Extended numerical science (experimental)

This extends the existing rusolver, ruintegrate and opt-in ruda-autodiff/solver-host.

Execution contracts

• Host FP64: thin one-sided Jacobi SVD, pseudoinverse and minimum-norm solve; complex LU/adjoint solve, Hermitian Cholesky, thin Householder QR; sparse LU with partial row pivoting and explicit fill budget; row-partitioned CG; analytic solve/Cholesky/SVD/symmetric-eigen/QR reverse-mode pullbacks.
• Existing Ruda autodiff graph: optional FP64 Autodiff<Host> functions solve_host, cholesky_host, singular_values_host, symmetric_eigenvalues_host. First-order only. GPU tensors cannot be silently transferred to this API. Full SVD/QR/eigenvector VJPs are explicit low-level pullbacks, not multi-output graph wrappers yet.
• Native device FP32: LU+solve (n<=32, nrhs<=8), QR (n<=32,m<=64,m>=n), real symmetric eigen (n<=32), dense SPD CG (n<=128, one RHS). All are contiguous small-batch, one-thread-per-system baselines, no host numerical fallback. They complement the existing device Cholesky. Call check_status_sync(); kernel submission is not success.
• Integrators: adaptive backward Euler/BDF1 with Newton and analytic or finite-difference Jacobian; sign-change events for RK45/BDF1; rational half/infinite-domain transformations over the existing GK15 integrator. Host FP64 only. ruintegrate now depends on local host rusolver for LU.

Complex64 uses TWO f64 components (128 storage bits), unlike CUDA's 64-bit single-precision complex storage name. SVD returns thin U(m,k), s(k), VT(k,n). Default rank cutoff is max(abs, rel * FrobeniusNorm(A)); truncated singular values are explicitly zero. Rank-deficient factors may thus be approximations. A nonconverged Jacobi sweep budget is an error, not an accepted factorization.

Sparse LU never allocates a dense n*n factor; BTreeMap fill and row pivot scans are not equivalent to an optimized supernodal/ordering-based solver. The fill limit counts factor nonzeros, not total allocator bytes. sparse optionally borrows existing ruSPARSE CSR formats. Distributed CG keeps matrix rows local, all-gathers full search vectors and reduces scalars through ordered all-gather. It is host FP64, O(n) vectors/rank, not a GPU halo-exchange or distributed direct factorization. collective reuses existing ruCCL TCP via RucclCommunicator. It needs a dedicated session; failures abort the group. Retry with a new session.

Spectral gradients reject repeated/near-degenerate spectra; SVD additionally requires full rank and separated nonzero singular values. QR differentiates with its forward pivot permutation fixed, not the discrete pivot selection. Symmetric input gradients use the full symmetric-matrix convention. Graph backward uses the existing non-fallible Backward contract; numerical errors panic explicitly, not return fabricated zero gradients. No complex/sparse/distributed/GPU graph integration or higher derivatives are provided by this patch.

BDF1 uses a full step vs two half steps for error estimation, and accepts the two half steps without extrapolation. It is first order, dense Newton/LU, default max dimension 256, not variable-order BDF/Radau. Event location uses cubic Hermite interpolation of accepted step endpoint states/derivatives and bisection. Direction is physical-time direction even when integrating backward. Tangencies and multiple crossings inside a step may be missed; limit max_step as needed. Root tolerance is not a global ODE error bound. Infinite transforms integrate the two tails separately, never claim Cauchy principal-value cancellation as convergence. Estimated errors do not prove integrability.

Examples: advanced-solver, advanced-integrate, solver-cuda-advanced and distributed-cg. Existing numerical and tensor-LU APIs remain in place. OdeStatus adds Event: downstream exhaustive matches must add a case.

Mathematical references (not source-code copies): LAPACK Jacobi SVD/DGESVJ and complex Householder QR documentation; SciPy solve_ivp/quad references for scope and validation; PyTorch linalg autograd documentation for spectral restrictions. See THIRD_PARTY_NOTICES.md and 中文完整指南.