[feat] squareroot matrix function - #278
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The exponential tests lived in three places: `common/exponential.jl` for the BLAS floats and one file each in the `genericschur` and `genericlinearalgebra` groups for the generic ones. Move the assertions into the shared TestSuite and leave a single driver that covers every element type, which removes both generic groups. Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
`squareroot` computes the principal square root, through `LinearAlgebra`, an (hermitian) eigenvalue decomposition, or the `Diagonal` fast path. The scalar type of the output matches the input, so a real matrix whose spectrum reaches the negative real axis raises a `DomainError`; eigenvalues that are negative only within `domain_atol` are clamped onto zero instead. `MatrixFunctionViaEig` and `MatrixFunctionViaEigh` gain a `domain_atol` field for this, and `src/implementations/matrixfunctions.jl` collects the input handling, reconstruction and domain helpers that the other matrix functions will share. Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
The assertions are invariants — `sqrt(A)^2 ≈ A`, hermiticity, the elementwise action on the spectrum — so that they also apply on GPU and to downstream array types, with a host-only cross-check against `LinearAlgebra`. The domain tests prescribe the full spectrum, and probe a genuinely negative eigenvalue, one that is negative by roundoff, and one in between that only an explicit `domain_atol` admits. Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
Describe the shared matrix function algorithms once, and add a section on the domain of `squareroot`: what `domain_atol` clamps, why an accepted result can be off by more than the tolerance, and why the tolerance of `MatrixFunctionViaLA` measures something else entirely. Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
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This is an attempt to move forward with #261 in smaller pieces, mostly since I got stuck a bit with the various different domain tolerances. Additionally I actually wanted to try out the actual square root dedicated algorithms to have something to go beyond the normal linear algebra routines, so here is that progress. What's nice is that this approach actually already works for non-blas paths, and seems to again outperform the linearalgebra routines.
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