Eigen vs. BLAS/LAPACK libraries, AMD Zen 5 limited to AVX2 (October 2026)

The cells of the AVX-512 page measured again with Eigen compiled for AVX2 and FMA and every library held to its AVX2 kernels, on the same machine: what the comparison looks like on a CPU without AVX-512.

This page was first published with a measurement of 2026-09-14 to 2026-09-16 at Eigen master 15f227178 on Ubuntu 24.04. It now shows a full re-measurement of Eigen and of every library, taken after the host moved to Ubuntu 26.04 with GCC 15.2, at Eigen master 674e4a4e5, and adds five operations: TRMM, SYMM, PSTRF, SYTRF and GELS. Against September’s last Eigen measurement, on the cells both pages share, Eigen’s AVX2 rates repeat within 8 % (geometric mean per operation and scalar) except AXPY in float (18 % faster), GEQRF (11 % faster), GETRF (8 to 18 % faster), GEEV (17 to 31 % faster) and TRSM (50 % faster in double, 78 % in float). The September version of the page and its result files are in the history of the website repository.

Conditions

The OpenBLAS and AOCL runs were repeated: a first pass of the 34 runs of 2026-10-25 and 2026-10-26 did not set the two switches above, and OpenBLAS reported its Cooperlake kernel target in it. Its results are not used. In the repeat the limits took effect: GEMM in double at n = 1024 ran at 74 and 75 GFLOP/s in OpenBLAS and AOCL, against 73 for Eigen; OpenBLAS reports Zen.

How to read the numbers. Every rate is nominal GFLOP/s: the textbook operation count of the operation, stated in each section below, divided by the median wall time of one call. Both arms of a comparison are built into the same binary and run on the same freshly randomized operands, and the reference library’s result is validated against Eigen’s before it is timed. A ratio divides Eigen’s rate by the library’s rate from the same binary and the same run. Each of the three binaries, one per library, also measured every Eigen cell, so Eigen was measured three times, the oneMKL pass a day apart from the other two: the Eigen line on each chart is the median of the three passes and the shaded band is their spread, the run-to-run noise on this host. The spread is within 3 % for 81 % of the cells and above 10 % for 4 %. In September a few level-3 cells ran at one of two rates up to 2.5 times apart, depending on the binary (SYRK and GEMM at n = 192, and in float at n = 256; see libeigen/eigen#3201); in this measurement no GEMM or SYRK cell differs by more than 25 % between the three passes. For the eigenvalue and singular value decompositions the count is the one of the classical algorithm, so a divide-and-conquer or blocked implementation is credited with work it does not do; the ratios between arms are unaffected.

Each run covered one operation against one library and ran under a fleet-quiet lease that kept other work off the host. It started only after the host passed a gate (no compilers or containers, the guest’s busy-core count below 0.5 and its one-minute load below 0.9, the Windows host’s CPU load at most 8 %) and was discarded and repeated if the guest averaged 1.8 busy cores or more in any 3 s sample, or a container started. The Windows host’s load is invisible inside the guest.

The dotted vertical lines on the charts mark the sizes at which the operands of the call, inputs and outputs together, fill the core’s L1 data cache (48 KiB), its L2 cache (1 MiB) and the L3 cache as the guest sees it (32 MiB).

AXPY

y += alpha * x against ?axpy, alpha = 0.75; (2n) flops. Lengths run from 2 to (2^{24}) (128 MiB per double vector), through every cache level into DRAM.

Short vectors. Up to about n = 32 the AXPY and DOT rates measure the cost of a call rather than the kernel. Eigen’s expression compiles inline into the calling loop, while a call into a shared library pays a cost of its own before it reads an element. That cost is fixed per call while the cost per element is similar in every arm, so the ratios at the small end are ratios of fixed costs. The DOT cells in double at n = 2 show it directly: 0.52 ns per call for Eigen against 4.0 ns for OpenBLAS, 3.9 ns for oneMKL and 8.3 ns for AOCL. Eigen still computes the whole operation on every call; nothing is hoisted out of the timed loop.

AXPY’s lead is smaller because every call adds into the y written by the call before it, and reading that memory back puts a floor under every arm: up to n = 32, Eigen’s AXPY holds at 2.6 to 3.0 ns per call where n is a multiple of the vector width, 4 in double and 8 in float, and at 4.2 ns where it is not and the remaining elements are written and reloaded through masked moves, while a September test on this machine that cycled the same kernel through 64 vectors instead of one measured 0.8 to 1.4 ns.

DOT

x.dot(y) against ?dot; (2n) flops, the same lengths. Real scalars only: the Fortran convention for returning a complex function value is not fixed by the platform ABI and differs between library builds, so ?dotc cannot be declared portably.

Up to about n = 32 the rates measure the cost of a call; see AXPY.

GEMV

y.noalias() += A * x against ?gemv, column-major, no transposition; (2mn) flops.

Square matrices, then the short-fat (n = 10000, m grows) and tall-skinny (m = 10000, n grows) sweeps.

GEMM

C.noalias() += A * B against ?gemm, column-major, no transposition; (2mnk) flops.

Square matrices first; then the two tall-skinny sweeps, where m stays at 4096 or 10000 while n = k grows; then the assorted non-square shapes that fit neither.

TRSM

A.triangularView<Lower>().solveInPlace(X) against ?trsm with the m-by-m lower triangle on the left and n right-hand sides, no transposition, non-unit diagonal; (m(m+1)n) flops. The solve is destructive, so the timed region includes one copy of B for both arms.

Square shapes (m = n), then the sweeps with a fixed triangle of order 1024 or 4096 and a growing number of right-hand sides, then the remaining shapes with 4096 right-hand sides against a small triangle.

TRMM

X.noalias() = A.triangularView<Lower>() * B against ?trmm with the m-by-m lower triangle on the left and n columns in B, no transposition, non-unit diagonal; (m(m+1)n) flops, TRSM’s count. ?trmm overwrites B, so the reference arm copies B into the destination and multiplies in place; the Eigen arm writes the product to the destination, which noalias() zero-fills before accumulating. Each arm makes the one pass over the destination its interface requires, and neither allocates inside the timed region.

The same shapes as TRSM: square, then the sweeps with a fixed triangle of order 1024 or 4096 and a growing number of columns, then the remaining shapes with 4096 columns against a small triangle.

SYMM

C.noalias() += A.selfadjointView<Lower>() * B against ?symm (?hemm for a complex scalar) with the m-by-m symmetric matrix stored in its lower triangle, on the left, and n columns in B; (2m^2n) flops, the count of the general product of the same shape. C accumulates across iterations and is never reset, as in GEMM.

The same shapes as TRSM.

SYRK

C.selfadjointView<Lower>().rankUpdate(A) against ?syrk, lower triangle, no transposition, with C n-by-n and A n-by-k; (n(n+1)k) flops.

Square shapes (n = k), then the sweeps: a 4096-by-4096 matrix updated by a growing number of columns, and the Gram matrix of a short, wide factor with 4096 columns.

POTRF

LLT<Ref<MatrixXd>> in place against ?potrf, lower triangle, symmetric positive definite operand; (n^3/3) flops. The timed region includes one copy of the operand for both arms, since the factorization is destructive.

PSTRF

LDLT<Ref<MatrixXd>> in place against ?pstrf, lower triangle, symmetric positive definite operand; (n(n-1)(n-2)/3 + 2n(n-1)) flops for both, (n^3/3) to leading order. The two compute different factors of the same pivoted matrix, (P^T A P = L L^T) for LAPACK and (L D L^T) for Eigen, and pivot on the same entry, the largest diagonal element of the remaining Schur complement. Both arms copy the operand into a destination allocated once and factorize the copy, as GETRF does. LDLT::compute also evaluates the operand’s L1 norm for rcond(), which ?pstrf does not. Each result is checked by applying the stored factors to a random vector and comparing with (Ax).

SYTRF

BunchKaufman<Ref<MatrixXd>> in place against ?sytrf (?hetrf for a complex scalar), lower triangle, symmetric indefinite operand; (n(n-1)(n-2)/3 + 2n(n-1)) flops for both, (n^3/3) to leading order. Both arms copy the operand into a destination allocated once and factorize the copy, as GETRF does; pivot and workspace arrays are set up once per shape. Each result is checked by applying the stored factors, with the pivot blocks of D, to a random vector and comparing with (Ax).

GETRF

PartialPivLU<Ref<MatrixXd>> in place against ?getrf, square operand; (2n^3/3) flops. Both arms copy the operand into a destination allocated once and factorize the copy; neither allocates inside the timed region.

GEQRF

HouseholderQR<Ref<MatrixXd>> in place against ?geqrf, m ≥ n; (2mn^2 - 2n^3/3) flops, the same copy-then-factorize protocol as GETRF.

Square shapes, then the tall-skinny sweeps with m fixed at 4096, 10000 and 100000.

GELS

Full-rank least squares (\min \lVert AX - B \rVert) through Householder QR with A m-by-n, m ≥ n, and B m-by-nrhs: HouseholderQR<Ref<MatrixXd>> followed by qr.solve(B) against ?gels (trans = 'N'); (2mn^2 - 2n^3/3) flops for the factorization plus (\text{nrhs},(4mn - n^2)) for applying (Q^T) and solving with R. ?gels overwrites A and B, so the reference arm copies both into destinations allocated once. The Eigen arm factorizes a copy of A through the in-place HouseholderQR and writes qr.solve(B) to a preallocated X; solve() copies B into a temporary it allocates, which is the cost of the expression a user writes, so it is timed. The solution is checked through the normal equations on a well-conditioned operand.

Square shapes with one right-hand side, then the tall-skinny sweeps with m fixed at 4096, 10000 and 100000, then right-hand-side sweeps: 16 to 256 columns of B at m = n = 1024 and 16 to 1024 at 4096 by 256.

SYEV

SelfAdjointEigenSolver with eigenvectors against ?syev (jobz = 'V', lower triangle), the like-for-like pair: tridiagonalization followed by implicit QR iteration on both sides; (9n^3) flops, the count of that algorithm (Golub & Van Loan, section 8.3.3). Neither works in place: the solver copies the operand internally and the reference arm pays the same one copy. Sizes to 2048.

GESDD

BDCSVD<MatrixXd, ComputeThinU | ComputeThinV> against ?gesdd with jobz = 'S', divide and conquer on both sides, m ≥ n, the thin U and V computed; (14mn^2 + 8n^3) flops, the Golub-Reinsch count (Golub & Van Loan, table 8.6.1). Square sizes to 2048, then tall-skinny sweeps with m fixed at 4096 and 10000.

GEEV

EigenSolver with eigenvectors against ?geev (jobvr = 'V'), a dense random nonsymmetric operand: Hessenberg reduction, QR iteration to the Schur form and back-substitution for the eigenvectors on both sides; (25n^3) flops, the count of the Hessenberg QR algorithm with Schur vectors (Golub & Van Loan, section 7.5.6). Sizes to 1024, where a call of EigenSolver takes 2.2 s in double.

Summary

Geometric mean of the Eigen ÷ library rate ratio over the square shapes in each size band. Above 1, Eigen is faster. Only the vector operations reach the last band.

Observations

• Level with the capped libraries on the large products. GEMM is 0.97 to 1.04 times OpenBLAS, oneMKL and AOCL in double from n = 192, TRSM 0.94 to 1.16 times OpenBLAS and AOCL, and SYMM and SYRK 0.94 to 1.09 against OpenBLAS and AOCL. oneMKL, on its generic path, is 1.1 to 2.4 times behind Eigen on TRSM, TRMM and SYRK.
• TRMM holds its rate with AVX2. Eigen runs at 62 to 68 GFLOP/s from m = 1024 to 4096 (0.88 to 0.98 times OpenBLAS), where the AVX-512 build falls from 128 to 37 GFLOP/s between m = 1024 and 4096 (see the AVX-512 page).
• The new factorizations are ahead of all three libraries from n = 24: SYTRF 1.02 to 1.7, PSTRF 1.18 to 1.71 and GELS 1.07 to 2.27 times in double.
• GEMV trails the libraries above n = 1536 (0.75 to 0.94), and GEEV at n = 192 to 1024 (0.68 to 0.80), as it did in September; GESDD is 0.76 to 0.85 at n = 24 to 128.

Caveats

• Linux runs under WSL2, so absolute rates are a lower bound on bare-metal Linux.
• oneMKL as shipped does not use its AVX-512 kernels on AMD CPUs, so its results here are those of its generic path, as on the AVX-512 page.
• The libraries’ AVX2 limits are their own documented environment switches; a library may still use instructions outside AVX2 and FMA that those switches do not name.
• Ubuntu 26.04’s GCC 15.2 targets x86-64-v3 by default, which already includes AVX2 and FMA.
• The OpenBLAS LAPACK arm is the reference LAPACK that OpenBLAS bundles, on top of OpenBLAS’s BLAS kernels; LAPACK ratios compare Eigen’s factorizations with different libraries’ blocking as well as their kernels.
• PSTRF compares two algorithms that pivot on the same entry but produce different factors, (L L^T) and (L D L^T), and charges both the same flop count.

Full tables

GFLOP/s per shape and library, with the Eigen ÷ library ratio. The Eigen column is the median over its three passes; each ratio is formed within the library’s own binary and run.