Restore dense pinv fallback for inverse Jacobian init

[ChrisRackauckas-Claude]

Ignore until reviewed by @ChrisRackauckas.

Summary

• restore dense StridedMatrix inverse-Jacobian initialization to the previous LU/inverse path with pinv fallback for singular inputs
• keep the LinearSolve workspace path for sparse and structured matrix inputs
• update linsolve_identity!! tests to pin dense singular pinv behavior while preserving sparse/structured workspace coverage

Root cause

#1039 moved the dense quasi-Newton inverse initialization path from the old maybe_pinv!! semantics to linsolve_identity!! through LinearSolve. For rank-deficient reset Jacobians, that pivoted-QR generalized inverse can differ from the SVD pinv result enough to destabilize Broyden with true-Jacobian bad updates. The reproduced symptom was Generalized Rosenbrock function | alg #4 returning residual norm 4.3999999999999995 instead of satisfying <= 1e-3, plus Brown almost linear function | alg #4 becoming an unexpected pass.

Validation

• env JULIA_DEPOT_PATH=/home/crackauc/sandbox/tmp_20260708_121627_10236/nls_broyden_bisect_depot JULIA_NUM_THREADS=1 JULIA_PKG_PRECOMPILE_AUTO=no JULIA_PKG_SERVER_REGISTRY_PREFERENCE=eager timeout 3600 julia +1 --startup-file=no --project=lib/NonlinearSolveBase -e 'import Pkg; Pkg.test()'\n - linsolve_identity!! workspace (#1020) | 56 Pass\n - Linear solver routing | 12 Pass\n - Dense LU refactorization allocations | 14 Pass\n - Testing NonlinearSolveBase tests passed\n- env JULIA_DEPOT_PATH=/home/crackauc/sandbox/tmp_20260708_121627_10236/nls_broyden_bisect_depot JULIA_NUM_THREADS=1 JULIA_PKG_PRECOMPILE_AUTO=no JULIA_PKG_SERVER_REGISTRY_PREFERENCE=eager timeout 3600 julia +1 --startup-file=no --project=/home/crackauc/sandbox/tmp_20260708_121627_10236/nls_broyden_bisect_env -e 'using Test; include("test/Core/23_test_problems_tests__item7.jl")'\n - completed with exit code 0; Generalized Rosenbrock function | alg #4 passed and Brown almost linear function | alg #4 was broken, not an unexpected pass\n- env NONLINEARSOLVE_TEST_GROUP=Core ... timeout 3600 julia +1 --startup-file=no --project=. -e 'import Pkg; Pkg.test()'\n - Broyden section passed: 23 Test Problems: Broyden | 95 Pass, 20 Broken, 115 Total\n - full Core then failed later in test/Core/homotopy_alloc_tests__item1.jl on allocation thresholds; I reproduced that same homotopy allocation failure on untouched origin/master, so it is separate from this patch\n- Runic check passed on the touched files\n- git diff --check passed

[ChrisRackauckas-Claude]

CI triage update:

• tests / Core (julia 1, ubuntu-latest) reached and passed Broyden: 23 Test Problems: Broyden | 95 Pass, 20 Broken, 115 Total. It failed later at test/Core/homotopy_alloc_tests__item1.jl with allocation thresholds 154.4 < 48 and 80.0 < 48. I separately reproduced the same homotopy allocation test class on clean origin/master at b491a4e8; the direct clean-master include hit sweep_per_step(prob_big) == 154.4 before stopping.
• tests / Core (julia lts, ubuntu-latest) passed.
• sublibraries / lib/NonlinearSolveBase / Julia lts/1/pre / Tests - Core all passed.
• sublibraries / lib/NonlinearSolveQuasiNewton / Julia 1 / Tests - Core and QA both passed.
• sublibraries / lib/NonlinearSolveBase [QA] failed, but the same QA failure reproduces on clean master: JacobianCache ambiguity and SingleCacheStorage public-access complaint.
• Several jobs failed before checkout while resolving Actions downloads (Service Unavailable, Bad Gateway, or Internal Server Error resolving actions/checkout / julia-actions); those are infrastructure failures, not code/test failures from this PR.

[ChrisRackauckas-Claude]

Rebased onto current master at 41681c5e4 after #1060 merged and reran validation on the final tree before updating the remote branch.

Final-tree validation on Julia 1.12:

• NonlinearSolveBase tests passed: workspace 56/56, routing 12/12, operator dispatch 1/1, dense-LU allocation coverage 14/14.
• Exact root Broyden file passed with local path assertions: 95 Pass, 20 Broken, 115 Total; the expected broken cases remained broken, with no unexpected pass.
• Whole-repository Runic check passed.
• git diff --check upstream/master...HEAD passed.

The rebased head is ac8b3d5a361d58b7eb5f02b5ff2897526ba13461; CI has restarted on that commit.

[ChrisRackauckas-Claude]

The two early red downgrade jobs on rebased head ac8b3d5a3 are the same shared-action resolver failure already classified on #1063, not Broyden test failures:

• Root downgrade fixes in-tree NonlinearSolveQuasiNewton 1.14.1, which requires SciMLOperators >= 1.24, but the generated downgrade manifest explicitly pins SciMLOperators = 1.22.0.
• SimpleNonlinearSolve downgrade fixes in-tree NonlinearSolveBase 2.34.3, which also requires >= 1.24, against the same explicit 1.22.0 pin.

Both jobs stop at Unsatisfiable requirements before any test starts. The shared path-source handling is addressed by stacked julia-downgrade-compat drafts #54 and #55. Remaining checks are still running.

[ChrisRackauckas-Claude]

Additional rebased-head CI classification from the exact logs:

• NonlinearSolveBase QA reports only the existing FunctionWrappersWrappers.SingleCacheStorage non-public access; the owning-package public-API prerequisite is FunctionWrappersWrappers#70.
• Root Core on Julia current and Julia prerelease stops at the clean-master homotopy allocation regression (154.4 < 48 and 80.0 < 48), addressed separately by Eliminate dense continuation step allocations on Julia 1.12 #1061.

Neither failure enters this PR's dense pseudoinverse/Broyden regression. No public-API exception or allocation threshold is changed here; the post-rebase Base suite and exact Broyden target were run locally on the current graph as recorded above.

[ChrisRackauckas-Claude]

Additional CI follow-up: I inspected the newly completed macOS current and prerelease Core logs. Both pass through the Broyden robustness section and then fail only the two clean-master homotopy allocation assertions addressed by #1061. The target NonlinearSolveBase dense/pseudoinverse regressions remain green. The sole outstanding check is the queued SimpleNonlinearSolve CUDA lane.

[ChrisRackauckas-Claude]

Closing this in favor of #1069.

The dense pinv fallback restoration was too broad: it changes inverse-Jacobian initialization semantics globally to satisfy one numerically fragile bad-Broyden robustness expectation. Bad Broyden with true-Jacobian resets can reasonably fail on the generalized Rosenbrock start, and Brown almost-linear now reaches the residual tolerance. #1069 updates that narrow test contract instead.