fix(#182): linear-algebra API was unconstructible and SVD returned wrong results

[mivertowski]

Round 8 of #182. The reporter fixed their N-body data race, moved on to SVD, and found the linear-algebra API unusable. Every observation was a real defect — including one that silently returned wrong math.

1. GPULinearAlgebraProvider was impossible to construct

Its constructor required an IKernelManager — an interface with no implementation anywhere in the repo — so the README's headline example could never compile. Worse, AddDotComputeAlgorithms() registered GpuMatrixOperations, which had the same required dependency, so that registration could not be resolved either.

IKernelManager is now optional on both (it is only needed for custom GPU kernel execution). The two GPU kernel call sites raise a descriptive error, which the provider already catches to fall back to CPU. Both types are registered via factories that resolve IKernelManager optionally.

2. SVD returned mathematically wrong results 🔴

GpuMatrixOperations.SVDAsync and GpuOptimizationStrategies.FallbackSVDAsync each carried their own "simplified eigendecomposition of A^T·A" — its own comment admitted "not numerically stable". On a 6×6 matrix:

	singular values	‖U·S·Vᵀ − A‖
before	0.6724, 0.0517, 0.4761 (unsorted, wrong)	2.8
after	4.6418, 3.2217, 2.8212	1.7e-6

Both now delegate to the existing, correct one-sided Jacobi SVD (MatrixDecomposition.ComputeJacobiSVD), so every entry point shares one implementation. MatrixMath.SVDAsync was already correct — the two now agree.

3. The README documented APIs that do not exist

• IAcceleratorService — no such type anywhere.
• LinearAlgebraKernels.GetKernelSource — the type is LinearAlgebraKernelLibrary.
• AdvancedLinearAlgebraKernels — no such file or type.
• GetOptimizedParameters — internal, not usable.

Usage sections rewritten against the current build, and they now answer the reporter's actual question: how to get an IAccelerator (from DI — a kernel name is only needed for IComputeOrchestrator, which runs user [Kernel] methods).

Verification

Everything documented was executed first: provider SVD reconstruction 1.7e-6, Multiply 1.2e-7, QR 3.0e-8, Solve 5.6e-8, and provider ≡ MatrixMath. New regression tests cover constructibility, DI resolution, SVD reconstruction, descending singular values, and provider/MatrixMath agreement. 262 Algorithms tests pass; solution builds clean.

Known remaining inconsistency (not addressed here)

The reporter also noticed duplication, and they're right: there are two Matrix classes (…LinearAlgebra.Matrix and …Types.LinearAlgebra.Matrix) and two LinearAlgebraOperation enums (…Algorithms and …Types). Consolidating those is a public-API breaking change and deserves its own issue rather than being bundled into a bug fix.

🤖 Generated with Claude Code