since it is the most natural. Sorry for a somewhat big change so far along.

This paper is also somewhat relevant: journal version; [arXiv version](https://arxiv.org/abs/2202.04012v1.

src/sage/matrix/matrix2.pyx

yes, just forgot to update this Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

add two examples of a Hermitian decomposition, one of which is not upper/lower triangular (so would be impossible with Cholesky decomposition)

replace always True with explicit conditional Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

remove this break conditional in favor of conditional for while loop Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

add a SEEALSO since this is an extended form of the Cholesky decomposition Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

jacksonwalters · 2025-01-17T19:25:38Z

I'm happy to have this accessed via cholesky(extended=True). That's really how we came upon it, trying to do the Cholesky decomposition over finite fields but realizing it didn't work for some matrices. It also makes it more likely other people will see it and use it. I'll attempt to move it.

this method for decomposition a Hermitian matrix U = AA* is similar in spirit to the Cholesky decomposition, but extends it to work over finite fields of square order.

…alters/sage into hermitian_decomposition

jacksonwalters · 2025-01-17T19:50:20Z

Okay, I have moved the code into cholesky(). It may require some changes. I believe Dima and I did look at that paper when trying to figure this out, but it didn't have exactly this method. Dima found it in GAP.

github-actions bot added the s: needs review label Dec 24, 2024

jacksonwalters mentioned this pull request Dec 24, 2024

unitary DFT for symmetric group algebra #38455

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jacksonwalters reopened this Dec 27, 2024

github-actions bot removed the s: needs review label Dec 27, 2024

jacksonwalters force-pushed the hermitian_decomposition branch from caed579 to f2f35b7 Compare December 27, 2024 14:38

github-actions bot added the s: needs review label Dec 27, 2024

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Update src/sage/matrix/matrix2.pyx

a8cd89e

add hermitian_decomposition method given a Hermitian matrix U, returns a matrix A such that U = AA* use translated GAP source code for BaseChangeCanonical Co-Authored-By: Travis Scrimshaw <clfrngrown@aol.com>

jacksonwalters force-pushed the hermitian_decomposition branch from 6300bae to a8cd89e Compare December 30, 2024 00:54

jacksonwalters and others added 10 commits December 29, 2024 19:49

handle 1x1 case

77ec178

Update src/sage/matrix/matrix2.pyx

1a72eed

Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

Update src/sage/matrix/matrix2.pyx

b1e2e36

Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

Update src/sage/matrix/matrix2.pyx

4ca915e

Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

Update src/sage/matrix/matrix2.pyx

a1d5df9

Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

Update src/sage/matrix/matrix2.pyx

93441a4

Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

use Matrix

8aea345

Merge branch 'hermitian_decomposition' of https://github.com/jacksonw…

3487c68

…alters/sage into hermitian_decomposition

remove blank line

53c1eb3

Trigger CI tests

7969772

jacksonwalters added 2 commits January 2, 2025 15:53

use self.__class__ within matrix.pyx file

6a3b1e3

Trigger CI tests

ee20021

tscrim requested changes Jan 4, 2025

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jacksonwalters and others added 3 commits January 15, 2025 10:27

Update src/sage/matrix/matrix2.pyx

ab9c846

change \ast to * for consistency Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

Update src/sage/matrix/matrix2.pyx

a55f4c0

add word "package" to GAP ``forms`` Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

initialize row to -1

d6b2fd1

tscrim reviewed Jan 17, 2025

View reviewed changes

jacksonwalters and others added 5 commits January 17, 2025 13:58

Update src/sage/matrix/matrix2.pyx

7e10900

yes, just forgot to update this Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

add two examples

ab0bbe5

add two examples of a Hermitian decomposition, one of which is not upper/lower triangular (so would be impossible with Cholesky decomposition)

Update src/sage/matrix/matrix2.pyx

af343af

replace always True with explicit conditional Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

Update src/sage/matrix/matrix2.pyx

d6f918d

remove this break conditional in favor of conditional for while loop Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

Update src/sage/matrix/matrix2.pyx

db1b181

add a SEEALSO since this is an extended form of the Cholesky decomposition Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

jacksonwalters added 4 commits January 17, 2025 14:44

move hermitian_decomposition into cholesky

70a6b5c

this method for decomposition a Hermitian matrix U = AA* is similar in spirit to the Cholesky decomposition, but extends it to work over finite fields of square order.

Merge branch 'hermitian_decomposition' of https://github.com/jacksonw…

28b2ade

…alters/sage into hermitian_decomposition

remove hermitian_decomposition

6ef7df3

default should be extended=False

553ddf9

3 trailing whitespace

704ae31

jacksonwalters changed the title ~~add hermitian_decomposition method~~ extend Cholesky decomposition method for finite fields of square order Jan 18, 2025

jacksonwalters changed the title ~~extend Cholesky decomposition method for finite fields of square order~~ extend Cholesky decomposition method for finite fields Jan 18, 2025

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extend Cholesky decomposition method for finite fields #39200

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extend Cholesky decomposition method for finite fields #39200

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