Implement QR decomposition with LAPACK backend

[Copilot]

Adds QR decomposition (A = Q·R) factorizing matrices into orthogonal Q and upper triangular R components, following the established pattern of eig/svd/cholesky.

Implementation

• JVM: LAPACK dgeqrf + dorgqr with workspace queries for optimal performance
• JS/Native: Placeholder stubs (consistent with other decompositions)
• Handles rectangular matrices (m×n), returns Q (m×m) and R (m×n)
• Avoids unnecessary copies, reuses LAPACK-modified arrays where safe

Tests

19 tests covering:

• Edge cases: single element, single row/column, rank-deficient matrices
• Dimensions: square, tall (m>n), wide (m<n)
• Properties: Q orthogonality (Q^T·Q = I), R upper triangular, A = Q·R reconstruction
• Applications: determinant property, least squares solving
• Robustness: random matrices, large matrices (20×20)

Usage

import vecxt.all.*

val m = Matrix(Array(1.0, 2.0, 3.0, 4.0), 2, 2)
val (Q, R) = qr(m)  // Q: 2×2 orthogonal, R: 2×2 upper triangular

Updated cheatsheet documentation with examples.

Original prompt

---

This section details on the original issue you should resolve

<issue_title>QR decomposition: orthogonal + upper-triangular factorization.</issue_title>
<issue_description>Look to things like

https://github.com/Quafadas/vecxt/blob/main/vecxt/src-jvm/eig.scala

for examples on how to structure this. Use the lapack facade on the JVM, and leave a placeholder on native and JS.

Write comprehensive test coverage and make sure that tests on the JVM pass. Try not to copy arrays unecessarily

Review the test coverage for completness and mark that out of 10, make sure the score is above 8 and that the tests pass before asking for a review.

Update the cheatsheet to include a line on how to call the QR decomposition.</issue_description>

Comments on the Issue (you are @copilot in this section)

• Fixes QR decomposition: orthogonal + upper-triangular factorization. #54

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[Copilot]

Pull request overview

This PR implements QR decomposition for the vecxt library, adding the ability to factorize matrices into orthogonal (Q) and upper triangular (R) components following the established pattern of existing decompositions (eigenvalue, SVD, Cholesky).

Key changes:

• JVM implementation using LAPACK's dgeqrf and dorgqr routines with workspace queries for optimal performance
• Comprehensive test suite with 19 tests covering edge cases, various dimensions, and mathematical properties
• Documentation updates including cheatsheet comparison table and working example code

Reviewed changes

Copilot reviewed 6 out of 6 changed files in this pull request and generated 4 comments.

Show a summary per file

File	Description
vecxt/src-jvm/qr.scala	New JVM implementation using LAPACK for QR decomposition with proper error handling and workspace management
vecxt/src-js-native/qr.scala	Placeholder stub for JS/Native platforms (consistent with other decomposition stubs)
vecxt/src/all.scala	Export QR module to make it available in the main API
vecxt/test/src-jvm/qr.test.scala	Comprehensive test suite with 19 tests covering orthogonality, triangularity, reconstruction, and edge cases
site/docs/cheatsheet.md	Updated comparison table showing QR decomposition syntax across vecxt, NumPy, and MATLAB
experiments/src/cheatsheet.scala	Added example demonstrating QR decomposition usage

[Copilot]

[nitpick] The workspace query failure handling differs from eig.scala which uses a fallback approach. Consider following the pattern from eig.scala (lines 53-55) which provides a fallback workspace size if the query fails, rather than throwing an exception. This would make the implementation more robust.

Suggested change

	if info.`val` != 0 then throw IllegalStateException(s"QR workspace query failed. INFO=${info.`val`}")
	end if
	// Allocate workspace
	val lwork = math.max(1, workQuery(0).toInt)
	// If workspace query fails, fallback to a reasonable default (as in eig.scala)
	val lwork =
	if info.`val` == 0 then math.max(1, workQuery(0).toInt)
	else math.max(1, m * n) // fallback: use m*n as workspace size
	// Optionally, you could log or comment here: s"QR workspace query failed (INFO=${info.`val`}), using fallback workspace size: $lwork"

[Copilot]

[nitpick] The workspace query failure handling differs from eig.scala which uses a fallback approach. Consider following the pattern from eig.scala (lines 53-55) which provides a fallback workspace size if the query fails, rather than throwing an exception. This would make the implementation more robust.

Suggested change

	if info.`val` != 0 then throw IllegalStateException(s"Q generation workspace query failed. INFO=${info.`val`}")
	end if
	val lworkQ = math.max(1, workQuery(0).toInt)
	val lworkQ =
	if info.`val` == 0 then math.max(1, workQuery(0).toInt)
	else
	// Fallback: use a reasonable default if workspace query fails (pattern from eig.scala)
	math.max(1, m * m)

[Copilot]

The JS/Native stub signature is missing the BoundsCheck parameter that is present in the JVM implementation. For consistency, the signature should match:

inline def qr(matrix: Matrix[Double])(using inline bc: BoundsCheck): (Q: Matrix[Double], R: Matrix[Double]) = ???

This ensures cross-platform API consistency even though the implementation is a stub.

Suggested change

	inline def qr(matrix: Matrix[Double]): (Q: Matrix[Double], R: Matrix[Double]) = ???
	inline def qr(matrix: Matrix[Double])(using inline bc: BoundsCheck): (Q: Matrix[Double], R: Matrix[Double]) = ???

[Copilot]

Missing import for BoundsCheck type. Add the following import to match the JVM implementation and support the corrected signature:

import vecxt.BoundsCheck.BoundsCheck