Core: Vectorize triangular solves across right-hand sides

This adds a shared SIMD kernel for TRSM, processing independent right-hand sides in packet lanes. GEBP’s register budget determines the RHS packet group, and Eigen’s configured L1 cache size controls when a solve bypasses matrix-multiply packing. Fresh measurements cover GCC and Clang × SSE2 and AVX2+FMA, using the same source revision and benchmark in all four configurations.

Measured revision: eba485c30, before the branch was rebased to 015451069. The TRSM kernel, benchmark and focused test sources are byte-identical after rebasing; the rebased tree has not been timed.

Overall speedup: pre-optimization Eigen 15f227178 → eba485c30. Geometric means over seven square sizes, 16–2048, for each scalar. Column-major lower-triangular left solve, including the RHS copy; greater than 1× is faster.

Compiler / ISA	float geomean	double geomean	Range across 14 cases
GCC / SSE2	1.548×	1.282×	1.005–3.683×
Clang / SSE2	1.549×	1.388×	1.024–3.964×
GCC / AVX2+FMA	2.548×	1.765×	1.085–7.924×
Clang / AVX2+FMA	2.859×	1.841×	1.092–12.262×

Review-fix cost: 0c7bdb309 → eba485c30. The same 44-case grid in each configuration, including both triangle layouts, square and narrow RHS cases. The Clang/AVX2 review row retains the earlier valid run; its attempted refresh was interrupted by background work and excluded. Both rebuilt binaries are byte-identical to the earlier pair.

Compiler / ISA	Column-major geomean (22)	Row-major geomean (22)	Lowest observed ratio
GCC / SSE2	0.990×	0.983×	double 64×17: 0.878×; 20.3% variation
Clang / SSE2	0.996×	1.001×	double 8×8 (row-major A): 0.972×; 1.6% variation
GCC / AVX2+FMA	1.012×	0.985×	float 16×16 (row-major A): 0.897×; 1.3% variation
Clang / AVX2+FMA (earlier run)	1.012×	0.997×	double 16×16 (row-major A): 0.965×; 4.3% variation

Variation in the last column is the maximum CV or run spread, not a confidence interval. The GCC/SSE2 0.878× case has 20.3% variation and is inconclusive. Tiny row-major float solves retain a cost: GCC/SSE2 8×8 takes 5.6% longer, and GCC/AVX2 16×16 takes 11.5% longer (0.0713 → 0.0795 µs), consistent with the earlier observed guard cost. Large SSE2 double gains are only about 1–3% and generally near the observed variability. Earlier localized 3–6% generalization costs against 76c9fb280 remain limitations. No additional targeted repeats were performed.

The kernel transposes a bounded RHS tile, solves register blocks of rows, and transposes the results back. Four register rows and at most two RHS packets preserve the established AVX2 schedule. WorkspaceRows = 128 bounds stack capacity separately from the cache policy: the direct-solve footprint estimate is n * (packets * PacketSize + RegisterRows + 1) * sizeof(Scalar), plus one packet transpose, against half of L1. Cache discovery and overrides use Eigen’s existing machinery; outer L2/L3 blocking remains in place. These are conservative defaults, not optimal tuning claims for every platform.

The fast path requires vectorized built-in IEEE binary float/double and compile-time unit inner stride. Partial RHS packets, complex/custom scalars and non-unit strides retain their existing paths. AVX-512 builds select their existing specialized TRSM kernel by default; unifying that implementation remains separate work.

Update: Thanks to @florian360 for the finite-result overflow example and @onalante-ebay for the expression/style suggestions. eba485c30 checks the final solved row of each row-major RHS tile before writeback, and retries a nonfinite tile using the original row-dot-product accumulation order. Every later row consumes all earlier solved rows, so nonfinite lanes propagate to the final row. Integer exponent-bit classification and Eigen’s fast-math barrier keep the check effective under Clang -ffast-math. Reciprocal initialization now uses a map of the active diagonal; the other suggestions use EIGEN_IF_CONSTEXPR and numext::round_down with an Index conversion.

The retry restores the previous diagonal-kernel behavior; it is not a general overflow-safe solver. The existing default AVX-512 specialization also overflows on the example before this MR and is unchanged.

Measurements: 2026-09-17, 13th Gen Intel(R) Core(TM) i7-13700HX under WSL2, GCC 13.3.0 / Clang 18.1.3. CPU-time medians, one process pinned to CPU 2, A/B/A/B invocations, eight repetitions per case. Windows clocks, scheduling and thermal policy were uncontrolled. Full flags, protocol and limitations are in Appendix E.

GCC and Clang correctness validation passed, including the additional SSE2 runs. The regression and prior shared-kernel AVX-512/NEON validation are detailed in Appendix E.

Appendix A: GCC / SSE2 — overall and review comparisons

Overall: 15f227178 → eba485c30 (14 fresh cases)

Case	Before (µs)	After (µs)	Speedup	CV before / after	Run spread before / after
float 16×16	0.5576	0.1514	3.683×	3.1% / 1.6%	1.7% / 0.6%
float 64×64	13.7400	6.8228	2.014×	1.3% / 1.6%	0.8% / 0.5%
float 128×128	91.2179	52.3067	1.744×	1.0% / 1.2%	0.1% / 0.3%
float 256×256	561.0331	473.9017	1.184×	1.4% / 1.7%	1.7% / 0.6%
float 512×512	4234.2389	3623.6479	1.169×	1.9% / 2.1%	1.9% / 1.0%
float 1024×1024	31527.6154	27794.8264	1.134×	0.7% / 1.6%	2.4% / 2.1%
float 2048×2048	227246.9757	216860.4282	1.048×	0.9% / 1.8%	1.6% / 0.5%
double 16×16	0.5667	0.2881	1.967×	6.5% / 1.2%	16.2% / 1.2%
double 64×64	22.4007	14.1618	1.582×	2.0% / 1.2%	0.7% / 0.9%
double 128×128	181.7513	110.6166	1.643×	1.9% / 1.7%	1.9% / 0.7%
double 256×256	1158.5543	1091.9134	1.061×	1.1% / 2.9%	1.3% / 1.9%
double 512×512	8358.3884	8231.0639	1.015×	2.4% / 2.6%	1.0% / 0.4%
double 1024×1024	60585.0261	59018.6196	1.027×	2.7% / 1.6%	0.1% / 0.8%
double 2048×2048	451579.4922	449198.4952	1.005×	1.3% / 2.4%	0.5% / 0.2%

Review: 0c7bdb309 → eba485c30 (44 cases)

Case	Before (µs)	After (µs)	Speedup	CV before / after	Run spread before / after
float 8×8	0.0345	0.0345	0.999×	2.3% / 1.5%	1.0% / 0.0%
float 16×16	0.1527	0.1536	0.994×	1.5% / 1.3%	2.1% / 0.2%
float 64×64	6.8934	6.9861	0.987×	2.6% / 2.8%	1.8% / 3.6%
float 128×128	52.7329	52.8021	0.999×	1.1% / 1.4%	1.1% / 0.2%
float 256×256	484.8405	486.8830	0.996×	3.8% / 1.9%	3.7% / 5.1%
float 512×512	3641.6702	3660.9862	0.995×	1.2% / 1.5%	2.0% / 2.7%
float 1024×1024	27734.8454	28237.3886	0.982×	1.3% / 1.7%	1.3% / 0.1%
float 64×8	0.8637	0.8777	0.984×	1.4% / 3.7%	0.1% / 1.0%
float 64×17	2.2002	2.2131	0.994×	3.4% / 5.2%	1.7% / 0.3%
float 128×8	3.2751	3.2834	0.997×	1.3% / 0.9%	0.8% / 1.4%
float 128×17	7.7792	7.6997	1.010×	2.4% / 1.1%	0.4% / 1.2%
double 8×8	0.0620	0.0567	1.094×	7.6% / 1.0%	19.8% / 1.2%
double 16×16	0.2920	0.2906	1.005×	2.6% / 3.2%	2.8% / 3.5%
double 64×64	14.3612	14.3880	0.998×	1.3% / 1.8%	0.8% / 2.4%
double 128×128	111.0311	112.8277	0.984×	1.1% / 1.9%	0.6% / 2.3%
double 256×256	1072.1497	1084.9934	0.988×	1.4% / 1.9%	0.1% / 0.1%
double 512×512	8000.9396	8078.9894	0.990×	1.4% / 4.2%	1.7% / 0.2%
double 1024×1024	59348.5789	59608.0896	0.996×	2.3% / 1.5%	0.5% / 1.4%
double 64×8	1.7688	1.8005	0.982×	1.2% / 2.7%	0.3% / 4.0%
double 64×17	3.8956	4.4344	0.878×	1.1% / 10.0%	0.1% / 20.3%
double 128×8	6.8641	6.9570	0.987×	1.4% / 6.1%	0.0% / 0.3%
double 128×17	15.1213	15.8217	0.956×	2.2% / 1.9%	0.2% / 0.8%
float 8×8 (row-major A)	0.0351	0.0371	0.947×	1.1% / 1.8%	0.2% / 1.4%
float 16×16 (row-major A)	0.1523	0.1549	0.984×	2.3% / 1.4%	1.7% / 1.8%
float 64×64 (row-major A)	7.3920	7.0280	1.052×	10.1% / 3.1%	15.7% / 2.5%
float 128×128 (row-major A)	52.2264	52.8342	0.988×	6.3% / 2.2%	0.3% / 1.2%
float 256×256 (row-major A)	477.2216	483.9215	0.986×	1.3% / 1.4%	0.5% / 1.2%
float 512×512 (row-major A)	3599.2099	3976.9805	0.905×	1.1% / 6.8%	0.4% / 17.1%
float 1024×1024 (row-major A)	27846.4506	28681.4812	0.971×	1.8% / 3.1%	1.0% / 3.7%
float 64×8 (row-major A)	0.8641	0.8646	0.999×	0.9% / 1.1%	1.8% / 0.5%
float 64×17 (row-major A)	2.1874	2.2039	0.993×	0.8% / 3.2%	0.5% / 1.5%
float 128×8 (row-major A)	3.2616	3.2842	0.993×	1.1% / 1.3%	0.1% / 2.0%
float 128×17 (row-major A)	8.4872	8.5835	0.989×	1.1% / 1.9%	0.4% / 1.6%
double 8×8 (row-major A)	0.0606	0.0589	1.030×	11.8% / 1.6%	14.5% / 0.9%
double 16×16 (row-major A)	0.2815	0.2823	0.997×	6.2% / 1.4%	2.9% / 0.9%
double 64×64 (row-major A)	14.3667	14.4944	0.991×	1.4% / 1.3%	1.2% / 0.5%
double 128×128 (row-major A)	111.1262	112.0957	0.991×	3.6% / 1.2%	1.4% / 0.5%
double 256×256 (row-major A)	1079.7081	1202.0414	0.898×	2.0% / 8.9%	1.2% / 12.0%
double 512×512 (row-major A)	8034.1150	8083.5608	0.994×	2.0% / 1.5%	1.1% / 0.5%
double 1024×1024 (row-major A)	59770.7063	61668.9123	0.969×	1.8% / 11.7%	0.7% / 0.7%
double 64×8 (row-major A)	1.8216	1.8725	0.973×	1.5% / 8.7%	1.2% / 5.9%
double 64×17 (row-major A)	4.0648	4.2599	0.954×	2.0% / 7.6%	0.6% / 7.5%
double 128×8 (row-major A)	7.0935	6.9709	1.018×	1.6% / 1.6%	2.7% / 0.1%
double 128×17 (row-major A)	16.1915	16.0425	1.009×	2.4% / 0.8%	1.8% / 0.5%

Appendix B: Clang / SSE2 — overall and review comparisons

Overall: 15f227178 → eba485c30 (14 fresh cases)

Case	Before (µs)	After (µs)	Speedup	CV before / after	Run spread before / after
float 16×16	0.6663	0.1681	3.964×	7.3% / 3.5%	7.9% / 15.2%
float 64×64	14.0409	7.6860	1.827×	4.9% / 6.0%	0.7% / 17.1%
float 128×128	91.6284	53.4162	1.715×	4.3% / 2.8%	5.5% / 2.1%
float 256×256	592.0898	456.4255	1.297×	8.0% / 1.3%	14.8% / 0.3%
float 512×512	3923.3726	3447.7765	1.138×	3.9% / 2.7%	1.0% / 0.1%
float 1024×1024	29448.5509	27023.4574	1.090×	1.7% / 3.6%	2.0% / 3.7%
float 2048×2048	221602.5165	206758.0272	1.072×	6.8% / 3.0%	2.9% / 0.5%
double 16×16	0.6979	0.2851	2.448×	4.3% / 1.3%	18.1% / 1.1%
double 64×64	24.3558	14.2676	1.707×	3.6% / 1.2%	16.3% / 1.3%
double 128×128	187.7486	111.6513	1.682×	6.3% / 1.8%	15.1% / 1.1%
double 256×256	1236.0550	1049.8741	1.177×	5.8% / 1.3%	18.4% / 0.2%
double 512×512	9015.4704	7908.4092	1.140×	3.1% / 2.6%	21.2% / 3.4%
double 1024×1024	60647.5749	59039.0686	1.027×	1.4% / 2.4%	2.6% / 2.3%
double 2048×2048	459841.1825	448971.9905	1.024×	7.0% / 5.0%	2.4% / 0.1%

Review: 0c7bdb309 → eba485c30 (44 cases)

Case	Before (µs)	After (µs)	Speedup	CV before / after	Run spread before / after
float 8×8	0.0351	0.0352	0.997×	1.2% / 1.4%	0.4% / 3.0%
float 16×16	0.1539	0.1547	0.995×	1.0% / 1.4%	0.5% / 1.4%
float 64×64	6.8761	6.9534	0.989×	1.0% / 1.2%	0.3% / 0.8%
float 128×128	52.8017	52.9650	0.997×	1.2% / 1.6%	1.2% / 0.1%
float 256×256	457.7716	459.6645	0.996×	5.4% / 6.9%	0.2% / 1.9%
float 512×512	3475.3373	3476.3559	1.000×	1.4% / 1.3%	0.8% / 0.4%
float 1024×1024	26646.3233	26526.2724	1.005×	2.2% / 6.7%	1.9% / 0.3%
float 64×8	0.8711	0.8809	0.989×	1.4% / 4.3%	0.4% / 1.7%
float 64×17	2.0853	2.0740	1.005×	3.5% / 4.9%	1.1% / 0.7%
float 128×8	3.2885	3.2787	1.003×	1.2% / 1.2%	0.6% / 2.1%
float 128×17	7.5892	7.6158	0.997×	0.9% / 1.8%	0.9% / 0.9%
double 8×8	0.0574	0.0578	0.992×	0.9% / 1.6%	1.9% / 1.5%
double 16×16	0.2865	0.2878	0.996×	1.7% / 2.4%	0.4% / 1.5%
double 64×64	14.2430	14.4612	0.985×	3.4% / 3.7%	0.4% / 3.1%
double 128×128	111.1773	114.0092	0.975×	1.6% / 1.6%	0.0% / 0.6%
double 256×256	1071.4788	1076.8173	0.995×	4.5% / 2.0%	2.8% / 3.6%
double 512×512	7981.5099	7974.0080	1.001×	1.8% / 5.4%	2.5% / 3.5%
double 1024×1024	60038.1918	59618.3565	1.007×	3.0% / 1.5%	0.6% / 1.2%
double 64×8	1.7971	1.8094	0.993×	1.4% / 2.1%	2.9% / 1.0%
double 64×17	4.0454	4.0298	1.004×	1.7% / 1.5%	4.3% / 1.2%
double 128×8	7.1518	7.1595	0.999×	3.9% / 1.5%	1.9% / 1.9%
double 128×17	15.5134	15.7093	0.988×	1.0% / 1.7%	0.3% / 1.8%
float 8×8 (row-major A)	0.0368	0.0366	1.005×	2.8% / 0.9%	4.7% / 1.3%
float 16×16 (row-major A)	0.1534	0.1539	0.997×	1.4% / 1.4%	0.6% / 0.1%
float 64×64 (row-major A)	6.9136	6.8988	1.002×	1.2% / 1.5%	0.2% / 0.5%
float 128×128 (row-major A)	52.0302	52.4365	0.992×	1.7% / 1.9%	0.2% / 0.6%
float 256×256 (row-major A)	468.0477	465.6385	1.005×	3.4% / 1.0%	1.0% / 1.5%
float 512×512 (row-major A)	3563.3720	3490.4749	1.021×	7.6% / 1.1%	2.3% / 0.1%
float 1024×1024 (row-major A)	27218.0376	26813.7949	1.015×	2.6% / 1.6%	5.2% / 0.0%
float 64×8 (row-major A)	0.8861	0.8724	1.016×	4.2% / 2.0%	3.4% / 0.7%
float 64×17 (row-major A)	2.1651	2.1744	0.996×	1.6% / 4.2%	0.2% / 1.3%
float 128×8 (row-major A)	3.2273	3.2611	0.990×	1.3% / 1.4%	0.3% / 0.8%
float 128×17 (row-major A)	8.5284	8.5458	0.998×	1.1% / 1.4%	0.6% / 0.0%
double 8×8 (row-major A)	0.0587	0.0605	0.972×	1.6% / 0.9%	0.6% / 0.1%
double 16×16 (row-major A)	0.2913	0.2930	0.994×	1.0% / 1.8%	0.9% / 0.2%
double 64×64 (row-major A)	14.3652	14.4330	0.995×	2.4% / 1.5%	0.5% / 0.3%
double 128×128 (row-major A)	110.8081	111.5462	0.993×	2.2% / 1.5%	1.5% / 1.6%
double 256×256 (row-major A)	1052.4882	1041.6074	1.010×	2.8% / 1.2%	3.5% / 0.2%
double 512×512 (row-major A)	7878.6308	7896.4791	0.998×	2.3% / 3.7%	1.7% / 0.1%
double 1024×1024 (row-major A)	59710.5261	59372.7654	1.006×	3.2% / 1.3%	0.6% / 0.4%
double 64×8 (row-major A)	1.8327	1.8034	1.016×	6.8% / 1.0%	0.9% / 0.2%
double 64×17 (row-major A)	4.0394	4.0292	1.003×	10.4% / 1.5%	2.3% / 1.3%
double 128×8 (row-major A)	6.9353	6.9864	0.993×	1.4% / 1.2%	0.6% / 0.5%
double 128×17 (row-major A)	15.8899	15.9444	0.997×	1.2% / 2.4%	1.0% / 0.9%

Appendix C: GCC / AVX2+FMA — overall and review comparisons

Overall: 15f227178 → eba485c30 (14 fresh cases)

Case	Before (µs)	After (µs)	Speedup	CV before / after	Run spread before / after
float 16×16	0.5707	0.0720	7.924×	1.4% / 1.3%	0.9% / 0.3%
float 64×64	10.6330	2.4777	4.291×	1.2% / 2.6%	0.3% / 2.3%
float 128×128	52.9245	18.2985	2.892×	2.9% / 2.3%	0.5% / 1.9%
float 256×256	277.5852	143.2835	1.937×	1.1% / 8.7%	0.8% / 1.3%
float 512×512	1881.8084	1103.0460	1.706×	2.6% / 1.9%	0.3% / 1.9%
float 1024×1024	13866.7695	8406.2973	1.650×	2.4% / 1.5%	1.1% / 1.0%
float 2048×2048	85655.7687	65872.5515	1.300×	1.8% / 1.7%	1.1% / 1.2%
double 16×16	0.5137	0.1212	4.240×	1.3% / 1.7%	4.3% / 3.0%
double 64×64	11.9751	5.1286	2.335×	5.8% / 7.9%	0.6% / 0.1%
double 128×128	66.5713	36.8364	1.807×	1.7% / 1.4%	0.6% / 0.4%
double 256×256	481.8207	310.4811	1.552×	1.3% / 2.3%	0.3% / 1.9%
double 512×512	3344.5898	2314.2324	1.445×	1.2% / 1.7%	1.4% / 0.6%
double 1024×1024	21422.2124	17475.8528	1.226×	2.1% / 1.9%	0.1% / 0.6%
double 2048×2048	147848.5783	136205.5195	1.085×	2.0% / 8.6%	1.1% / 1.9%

Review: 0c7bdb309 → eba485c30 (44 cases)

Case	Before (µs)	After (µs)	Speedup	CV before / after	Run spread before / after
float 8×8	0.0264	0.0252	1.045×	1.7% / 0.9%	0.1% / 0.1%
float 16×16	0.0714	0.0720	0.992×	0.9% / 0.9%	0.0% / 0.2%
float 64×64	2.4884	2.4199	1.028×	1.4% / 1.2%	0.0% / 1.2%
float 128×128	18.5123	18.1125	1.022×	1.0% / 1.1%	0.7% / 0.8%
float 256×256	143.6751	142.7290	1.007×	1.3% / 2.0%	0.4% / 0.5%
float 512×512	1095.1285	1103.2076	0.993×	1.1% / 2.1%	0.5% / 0.6%
float 1024×1024	8693.2693	8437.8220	1.030×	3.6% / 1.8%	6.9% / 0.5%
float 64×8	0.4198	0.4138	1.015×	8.3% / 1.0%	2.8% / 0.3%
float 64×17	0.9630	0.9545	1.009×	1.1% / 1.3%	0.4% / 0.1%
float 128×8	1.6742	1.6627	1.007×	1.7% / 1.0%	3.0% / 0.5%
float 128×17	3.3206	3.2641	1.017×	0.8% / 2.4%	0.1% / 0.1%
double 8×8	0.0356	0.0353	1.008×	1.8% / 1.1%	1.6% / 0.4%
double 16×16	0.1214	0.1201	1.011×	1.6% / 1.6%	3.2% / 0.6%
double 64×64	5.0954	5.1248	0.994×	0.9% / 1.4%	0.1% / 1.8%
double 128×128	36.8140	36.3208	1.014×	1.6% / 1.2%	0.9% / 0.3%
double 256×256	322.4672	316.7923	1.018×	4.7% / 1.9%	6.8% / 3.4%
double 512×512	2316.2845	2303.0798	1.006×	1.6% / 2.1%	0.2% / 2.3%
double 1024×1024	17909.0610	17630.6995	1.016×	2.3% / 2.1%	0.6% / 1.0%
double 64×8	0.5990	0.5959	1.005×	1.4% / 0.8%	0.4% / 0.6%
double 64×17	1.6419	1.5930	1.031×	2.4% / 0.6%	1.0% / 0.4%
double 128×8	2.2146	2.2286	0.994×	1.7% / 1.0%	1.1% / 0.3%
double 128×17	5.9656	5.8751	1.015×	1.9% / 1.2%	1.1% / 0.2%
float 8×8 (row-major A)	0.0276	0.0277	0.995×	1.1% / 1.5%	0.1% / 1.5%
float 16×16 (row-major A)	0.0713	0.0795	0.897×	1.3% / 0.8%	0.9% / 0.2%
float 64×64 (row-major A)	2.5523	2.5327	1.008×	10.1% / 0.8%	0.3% / 1.1%
float 128×128 (row-major A)	19.1622	19.2573	0.995×	1.2% / 1.1%	0.4% / 1.4%
float 256×256 (row-major A)	145.2083	144.6909	1.004×	1.2% / 0.8%	0.4% / 0.2%
float 512×512 (row-major A)	1104.7704	1118.2685	0.988×	1.2% / 1.8%	1.8% / 0.2%
float 1024×1024 (row-major A)	8444.9349	8436.3459	1.001×	1.6% / 0.8%	2.1% / 0.4%
float 64×8 (row-major A)	0.4072	0.4137	0.984×	0.9% / 1.8%	0.4% / 0.6%
float 64×17 (row-major A)	1.1023	1.1158	0.988×	1.1% / 2.3%	0.2% / 1.3%
float 128×8 (row-major A)	1.6629	1.6852	0.987×	1.2% / 1.9%	0.1% / 1.1%
float 128×17 (row-major A)	4.2747	4.3057	0.993×	1.0% / 1.0%	1.5% / 0.7%
double 8×8 (row-major A)	0.0353	0.0369	0.957×	10.0% / 1.0%	0.8% / 0.3%
double 16×16 (row-major A)	0.1206	0.1238	0.974×	7.2% / 0.7%	2.4% / 1.8%
double 64×64 (row-major A)	5.3403	5.4640	0.977×	2.7% / 0.9%	0.3% / 1.3%
double 128×128 (row-major A)	38.5046	39.2370	0.981×	1.4% / 2.9%	0.5% / 2.2%
double 256×256 (row-major A)	318.9094	320.1195	0.996×	1.6% / 1.8%	0.8% / 0.5%
double 512×512 (row-major A)	2319.8356	2343.3450	0.990×	1.3% / 6.3%	1.0% / 0.2%
double 1024×1024 (row-major A)	17693.2769	17645.9338	1.003×	2.0% / 1.4%	0.3% / 1.3%
double 64×8 (row-major A)	0.6260	0.6461	0.969×	1.2% / 4.4%	0.8% / 4.6%
double 64×17 (row-major A)	1.7756	1.7852	0.995×	1.3% / 0.8%	0.4% / 1.5%
double 128×8 (row-major A)	2.3421	2.3685	0.989×	1.5% / 0.9%	1.6% / 0.2%
double 128×17 (row-major A)	6.8608	6.8619	1.000×	1.2% / 2.9%	0.2% / 0.6%

Appendix D: Clang / AVX2+FMA — overall and review comparisons

Overall: 15f227178 → eba485c30 (14 fresh cases)

Case	Before (µs)	After (µs)	Speedup	CV before / after	Run spread before / after
float 16×16	0.8516	0.0695	12.262×	1.7% / 1.5%	0.2% / 1.2%
float 64×64	15.6504	2.5945	6.032×	1.5% / 2.2%	0.7% / 1.5%
float 128×128	74.1042	18.8137	3.939×	0.7% / 3.7%	0.0% / 1.4%
float 256×256	378.7491	177.9588	2.128×	1.7% / 1.1%	0.4% / 0.6%
float 512×512	2286.4058	1430.5906	1.598×	1.5% / 1.3%	0.1% / 1.0%
float 1024×1024	15176.6062	11345.0487	1.338×	1.0% / 1.2%	0.1% / 0.2%
float 2048×2048	108718.5517	92347.0618	1.177×	5.2% / 1.9%	0.4% / 1.4%
double 16×16	0.6933	0.1145	6.054×	1.6% / 1.7%	3.3% / 0.4%
double 64×64	14.1777	5.3982	2.626×	1.6% / 2.6%	0.4% / 7.2%
double 128×128	77.0706	37.3305	2.065×	1.0% / 1.5%	0.6% / 1.0%
double 256×256	505.4704	370.2104	1.365×	1.7% / 1.6%	1.3% / 0.9%
double 512×512	3672.2243	2886.6571	1.272×	1.9% / 3.1%	0.7% / 0.8%
double 1024×1024	26642.6718	23104.5532	1.153×	1.3% / 5.1%	1.5% / 1.8%
double 2048×2048	200970.5788	184114.9500	1.092×	1.2% / 2.0%	0.4% / 1.2%

Review: 0c7bdb309 → eba485c30 (44 cases) — retained earlier valid run; attempted refresh excluded

Case	Before (µs)	After (µs)	Speedup	CV before / after	Run spread before / after
float 8×8	0.0263	0.0265	0.990×	3.1% / 2.6%	0.8% / 2.1%
float 16×16	0.0723	0.0727	0.994×	2.8% / 2.5%	1.1% / 3.1%
float 64×64	2.6447	2.6147	1.011×	2.2% / 2.1%	1.0% / 2.0%
float 128×128	20.1068	18.9734	1.060×	1.8% / 1.9%	1.2% / 0.4%
float 256×256	178.7853	179.8428	0.994×	1.1% / 1.6%	0.2% / 1.0%
float 512×512	1418.5618	1420.6191	0.999×	0.9% / 1.7%	0.6% / 1.4%
float 1024×1024	11275.5762	11261.5353	1.001×	1.2% / 0.9%	0.3% / 1.2%
float 64×8	0.4483	0.4442	1.009×	3.0% / 2.8%	4.9% / 1.9%
float 64×17	1.1414	1.1136	1.025×	6.3% / 3.9%	0.5% / 0.3%
float 128×8	1.7242	1.7860	0.965×	3.3% / 2.9%	0.8% / 4.1%
float 128×17	3.6715	3.5161	1.044×	2.2% / 1.9%	0.4% / 2.5%
double 8×8	0.0332	0.0333	0.997×	3.3% / 3.8%	4.5% / 0.8%
double 16×16	0.1231	0.1215	1.013×	2.6% / 3.0%	5.5% / 2.3%
double 64×64	5.3873	5.2220	1.032×	2.2% / 2.1%	0.6% / 3.0%
double 128×128	39.6703	38.3379	1.035×	1.6% / 2.6%	0.0% / 2.4%
double 256×256	368.2548	368.2284	1.000×	1.7% / 0.9%	0.6% / 0.5%
double 512×512	2856.9706	2855.3293	1.001×	1.7% / 2.1%	0.5% / 0.7%
double 1024×1024	22786.2134	22784.3764	1.000×	1.7% / 1.2%	1.0% / 0.2%
double 64×8	0.6452	0.6234	1.035×	2.1% / 2.0%	2.7% / 2.3%
double 64×17	1.6986	1.7008	0.999×	3.2% / 2.7%	0.5% / 1.7%
double 128×8	2.4676	2.3751	1.039×	2.6% / 2.7%	1.7% / 0.9%
double 128×17	6.1925	6.0064	1.031×	2.5% / 2.7%	2.2% / 2.2%
float 8×8 (row-major A)	0.0269	0.0272	0.992×	4.1% / 3.1%	6.9% / 1.2%
float 16×16 (row-major A)	0.0724	0.0736	0.984×	2.6% / 2.6%	2.8% / 4.0%
float 64×64 (row-major A)	2.8109	2.6711	1.052×	1.7% / 2.2%	2.0% / 1.9%
float 128×128 (row-major A)	20.8434	20.4351	1.020×	1.6% / 2.0%	1.6% / 0.1%
float 256×256 (row-major A)	179.3322	180.0101	0.996×	2.1% / 1.4%	1.0% / 0.0%
float 512×512 (row-major A)	1442.6603	1451.8157	0.994×	1.1% / 1.5%	0.1% / 0.5%
float 1024×1024 (row-major A)	11372.3660	11425.4480	0.995×	1.4% / 0.9%	0.4% / 0.3%
float 64×8 (row-major A)	0.4380	0.4394	0.997×	1.9% / 3.0%	5.2% / 0.3%
float 64×17 (row-major A)	1.3788	1.3556	1.017×	1.2% / 2.1%	0.2% / 0.6%
float 128×8 (row-major A)	1.7504	1.7250	1.015×	1.2% / 2.7%	1.2% / 0.8%
float 128×17 (row-major A)	6.0667	6.0013	1.011×	2.4% / 0.9%	1.1% / 0.2%
double 8×8 (row-major A)	0.0340	0.0347	0.979×	2.9% / 2.9%	1.1% / 5.8%
double 16×16 (row-major A)	0.1245	0.1291	0.965×	2.5% / 4.2%	4.3% / 1.5%
double 64×64 (row-major A)	5.4711	5.5156	0.992×	2.6% / 2.1%	1.8% / 1.1%
double 128×128 (row-major A)	40.4738	40.9422	0.989×	1.5% / 1.4%	1.2% / 1.2%
double 256×256 (row-major A)	376.2471	377.5466	0.997×	2.4% / 2.4%	1.9% / 1.3%
double 512×512 (row-major A)	2894.8179	2893.7299	1.000×	2.2% / 1.6%	2.8% / 1.7%
double 1024×1024 (row-major A)	23036.7122	23123.5302	0.996×	1.6% / 2.6%	0.3% / 0.6%
double 64×8 (row-major A)	0.6588	0.6623	0.995×	2.7% / 2.5%	0.1% / 0.5%
double 64×17 (row-major A)	1.9987	2.0610	0.970×	2.1% / 4.2%	2.2% / 4.8%
double 128×8 (row-major A)	2.4868	2.5158	0.988×	2.4% / 2.6%	0.8% / 0.3%
double 128×17 (row-major A)	8.3608	8.4229	0.993×	1.0% / 1.5%	0.3% / 0.4%

Appendix E: Provenance and validation details

Measurement conditions: 2026-09-17, 13th Gen Intel(R) Core(TM) i7-13700HX under WSL2; GCC 13.3.0, Clang 18.1.3, Google Benchmark 1.9.5. Common flags: -std=c++14 -O3 -DNDEBUG -march=x86-64 -mtune=generic. SSE2 adds -msse2 -mno-avx -mno-fma; AVX2+FMA adds -mavx2 -mfma. Compiler/Eigen macros were checked. Each pair uses identical sources, flags and filters, without external BLAS or OpenMP.

One benchmark ran at a time on CPU 2, in A/B/A/B order, with eight repetitions of ≥0.08 CPU seconds per case per invocation. Compilation and correctness runs finished before timing. System-wide process monitoring rejected sustained competing work; Windows scheduling, clocks and thermal policy were uncontrolled. All included cases passed the benchmark’s residual check. An interrupted Clang/SSE2 overall attempt was discarded and restarted before the request to omit additional repeats. The later interrupted Clang/AVX2 review refresh was discarded without rerunning it.

The reported ratio uses CPU time. Elapsed-time ratios differed by up to 4.3% in the overall grid and 7.0% in the review grid; small differences near this variability are inconclusive. No additional targeted repeats were performed; noisy cases remain inconclusive.

MR head at publication: 0154510695ec7051f030808e520fd3c750c88015 (rebased during the measurements). Revisions used for benchmarking: original 15f22717866165e381d36d52209400daaf39a386, previous MR 0c7bdb309a94d4a297abacf48900decbdb3a2f01, candidate eba485c3099b2a9712c76d0201a684d578f5dbcf. The timed expression is x = b; a.triangularView<Lower>().solveInPlace(x); setup and residual checks are excluded. RHS storage is column-major. Speedup = before/after, each time being the median of two invocation medians. CV is the maximum within-invocation coefficient of variation; run spread is max(invocation medians) / min(invocation medians) - 1.

These are local before/after results motivated by the Zen 5 benchmark; they are not new Zen 5 or vendor-relative measurements. The Zen 5 host was busy and was not timed in this round.

The retained Clang/AVX2 review comparison used -std=c++14 -O3 -DNDEBUG -mavx2 -mfma with the same benchmark source, machine, affinity, A/B/A/B order and repetition settings. Its rebuilt before and after executables, with the explicit generic x86-64 flags above, have identical SHA-256 hashes to that earlier pair. All four overall comparisons and the other three review comparisons are fresh.

Fresh SSE2 validation also passed trsm_packet and trsm_packet_cancellation with GCC 13.3 and Clang 18.1, both scalar parts, r2 s42. The AVX2 correctness results below apply to the same unchanged candidate source.

The new regression explicitly exercises the shared implementation. Its controlled cache settings keep cancelling terms within a diagonal panel: the harness’s synthetic one-row AVX-512 double panels instead split cancellation across separate GEMM updates, which also fails before this MR. Existing small-block tests remain enabled.

Validation passed with both GCC and Clang: six CMake targets per compiler (trsm_packet, cancellation, and fast-math cancellation, each split by scalar), plus C++14 -O0, triangular and Cholesky integration checks including complex scalars. Additional checks cover SSE2, scalar-only, an eight-register budget, alternate layout/index settings, and ASan/UBSan. Cancellation cases cover upper/lower, unit/nonunit, left/right, packet tails, mixed lanes, poisoned unused coefficients, and orders 4–256. The regression fails at the previous MR revision. Shared-kernel AVX-512 passed natively on AMD RYZEN AI MAX+ 395 with GCC and Clang; AArch64 NEON passed under QEMU with GCC 14. No timings were taken on that loaded remote machine. Other ISAs and MSVC were not validated in this round.

Disassembly of the current AVX2 binaries confirms integer exponent checks in the final-row guard (scalar integer checks with GCC, packed integer masks/comparisons with Clang). The guard and retry stay outside the substitution loops. This adds work for row-major triangles, most visible in tiny solves; measured costs are included above.