Status: current · RTX 4090 (sm_89), CUDA backend, saturating batches unless a section says otherwise.

ortho (src/ops/ortho/ortho.cc) orthogonalizes blocks of vectors for syevx_lobpcg, syevx_filtered and lanczos, with k in the tens. The Cholesky family (Cholesky, Chol2, ShiftChol3) forms the Gram matrix \(C = A^H A\) with syrk for real types at small k, and with GEMM otherwise. The level-3 side of this decision is on the level-3 page.

ortho: the Gram matrix through syrk, per precision

syrk computes \(C = A^H A\) at half the arithmetic of a GEMM. The table records the alternatives that were measured.

Alternative	Result	Verdict
syrk as a host loop over cublasXsyrk (no batched kernel)	96x slower than GEMM in float; 115 ms against 0.9 ms GEMM in double	rejected
syrk through syrk_gram_tiles (batched Gram tile kernel)	0.96x to 1.62x over GEMM, by k and precision (table below)	taken for k <= 64 (float) and k <= 128 (double)
herk through the Gram tile kernel (complex)	compute bound; GEMM plus Hermitian fold wins at every Gram shape	rejected; complex keeps GEMM

k. The single-tile Gram kernel covers k <= 128, but the useful limit differs by precision. End-to-end speed-up of syrk over GEMM, with m = 1024, batch 512, Chol2:

k	float	double
32	1.62x	1.02x
64	1.12x	1.20x
128	0.96x (loss)	1.34x

Float at k = 128 is a wash: the SGEMM it replaces is already at both the compute and the bandwidth roof. FP64 runs at 1/64 rate on this part, so double is compute bound and the halved arithmetic lands in full. Hence gram_max_k is 64 for float and 128 for double. Above those limits, double and complex stay on the host loop, which loses to GEMM by 2x at k = 256. Absolute times are on the level-3 page.

Invariants that the code keeps next to the call:

syrk writes only the lower triangle. potrf and trsm default to Uplo::Lower, and ShiftChol3's shift kernel reads only the diagonal.
svqb_alg keeps its GEMM. It scales the whole k x k matrix before syev, so a one-triangle C would multiply uninitialised workspace.
The gate is that syrk reaches the Gram tile kernel on this route (select::level3_tile_substitution_enabled, src/select/vendor.hh), not "this is NVIDIA".
BATCHLAS_ORTHO_GRAM=gemm pins GEMM, so the substitution can be measured from one binary.

ortho: host devices force Householder

On a non-GPU device, ortho replaces the requested algorithm with OrthoAlgorithm::Householder (ortho_force_householder). The test is on the device, not on Backend::NETLIB.

The rule is evaluated in the op and in all three size queries, and they must agree. Sizing for Chol2 and running Householder would overrun the BumpAllocator carved for tau, geqrf_ws and orgqr_ws.

Any ortho-dependent A/B on the NETLIB/CPU build therefore measures Householder, whatever it asks for. The LOBPCG soft-locking A/B hit exactly this (see soft locking).

ortho: open debts

The transposed CGS arm builds a view that does not describe the memory; see known defect 1.
The per-precision table has no recorded date or raw file. A re-measure should record both (see Documentation conventions).