Benchmarks

linalg benchmarks

The generated measurement tables for the linalg module — every case, the harness that measured it, and how to reproduce it.

linalg goes head-to-head with NumPy, whose array ops dispatch to BLAS/LAPACK — hand-tuned, vectorized, often multi-threaded Fortran. numpy_compare.py feeds the same fixed-seed, well-conditioned matrix to both sides, runs the same op many times with the result consumed, and flags any disagreement in the answers.

Read the stamp before the numbers: NumPy's speed depends on which BLAS it links, so the stamp records the resolved library, not the NumPy version. Here that is libblas.so.3 → the reference implementation, not OpenBLAS or MKL — a tuned BLAS would narrow the large-n rows. The small-n wins come from paying no per-call Python and dispatch overhead, which no choice of BLAS changes. The Eigen comparison is a separate Google Benchmark harness (eigen_compare_bench.cpp) with both sides compiled C++ on one thread; the two tables never share a column. Methodology and the reference machine: docs/performance.md.

vs NumPy

Measured by scripts/numpy_compare.py; reproduce with python3 scripts/numpy_compare.py --suite linalg --md docs/bench/linalg-vs-numpy.md.

op	operand dimensions	cheatah	NumPy	winner	band
`matmul`	4	0.09	0.79	cheatah 8.3x	7.20-8.48
`matmul`	16	0.54	2.24	cheatah 4.1x	3.56-4.29
`matmul`	32	3.22	13.43	cheatah 4.2x	3.94-4.60
`matmul`	64	25.60	116.37	cheatah 4.5x	4.30-4.71
`matmul`	96	90.65	318.41	cheatah 3.5x	3.38-3.63
`solve`	4	0.22	2.45	cheatah 11.0x	10.03-11.49
`solve`	16	1.06	4.34	cheatah 4.1x	3.81-4.35
`solve`	32	3.69	10.35	cheatah 2.8x	2.55-2.85
`solve`	64	18.02	50.82	cheatah 2.8x	2.72-3.10
`det`	4	0.11	2.05	cheatah 22.6x	15.44-28.51
`det`	16	0.81	3.76	cheatah 4.7x	3.94-5.39
`det`	32	2.80	9.58	cheatah 3.4x	3.18-3.53
`det`	64	14.17	47.05	cheatah 3.4x	3.22-3.51
`inv`	4	0.26	2.18	cheatah 8.2x	7.66-9.08
`inv`	16	1.56	6.35	cheatah 4.1x	3.18-4.40
`inv`	32	6.97	25.27	cheatah 3.6x	3.51-3.65
`inv`	64	46.20	142.12	cheatah 3.1x	2.95-3.51
`eigvalsh`	2	0.22	2.01	cheatah 9.2x	8.95-9.74
`eigvalsh`	3	0.48	2.35	cheatah 4.9x	4.67-5.01
`eigvalsh`	4	0.62	2.66	cheatah 4.3x	3.88-4.37
`eigvalsh`	6	1.43	3.49	cheatah 2.4x	2.34-2.48
`eigvalsh`	8	1.91	4.20	cheatah 2.2x	2.10-2.23
`eigvalsh`	16	6.73	9.20	cheatah 1.4x	1.32-1.39
`eigvalsh`	32	26.01	25.98	cheatah 1.0x	0.95-1.04
`eigvalsh`	64	106.80	149.84	cheatah 1.4x	1.26-1.43
`dot`	64	0.02	0.62	cheatah 34.4x	28.33-46.46
`dot`	1024	0.09	1.08	cheatah 11.6x	7.28-17.05
`dot`	16384	2.59	8.16	cheatah 3.1x	2.77-3.54
`ndarray.sqrt`	64	0.19	0.88	cheatah 4.7x	3.07-4.97
`ndarray.sqrt`	1024	0.96	1.77	cheatah 1.8x	1.52-2.09
`ndarray.sqrt`	16384	14.58	14.66	NumPy 1.0x	0.95-1.04
`ndarray.exp`	64	0.21	1.05	cheatah 5.0x	3.83-5.62
`ndarray.exp`	1024	1.00	4.03	cheatah 4.0x	2.52-4.76
`ndarray.exp`	16384	13.52	48.89	cheatah 3.6x	3.40-3.92
`ndarray.sin`	64	0.22	1.11	cheatah 5.0x	4.08-6.13
`ndarray.sin`	1024	0.95	5.52	cheatah 5.9x	4.23-6.25
`ndarray.sin`	16384	13.69	89.32	cheatah 6.5x	5.68-6.84
`ndarray.add`	64	0.11	0.62	cheatah 5.5x	4.58-6.05
`ndarray.add`	16384	2.48	2.88	cheatah 1.2x	0.78-1.23
`cholesky`	8	0.31	2.63	cheatah 8.4x	7.20-9.79
`cholesky`	32	3.77	7.35	cheatah 2.0x	1.68-2.15
`cholesky`	64	16.81	25.99	cheatah 1.5x	1.35-1.78
`qr`	8	1.07	8.88	cheatah 8.3x	7.53-8.64
`qr`	32	14.79	28.49	cheatah 2.0x	1.72-2.16
`qr`	64	102.62	137.47	cheatah 1.3x	1.16-1.36
`svdvals`	8	3.28	6.15	cheatah 1.9x	1.70-1.93
`svdvals`	32	49.70	42.83	NumPy 1.2x	0.82-0.96
`svdvals`	64	229.46	208.75	NumPy 1.1x	0.88-0.97
`svd (full)`	8	3.93	10.53	cheatah 2.7x	2.51-2.85
`svd (full)`	32	70.11	115.62	cheatah 1.6x	1.50-1.65
`svd (full)`	64	433.35	739.34	cheatah 1.7x	1.50-1.76
`pinv`	8	4.63	18.86	cheatah 4.1x	3.37-4.19
`pinv`	32	100.36	136.64	cheatah 1.4x	1.31-1.47
`pinv`	64	633.48	800.84	cheatah 1.3x	1.23-1.31
`cond`	8	3.23	10.21	cheatah 3.2x	3.07-3.31
`cond`	32	49.56	45.71	NumPy 1.1x	0.89-0.97
`cond`	64	217.89	195.27	NumPy 1.1x	0.89-0.95
`matrix_rank`	8	2.98	11.98	cheatah 4.0x	3.86-4.16
`matrix_rank`	32	46.95	51.17	cheatah 1.1x	0.98-1.12
`matrix_rank`	64	216.33	194.85	NumPy 1.1x	0.88-0.93
`slogdet`	8	0.20	3.15	cheatah 15.2x	11.78-17.67
`slogdet`	32	2.93	10.48	cheatah 3.6x	3.45-3.71
`slogdet`	64	14.03	48.95	cheatah 3.5x	2.20-3.56
`eigh`	8	2.48	6.52	cheatah 2.6x	2.32-4.14
`eigh`	32	37.96	64.51	cheatah 1.7x	1.62-1.72
`eigh`	64	238.38	401.26	cheatah 1.7x	1.61-1.72
`eigvals`	8	7.88	14.11	cheatah 1.8x	1.73-1.89
`matrix_power`	8	0.91	2.49	cheatah 2.7x	2.60-2.79
`matrix_power`	32	12.27	28.00	cheatah 2.3x	1.96-2.38
`matrix_power`	64	97.93	214.16	cheatah 2.2x	2.12-2.40
`trace`	32	0.01	1.20	cheatah 168.1x	120.84-178.23
`trace`	256	0.08	1.47	cheatah 20.2x	17.90-22.16
`norm(matrix)`	32	0.09	1.55	cheatah 17.8x	10.01-24.71
`norm(matrix)`	256	7.38	32.35	cheatah 4.4x	3.92-4.52
`outer`	64	0.34	4.05	cheatah 11.9x	7.13-13.10
`outer`	256	9.78	48.77	cheatah 5.0x	4.15-5.25
`kron`	8	1.71	13.12	cheatah 7.8x	7.24-8.74
`kron`	16	20.25	72.57	cheatah 3.6x	3.20-4.17
`kron`	32	347.09	950.98	cheatah 2.7x	2.49-2.81

vs Eigen

Measured by tests/benchmarks/eigen_compare_bench.cpp; reproduce with scripts/bench_run.sh publish linalg-vs-eigen.

case	cheatah	spread	vs	rival	spread	ratio	verdict
BM_dot/16384	1.96 µs	±249.61 ns IQR	eigen	2.58 µs	±71.26 ns IQR	1.32x	faster
BM_dot/64	11.66 ns	±1.57 ns IQR	eigen	9.08 ns	±0.37 ns IQR	0.78x	slower
BM_inv/32	6.82 µs	±1.34 µs IQR	eigen	11.86 µs	±133.38 ns IQR	1.74x	faster
BM_inv/64	40.48 µs	±8.56 µs IQR	eigen	67.73 µs	±1.32 µs IQR	1.67x	faster
BM_matmul/32	3.23 µs	±1.23 µs IQR	eigen	3.90 µs	±115.58 ns IQR	1.21x	faster
BM_matmul/96	58.11 µs	±32.71 µs IQR	eigen	94.56 µs	±2.47 µs IQR	1.63x	faster
BM_solve/32	3.59 µs	±175.70 ns IQR	eigen	4.49 µs	±71.85 ns IQR	1.25x	faster
BM_solve/64	16.73 µs	±1.91 µs IQR	eigen	20.77 µs	±525.30 ns IQR	1.24x	faster

Tally (a difference counts only above 1.15x AND 0.25 ns) — vs eigen: 7 faster / 0 parity / 1 slower.

Loss vs eigen: BM_dot/64 — cheatah 11.66 ns vs 9.08 ns (1.28x slower)

What the two tables say

vs NumPy/LAPACK cheatah wins across nearly the whole library — the products, the LU family, the full SVD, the symmetric eigensolver, outer, qr, kron and norm. The rows still behind are the SVD-threshold queries — svdvals and cond at 32 and 64, and matrix_rank at 64 — where LAPACK's bidiagonal solver edges it, and one elementwise tie.
vs Eigen 3.4 on one core cheatah leads inv, matmul, solve and the large dot, and loses the 64-element dot, where Eigen's kernel is the tighter one. Those four routines are the whole published Eigen suite, and its tally is its own — the two tables never divide into each other.

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