Eigen vs. BLAS/LAPACK libraries, Apple M4 with SME (October 2026)

The operations and shapes of the Zen 5 page measured on an Apple M4, with Eigen compiled for that CPU (-mcpu=apple-m4), which selects its SME kernels, against the three BLAS/LAPACK libraries available on macOS: Apple Accelerate, Arm Performance Libraries and OpenBLAS. The NEON page has the same cells with Eigen built without SME, at the same Eigen master.

This page was first published with a measurement of 2026-09-19 at Eigen master 46a2d0c56 on macOS 27.0, and updated on 2026-10-02 to a re-measurement at master 3cf994918 on macOS 27.0.1. It now shows a full re-measurement of every library and of Eigen at master 674e4a4e5, and adds five operations: TRMM, SYMM, PSTRF, SYTRF and GELS. Against the 2026-10-02 measurement, on the cells both share, Eigen’s rates repeat within 7 % (geometric mean per operation and scalar) except GEEV, 49 % (float) and 71 % (double) faster, and GEQRF, 11 to 14 % faster. The earlier versions of the page and their result files are in the history of the website repository.

Conditions

GEMM at n = 1024 ran at 431 GFLOP/s in double and 1671 in float in Eigen, against 385 and 1668 for Accelerate, 369 and 715 for Arm Performance Libraries, and 57 and 114 for OpenBLAS.

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 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 86 % of the cells and above 10 % for 3 %.

Each run covered one operation against one library. A run started only once the machine had been idle (other processes using less than 0.35 of a core over two 5 s windows, no keyboard or mouse input for 10 minutes) with no thermal or performance warning from pmset, and was discarded and repeated if the whole machine averaged 1.45 busy cores or more during it, or reached 3.0 in any 5 s window; 16 attempts were discarded this way. The CI virtual machine on the host was paused for the whole session. Background services of the logged-in sessions (malware scans, search and photo indexing, mail) kept the machine near those limits at times.

The limits were looser for the first hours. The session started with a rejection limit of 1.35 busy cores on average and 2.0 at the peak, the limits of the September measurement. Another account’s background session on the machine (0.35 to 1.0 busy cores while idle) kept the gate from opening, so the idle threshold at the start of a run was raised to 0.6 of a core from 13:55 on 2026-10-25, and the rejection limits to 1.45 and 3.0 from 14:29, then to 1.6 and 4.5 from 15:17, until the console applications of both accounts were quit at 15:37; the original idle threshold and the limits of 1.45 and 3.0 were then restored for the rest of the session. Three AXPY runs were measured in that window. The Accelerate run is attempt 3 of 7: the attempts agree cell by cell (median difference 0.7 % or less, 90th percentile 3.2 %), and attempt 3, with the lowest peak (2.18 busy cores, mean 1.43), was kept; the runs against Arm Performance Libraries (mean 1.44, peak 3.38) and OpenBLAS (mean 1.42, peak 2.57) were accepted on their first attempts.

For the eigenvalue and singular value decompositions the operation 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.

The dotted vertical lines on the charts mark the sizes at which the operands of the call, inputs and outputs together, fill a performance core’s L1 data cache (128 KiB) and the L2 cache that the four performance cores share (16 MiB); the M4 has no L3.

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, so the ratios at the small end are ratios of fixed costs. The DOT cells in double at n = 2 show it directly: 0.73 ns per call for Eigen against 2.7 ns for Accelerate, 4.9 ns for Arm Performance Libraries and 12.4 ns for OpenBLAS. Eigen still computes the whole operation on every call; nothing is hoisted out of the timed loop.

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. Eigen runs it on SME, except for small blocks, which take a NEON path.

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, where a call of SelfAdjointEigenSolver takes 2.9 s in double.

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 2.0 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

• Against Accelerate. Eigen’s SME GEMM matches Accelerate’s (1.04 to 1.16 in the bands from n = 192 in double) and is ahead of Arm Performance Libraries (1.2 to 1.5 from n = 192 in double, and 2.3 times in float at n = 1024). The vector operations favor Accelerate at the larger sizes: AXPY 0.41 and 0.32 in double from n = 192, DOT 0.58 from n = 1536. Eigen trails Accelerate on SYRK (0.38 to 0.40 up to n = 128), on GESDD (0.70 to 0.82 from n = 24) and on TRMM at every size (0.25 to 0.83).
• The new operations. PSTRF, SYTRF and GELS are ahead of all three libraries in every band: PSTRF 1.6 to 3.9, SYTRF 1.2 to 2.5 and GELS 1.3 to 4.6 times in double. SYMM is level with Accelerate from n = 192 (0.99 to 1.15). TRMM is 0.6 to 0.8 times Accelerate and 4.5 to 5.8 times Arm Performance Libraries and OpenBLAS above n = 192.
• OpenBLAS runs the Homebrew build’s vortexm4 kernels without SME; Eigen’s SME level-3 kernels are 5 to 8 times faster in double for GEMM, SYMM, SYRK and TRMM above n = 192.
• GEEV trails all three libraries at n = 192 to 1024 (0.51 to 0.61 in double); it leads at the small end.

Caveats

• macOS cannot bind a thread to a core, so a single-threaded rate is that of whichever core the scheduler chose; a busy foreground thread normally stays on a performance core.
• The machine is a desktop with logged-in sessions and their background services. Runs that they disturbed beyond the limits above were repeated; smaller disturbances remain in the numbers, and the first three AXPY runs were measured under the looser limits described above.
• Eigen’s SME kernels cover matrix products and, within size ranges set from its cache-size estimates, real DOT, AXPY and column-major GEMV. TRSM, TRMM, SYMM, SYRK and the blocked factorizations reach SME through their matrix-product updates; everything else is NEON. Eigen reads its L2 estimate from the operating system, which on this machine reports the efficiency cores’ 4 MiB L2, so the vector kernels hand back to NEON at 4 MiB per operand rather than at the performance cores’ 16 MiB.
• Accelerate is identified by the macOS build, since the framework reports no version of its own.
• 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.