Add gramcover: covers of A'*W*A from a cover of A#34
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gramcover(a, b, A[, w or W]) and its mutating form return a symmetric
cover s of the (weighted) Gram matrix A'*W*A directly from an
asymmetric cover of A, without forming the product:
s[j] = sqrt(Σ_{i,i'} a[i]*|W[i,i']|*a[i']) * b[j], accumulated over the
rows of j's connected component of A's support (components coupled by
off-diagonal entries of W are merged, since the product couples them).
Only abs.(W) enters the bound, so W need not be positive definite or
symmetric. Per-component accumulation makes s invariant under the
cover's per-component gauge — safe as an absolute scale where b alone
is not, e.g. Levenberg-Marquardt damping — and strictly tighter than
the global bound norm(a)*b on multi-component supports. Each sqrt is
inflated by a summation-roundoff bound so the returned cover holds
without verifying it against a formed Gram matrix.
The manual gains a section on the gauge sensitivity of bare cover
factors and the Gram cover's covariance under one-sided rescaling.
Assisted-by: Claude Fable 5 (claude-fable-5) <noreply@anthropic.com>
Codecov Report❌ Patch coverage is
Additional details and impacted files@@ Coverage Diff @@
## main #34 +/- ##
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+ Coverage 98.77% 98.78% +0.01%
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Files 13 14 +1
Lines 2196 2306 +110
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+ Hits 2169 2278 +109
- Misses 27 28 +1 ☔ View full report in Codecov by Harness. 🚀 New features to boost your workflow:
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On Julia 1.10, J' * Diagonal(w) * J with unit-carrying eltypes routes through LinearAlgebra's _tri_matmul, which materializes the Adjoint-Diagonal factor by writing unscaled values into an intermediate allocated with the product's units, throwing a DimensionError. Parenthesizing as J' * (Diagonal(w) * J) keeps every partial product unit-homogeneous. Assisted-by: Claude Fable 5 (claude-fable-5) <noreply@anthropic.com>
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gramcover(a, b, A[, w or W]) and its mutating form return a symmetric cover s of the (weighted) Gram matrix A'WA directly from an asymmetric cover of A, without forming the product: s[j] = sqrt(Σ_{i,i'} a[i]*|W[i,i']|*a[i']) * b[j], accumulated over the rows of j's connected component of A's support (components coupled by off-diagonal entries of W are merged, since the product couples them). Only abs.(W) enters the bound, so W need not be positive definite or symmetric. Per-component accumulation makes s invariant under the cover's per-component gauge — safe as an absolute scale where b alone is not, e.g. Levenberg-Marquardt damping — and strictly tighter than the global bound norm(a)*b on multi-component supports. Each sqrt is inflated by a summation-roundoff bound so the returned cover holds without verifying it against a formed Gram matrix.
The manual gains a section on the gauge sensitivity of bare cover factors and the Gram cover's covariance under one-sided rescaling.
Assisted-by: Claude Fable 5 (claude-fable-5) noreply@anthropic.com