Eigen vs. BLAS/LAPACK libraries, Apple M4 with SME (September 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 kernel for matrix products, against the three BLAS/LAPACK libraries available on macOS: Apple Accelerate, Arm Performance Libraries and OpenBLAS.

Conditions

GEMM at n = 1024 ran at 382 GFLOP/s in double and 1513 in float in Eigen, against 362 and 1665 for Accelerate, 364 and 712 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 87 % of the cells and above 10 % for 3 %; the widest are at the small end, such as SYRK in double at n = 24 (10.8 to 17.6 GFLOP/s across the three binaries), and in DOT, which at n = 2048 in double ran at 19.3 GFLOP/s in two binaries and 14.3 in the third.

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) with no thermal or performance warning from pmset, and was discarded and repeated if the whole machine averaged 1.35 busy cores or more during it, or reached 2.0 in any 5 s window; 5 of 41 runs were repeated. Spotlight indexing was switched off for the session.

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.

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 library call goes through the shared library’s symbol stub, the Fortran interface’s argument checks and a runtime kernel dispatch 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: 0.49 ns per call for Eigen against 2.7 ns for Accelerate, 5.0 ns for Arm Performance Libraries and 12.5 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. This is the one operation Eigen runs on SME here.

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.

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.

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.

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 2.8 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 3.4 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.

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 a logged-in session and its background services. Runs that they disturbed beyond the limits above were repeated; smaller disturbances remain in the numbers.
• Eigen has an SME kernel for GEMM only. TRSM, SYRK and the blocked factorizations reach it through their matrix-product updates; the vector and matrix-vector kernels are NEON.
• Accelerate is identified by the macOS build, since the framework reports no version of its own.

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.