Eigen vs. BLAS/LAPACK libraries, Raspberry Pi 5 (October 2026)

Eigen’s dense kernels measured head-to-head against two BLAS/LAPACK libraries on a Raspberry Pi 5, one thread, in 64-bit mode: the operations and shapes of the Zen 5 page, with Eigen built for the Armv8-A NEON baseline, against OpenBLAS and Arm Performance Libraries. This is the first measurement of a Raspberry Pi on this site. The same cells in 32-bit mode, against OpenBLAS, are on the 32-bit page.

Conditions

GEMM at n = 1024 ran at 16.0 GFLOP/s in double and 33.1 in float in Eigen, against 17.0 and 32.8 for OpenBLAS, and 15.9 and 32.8 for Arm Performance Libraries.

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 two binaries, one per library, also measured every Eigen cell, so Eigen was measured twice: the Eigen line on each chart is the median of the two passes and the shaded band is their spread, the run-to-run noise on this board: within 3 % for 92 % of the cells and above 10 % for 1 %. 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, under an exclusive benchmark lock that keeps other timing jobs off the board. It started only after an idle gate (no compilers, one-minute load below 0.5, less than 0.2 of a core busy, SoC temperature below 70 °C) and was discarded and repeated if the whole machine reached 1.8 busy cores in any 4 s sample, or if the firmware’s throttle flags (vcgencmd get_throttled) changed during the run: 2 of 36 attempts were discarded, one for a disturbance (AXPY against OpenBLAS) and one for another (SYEV against Arm Performance Libraries). The flags read 0x0, no under-voltage and no throttling, before and after every accepted run. The SoC temperature peaked at 71.6 °C during the session.

The dotted vertical lines on the charts mark the sizes at which the operands of the call, inputs and outputs together, fill a core’s L1 data cache (64 KiB), its L2 cache (512 KiB) and the shared L3 cache (2 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, so the ratios at the small end are ratios of fixed costs. The DOT cells in double at n = 2 show it directly: 1.98 ns per call for Eigen against 8.3 ns for OpenBLAS and 19.2 ns for Arm Performance Libraries. 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.

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 29.3 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 11.1 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 both libraries on the large level-3 products. GEMM is 0.94 to 1.05 times OpenBLAS and Arm Performance Libraries in double from n = 192, TRSM 0.84 to 1.10 and POTRF 0.92 to 1.12. Against OpenBLAS, GETRF is 0.86 to 0.87 above n = 192; against Arm Performance Libraries it is 1.02 to 1.38.
• Eigen leads at the small end. Up to n = 128 it is 1.1 to 5.8 times faster on AXPY and DOT and 1.5 to 4.6 times on GEMV and TRSM.
• GEMV falls behind once the matrix leaves the L2 cache. Eigen’s rate drops from 6.1 GFLOP/s at n = 256 (0.5 MiB) to 2.8 at 512 and 1.1 at 1536, while OpenBLAS holds 4.7 and then 3.6 to 3.3 up to n = 4096: ratios of 0.61 at n = 512 and about 0.3 from n = 768. Arm Performance Libraries sits between the two (1.5 to 2.5 GFLOP/s from n = 1536).
• TRMM and SYMM are 0.75 to 0.88 times both libraries above n = 192.
• The new factorizations are ahead of OpenBLAS’s C LAPACK and level with or ahead of Arm Performance Libraries: PSTRF 1.1 to 1.6, SYTRF 0.96 to 1.5 and GELS 1.1 to 1.7 in double.
• SYEV, GESDD and GEEV trail both libraries at the larger sizes: SYEV 0.74 (OpenBLAS) and 0.87 (Arm Performance Libraries) above n = 1536, GESDD 0.77 to 0.80 against OpenBLAS from n = 24 and 0.65 to 0.81 against Arm Performance Libraries to n = 1024, and GEEV 0.45 at n = 192 to 1024.

Caveats

• The Raspberry Pi 5 has 4 KiB pages and a small shared last-level cache (2 MiB), so rates at sizes beyond a few megabytes are largely set by the memory system; compare arms measured in the same session.
• OpenBLAS was built without a Fortran compiler, so its LAPACK routines are the C translation of the reference LAPACK rather than the Fortran one the other pages’ OpenBLAS builds use; BLAS kernels are unaffected. LAPACK ratios against OpenBLAS therefore compare Eigen with that translation.
• The board runs the performance governor at 2.4 GHz with an active cooler; its SoC peaked at 71.6 °C and the firmware reported no throttling or under-voltage during any accepted run.
• 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.