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Benchmarks: what we measure and what it means

Source: vignettes/articles/benchmarks.Rmd
benchmarks.Rmd

This is a pkgdown-only article, not a CRAN vignette. It performs no timing at build time: every number below is read from result directories stored in the repository under inst/benchmarks/results/, produced by the benchmark suite inst/benchmarks/run-suite.R. Rebuilding the site never changes a number; adding a result directory does.

What is measured and why

Each case is a deterministic problem (fixed generator and seed) with a wanted set: the k largest or smallest eigenvalues, the k eigenvalues of largest modulus, the k eigenvalues nearest a shift, the k largest singular triplets, or the full spectrum. Every method (eigencore, RSpectra, irlba, PRIMME when installed, and dense base R/LAPACK where it fits in memory) is run on the same matrix with the same requested tolerance (tol = 1e-8) and, for each case and method, the suite records:

  • wall time: one untimed warm-up call, then the median and minimum over reps timed calls (5 in the standard profile). eigencore’s time always includes building its certificate; the other packages do no certification.
  • operator applications (matvecs): the hardware-independent work each package reports about itself: eigencore’s work() record (operator_columns + adjoint_columns of the solve phase), RSpectra’s nops, irlba’s mprod, PRIMME’s numMatvecs. For shift-invert cases an application is a linear solve. eigencore’s explicit-Gram SVD paths do not apply the operator column by column, so they report no comparable count.
  • memory: peak resident-set growth of the process during the warm-up call (Linux, via VmHWM), which includes C/C++ heaps, and the R heap high-water mark. Both are in results.csv; neither is a cross-package claim.
  • accuracy, computed by the suite itself for every method’s output, defined in the next section, plus eigencore’s own certificate verdict.

Threads are a property of the run: each thread setting runs in a fresh R process with OPENBLAS_NUM_THREADS, OMP_NUM_THREADS, and options(eigencore.threads) all set to the same value, so “4 threads” means the whole process (BLAS, LAPACK and eigencore’s OpenMP kernels) had four. RSpectra and irlba are effectively single-threaded on sparse inputs, so their 4-thread rows mostly measure noise.

The independent accuracy check

The suite does not trust any solver’s own residuals. For every returned pair it recomputes

  • the residual A x - lambda B x (B = I for standard problems), and for SVDs both sides, sqrt(||A v - sigma u||^2 + ||A' u - sigma v||^2);
  • the normwise 2-norm backward error ||r|| / ((||A||_2 + |lambda| ||B||_2) ||x||) (SVD: / ||A||_2). This is the definition eigencore’s own certificates use since the C12 change, but the suite divides by a high-accuracy ||A||_2: exact when the spectrum is known or a dense factorisation is affordable, otherwise a converged Golub-Kahan-Lanczos estimate with full reorthogonalisation, computed once per case and cached. (eigencore’s certificate divides by a value that never exceeds ||A||_2, so its own number can only be larger.)
  • the orthogonality loss max |X' X - I| (X' B X - I for generalized problems, U and V for SVDs);
  • the value error against a trusted reference: the analytic spectrum (Laplacians, finite-element pencils, matrices built as Q diag(values) Q'), dense LAPACK when feasible, or else a tight iterative reference: RSpectra at tol = 1e-13 asking for k + 10 and k + 20 values with large subspaces, and eigencore at tol = 1e-12. Each candidate is verified with the independent backward error, and among the accurate ones the set reaching furthest into the target wins (a value with a tiny backward error is a genuine eigenvalue, so a set that reaches further cannot be wrong where a shorter one may have skipped values). Disagreement between candidates is stored in cases.csv. The error is scaled like the backward error: |lambda - lambda_ref| / (||A||_2 + |lambda_ref| ||B||_2).
  • whether the method returned the requested target set (target_ok): k values, each matched to a distinct reference value within a gap-aware tolerance, and every matched reference value inside the wanted set (ties at the boundary, such as repeated eigenvalues and conjugate pairs, count as inside). A small backward error does not imply the right target set: a Krylov method can converge accurately to the wrong eigenvalues, or return one copy of a repeated eigenvalue and skip the other.

In the tables below a time is marked † when the method did not return the wanted set, and ‡ when its independent backward error exceeds 1e-6. Such times are not comparable with correct ones.

Result sets

Environment of each stored result set (one column per run).
20261015-9d7130c3-standard 20261015-9d7130c3-scaling
run 20261015-9d7130c3-standard 20261015-9d7130c3-scaling
date 2026-10-15T15:54:12Z 2026-10-15T16:10:12Z
machine 9d7130c3 9d7130c3
cpu Intel(R) Xeon(R) Processor @ 2.80GHz (4 logical / 4 physical cores, 15.7 GB) Intel(R) Xeon(R) Processor @ 2.80GHz (4 logical / 4 physical cores, 15.7 GB)
R 4.3.3 (2024-02-29) 4.3.3 (2024-02-29)
platform x86_64-pc-linux-gnu x86_64-pc-linux-gnu
BLAS OpenBLAS (libopenblasp-r0.3.26.so) OpenBLAS (libopenblasp-r0.3.26.so)
eigencore 2.0.0 @ e560f8ad61 2.0.0 @ e560f8ad61
competitors RSpectra 0.16.1, irlba 2.3.5.1 RSpectra 0.16.1, irlba 2.3.5.1
threads 1,4 1,4
blas_threads 1:1 4:4 1:1 4:4
reps 5 3
load_start_end 1.02 / 1.99 1.99 / 1.25
wall 958 s 990 s
Cases of the standard profile.
case generator nnz reference values ||A||_2
sym_sparse_n20000_LA_k10 random sparse symmetric, n=20000, ~5 nnz/row, N(0,1) values 99,998 certified iterative: RSpectra(k=30, tol=1e-13), independent backward error 1.6e-14; candidates RSpectra(k=20, tol=1e-13) (bwd 7.4e-15), RSpectra(k=30, tol=1e-13) (bwd 1.6e-14), eigencore(k=20, tol=1e-12) (bwd 6.5e-13); max top-k disagreement 1.6e-14 Golub-Kahan-Lanczos (44 steps, full reorth.)
sym_sparse_n20000_SA_k10 random sparse symmetric, n=20000, ~5 nnz/row, N(0,1) values 99,998 certified iterative: RSpectra(k=30, tol=1e-13), independent backward error 1.2e-14; candidates RSpectra(k=20, tol=1e-13) (bwd 6.7e-15), RSpectra(k=30, tol=1e-13) (bwd 1.2e-14), eigencore(k=20, tol=1e-12) (bwd 6.5e-13); max top-k disagreement 9.9e-15 Golub-Kahan-Lanczos (44 steps, full reorth.)
nonsym_sparse_n20000_LM_k6 random sparse nonsymmetric, n=20000, ~5 nnz/row, N(0,1) values 100,000 certified iterative: RSpectra(k=16, tol=1e-13), independent backward error 9.4e-14; candidates RSpectra(k=16, tol=1e-13) (bwd 9.4e-14), RSpectra(k=26, tol=1e-13) (bwd 2.1e-14), eigencore(k=16, tol=1e-12) (bwd 2.4e-13); max top-k disagreement 3.8e-01 Golub-Kahan-Lanczos (60 steps, full reorth.)
svd_sparse_50000x2000_k20 random sparse 50000 x 2000, density 0.002, N(0,1) values 200,000 dense LAPACK eigen() of the Gram matrix (top singular values) exact (dense Gram eigen)
svd_sparse_50000x20000_k20 random sparse 50000 x 20000, density 0.0005, N(0,1) values 500,000 certified iterative: RSpectra(k=30, tol=1e-13), independent backward error 4.6e-14; candidates RSpectra(k=30, tol=1e-13) (bwd 4.6e-14), RSpectra(k=40, tol=1e-13) (bwd 2.4e-14), eigencore(k=30, tol=1e-12) (bwd 8.9e-13); max top-k disagreement 8.4e-15 Golub-Kahan-Lanczos (36 steps, full reorth.)
dense_sym_n1500_LA_k10 dense symmetric crossprod(X)/n, X 1500 x 1500 N(0,1) 2,250,000 dense LAPACK eigen() exact (dense eigen)
eig_full_n1500_vectors dense symmetric crossprod(X)/n, X 1500 x 1500 N(0,1) 2,250,000 dense LAPACK eigen() exact (dense eigen)
eig_full_n1500_values dense symmetric crossprod(X)/n, X 1500 x 1500 N(0,1) 2,250,000 dense LAPACK eigen() exact (dense eigen)
shift_invert_lap2d_g100_sigma4.01_k6 2-D 5-point Dirichlet Laplacian, 100 x 100 grid (n=10000); analytic spectrum, repeated eigenvalues 49,600 analytic spectrum exact (analytic spectrum)
generalized_fem2d_g50_SA_k6 generalized SPD FEM pencil (Q1 stiffness/mass), 50 x 50 grid (n=2500); analytic spectrum 21,904 analytic spectrum Golub-Kahan-Lanczos (80 steps, full reorth.)
pca_centered_50000x1000_k10 random sparse 50000 x 1000, density 0.01, Exp(1) values 500,000 dense LAPACK eigen() of the Gram matrix (top singular values) exact (dense Gram eigen)
banded_lap1d_n20000_SA_k8 1-D Dirichlet Laplacian (tridiagonal), n=20000; analytic spectrum 59,998 analytic spectrum exact (analytic spectrum)
laplacian2d_g150_SA_k10 2-D 5-point Dirichlet Laplacian, 150 x 150 grid (n=22500); analytic spectrum, repeated eigenvalues 111,900 analytic spectrum exact (analytic spectrum)
powerlaw_dense_n2000_LA_k10 dense symmetric Q diag(j^-1) Q’, n=2000; analytic spectrum 4,000,000 analytic spectrum exact (analytic spectrum)
clustered_dense_n2000_LA_k10 dense symmetric, n=2000, top spectrum 1 (x3), 1-1e-6..3e-6, 0.9 (x2), 0.85, 0.8, bulk U(0,0.7); analytic 4,000,000 analytic spectrum exact (analytic spectrum)
lowrank_noise_5000x500_k10 dense 5000 x 500, rank-12 signal (s = 10..0.83) + 0.01 N(0,1) noise 2,500,000 dense LAPACK eigen() of the Gram matrix (top singular values) exact (dense Gram eigen)

Standard profile results

Sparse symmetric, extreme eigenvalues

case threads eigencore RSpectra base eigencore / RSpectra matvecs ec / RS bwd err ec bwd err RS certified
sym_sparse_n20000_LA_k10 1 569 ms 291 ms n/a 1.96 253 / 230 8.1e-09 2.4e-09 yes
sym_sparse_n20000_LA_k10 4 620 ms 285 ms n/a 2.18 253 / 230 8.1e-09 2.4e-09 yes
sym_sparse_n20000_SA_k10 1 580 ms 312 ms n/a 1.86 283 / 236 5.4e-09 4.4e-09 yes
sym_sparse_n20000_SA_k10 4 667 ms 332 ms n/a 2.01 283 / 236 5.4e-09 4.4e-09 yes

Sparse nonsymmetric, largest modulus

case threads eigencore RSpectra base eigencore / RSpectra matvecs ec / RS bwd err ec bwd err RS certified
nonsym_sparse_n20000_LM_k6 1 7.55 s 2.80 s† n/a 2.69 3990 / 3947 2.1e-09 2.6e-09 (miss) yes
nonsym_sparse_n20000_LM_k6 4 7.05 s 3.15 s† n/a 2.24 3990 / 3947 2.1e-09 2.6e-09 (miss) yes

Sparse SVD

case threads eigencore RSpectra irlba base eigencore / RSpectra matvecs ec / RS bwd err ec bwd err RS certified
svd_sparse_50000x2000_k20 1 335 ms 201 ms 2.04 s n/a 1.67 488 / 486 8.3e-10 1.8e-09 yes
svd_sparse_50000x2000_k20 4 349 ms 188 ms 1.53 s n/a 1.85 488 / 486 8.3e-10 1.8e-09 yes
svd_sparse_50000x20000_k20 1 2.33 s 1.06 s 5.20 s n/a 2.19 706 / 602 2.3e-10 7.1e-09 yes
svd_sparse_50000x20000_k20 4 1.45 s 1.04 s 3.77 s n/a 1.39 706 / 602 2.3e-10 7.1e-09 yes

Dense symmetric, partial

case threads eigencore RSpectra base eigencore / RSpectra matvecs ec / RS bwd err ec bwd err RS certified
dense_sym_n1500_LA_k10 1 198 ms 105 ms 934 ms 1.89 166 / 169 6.1e-09 1.2e-09 yes
dense_sym_n1500_LA_k10 4 142 ms 47 ms 603 ms 3.03 166 / 169 6.1e-09 1.2e-09 yes

Full dense eigendecomposition

case threads eigencore base bwd err ec certified
eig_full_n1500_vectors 1 1.38 s 925 ms 2.0e-15 yes
eig_full_n1500_vectors 4 773 ms 696 ms 1.8e-15 yes
eig_full_n1500_values 1 438 ms 380 ms - -
eig_full_n1500_values 4 260 ms 210 ms - -

Interior eigenvalues by shift-invert

case threads eigencore RSpectra base eigencore / RSpectra matvecs ec / RS bwd err ec bwd err RS certified
shift_invert_lap2d_g100_sigma4.01_k6 1 116 ms 56 ms n/a 2.07 45 / 41 4.9e-14 6.4e-14 yes
shift_invert_lap2d_g100_sigma4.01_k6 4 146 ms 54 ms n/a 2.71 45 / 41 4.9e-14 6.4e-14 yes

Generalized symmetric-definite

case threads eigencore RSpectra base eigencore / RSpectra matvecs ec / RS bwd err ec bwd err RS certified
generalized_fem2d_g50_SA_k6 1 334 ms 645 ms 10.6 s 0.52 - / 1042 8.1e-09 6.0e-11 yes
generalized_fem2d_g50_SA_k6 4 380 ms 668 ms 6.63 s 0.57 - / 1042 8.1e-09 6.0e-11 yes

Centred sparse PCA (implicit centring)

case threads eigencore RSpectra irlba base eigencore / RSpectra matvecs ec / RS bwd err ec bwd err RS certified
pca_centered_50000x1000_k10 1 511 ms 533 ms 982 ms n/a 0.96 366 / 330 4.6e-11 6.1e-09 yes
pca_centered_50000x1000_k10 4 341 ms 497 ms 800 ms n/a 0.69 366 / 330 4.6e-11 6.1e-09 yes

Banded (1-D Laplacian), smallest

case threads eigencore RSpectra base eigencore / RSpectra matvecs ec / RS bwd err ec bwd err RS certified
banded_lap1d_n20000_SA_k8 1 61 ms 9.97 s† n/a 0.01 23 / 12020 1.2e-10 - (miss) yes
banded_lap1d_n20000_SA_k8 4 68 ms 10.4 s† n/a 0.01 23 / 12020 1.2e-10 - (miss) yes

2-D Laplacian, smallest (repeated eigenvalues)

case threads eigencore RSpectra base eigencore / RSpectra matvecs ec / RS bwd err ec bwd err RS certified
laplacian2d_g150_SA_k10 1 382 ms 10.0 s† n/a 0.04 61 / 6488 3.3e-16 3.1e-13 (miss) yes
laplacian2d_g150_SA_k10 4 439 ms 9.93 s† n/a 0.04 61 / 6488 5.9e-16 3.1e-13 (miss) yes

Power-law spectrum

case threads eigencore RSpectra base eigencore / RSpectra matvecs ec / RS bwd err ec bwd err RS certified
powerlaw_dense_n2000_LA_k10 1 278 ms 66 ms 2.27 s 4.25 50 / 40 2.2e-16 7.7e-13 yes
powerlaw_dense_n2000_LA_k10 4 149 ms 22 ms 1.09 s 6.72 50 / 40 2.6e-16 7.7e-13 yes

Clustered / repeated top eigenvalues

case threads eigencore RSpectra base eigencore / RSpectra matvecs ec / RS bwd err ec bwd err RS certified
clustered_dense_n2000_LA_k10 1 449 ms 264 ms 2.81 s 1.70 125 / 127 1.1e-11 8.0e-11 yes
clustered_dense_n2000_LA_k10 4 190 ms 53 ms 1.88 s 3.58 125 / 127 7.3e-12 9.7e-11 yes

Low rank plus noise SVD

case threads eigencore RSpectra irlba base eigencore / RSpectra matvecs ec / RS bwd err ec bwd err RS certified
lowrank_noise_5000x500_k10 1 117 ms 115 ms 99 ms 551 ms 1.02 - / 52 9.9e-16 3.8e-14 yes
lowrank_noise_5000x500_k10 4 77 ms 27 ms 27 ms 316 ms 2.87 - / 52 1.3e-15 3.8e-14 yes

Time relative to RSpectra

Ratios below one mean faster than RSpectra on the same machine and thread setting.

Operator applications

Operator applications are the hardware-independent part of the comparison: they do not change with the machine, the BLAS, or the load. A dash means the package reported no comparable count (dense LAPACK, explicit Gram paths).

Operator applications (A and A’ counted separately) at threads = 1.
case_id eigencore RSpectra irlba
sym_sparse_n20000_LA_k10 253 230 -
sym_sparse_n20000_SA_k10 283 236 -
nonsym_sparse_n20000_LM_k6 3990 3947 -
svd_sparse_50000x2000_k20 488 486 756
svd_sparse_50000x20000_k20 706 602 1096
dense_sym_n1500_LA_k10 166 169 -
eig_full_n1500_vectors - - -
eig_full_n1500_values - - -
shift_invert_lap2d_g100_sigma4.01_k6 45 41 -
generalized_fem2d_g50_SA_k6 - 1042 -
pca_centered_50000x1000_k10 366 330 390
banded_lap1d_n20000_SA_k8 23 12020 -
laplacian2d_g150_SA_k10 61 6488 -
powerlaw_dense_n2000_LA_k10 50 40 -
clustered_dense_n2000_LA_k10 125 127 -
lowrank_noise_5000x500_k10 - 52 48

Accuracy summary

Independent backward error / scaled value error / target set (“cert”: eigencore’s own certificate passed). threads = 1.
case_id eigencore RSpectra irlba base
sym_sparse_n20000_LA_k10 8.1e-09 / 1.6e-14 / ok, cert 2.4e-09 / 1.6e-14 / ok - n/a
sym_sparse_n20000_SA_k10 5.4e-09 / 9.3e-15 / ok, cert 4.4e-09 / 7.4e-15 / ok - n/a
nonsym_sparse_n20000_LM_k6 2.1e-09 / 1.3e-11 / ok, cert 2.6e-09 / 8.2e-02 / MISS - n/a
svd_sparse_50000x2000_k20 8.3e-10 / 3.3e-15 / ok, cert 1.8e-09 / 3.0e-15 / ok 8.6e-09 / 4.5e-15 / ok n/a
svd_sparse_50000x20000_k20 2.3e-10 / 8.4e-15 / ok, cert 7.1e-09 / 6.0e-15 / ok 8.2e-09 / 1.0e-14 / ok n/a
dense_sym_n1500_LA_k10 6.1e-09 / 5.8e-15 / ok, cert 1.2e-09 / 5.5e-15 / ok - 8.2e-16 / 5.3e-15 / ok
eig_full_n1500_vectors 2.0e-15 / 5.4e-15 / ok, cert - - 1.8e-15 / 5.3e-15 / ok
eig_full_n1500_values - / 5.5e-15 / ok - - - / 0.0e+00 / ok
shift_invert_lap2d_g100_sigma4.01_k6 4.9e-14 / 1.2e-15 / ok, cert 6.4e-14 / 8.9e-16 / ok - n/a
generalized_fem2d_g50_SA_k6 8.1e-09 / 1.2e-15 / ok, cert 6.0e-11 / 2.3e-13 / ok - 2.2e-15 / 2.5e-16 / ok
pca_centered_50000x1000_k10 4.6e-11 / 1.8e-15 / ok, cert 6.1e-09 / 1.8e-15 / ok 9.4e-09 / 4.0e-15 / ok n/a
banded_lap1d_n20000_SA_k8 1.2e-10 / 2.8e-17 / ok, cert - / - / MISS - n/a
laplacian2d_g150_SA_k10 3.3e-16 / 2.1e-17 / ok, cert 3.1e-13 / 3.1e-13 / MISS - n/a
powerlaw_dense_n2000_LA_k10 2.2e-16 / 1.2e-16 / ok, cert 7.7e-13 / 1.3e-15 / ok - 3.8e-16 / 1.7e-16 / ok
clustered_dense_n2000_LA_k10 1.1e-11 / 4.9e-16 / ok, cert 8.0e-11 / 2.0e-15 / ok - 1.4e-15 / 1.4e-15 / ok
lowrank_noise_5000x500_k10 9.9e-16 / 1.2e-16 / ok, cert 3.8e-14 / 3.9e-16 / ok 6.0e-15 / 7.7e-16 / ok 9.1e-15 / 7.1e-16 / ok

Rows where a method missed the wanted set or eigencore’s certificate failed (generated from the stored results):

  • nonsym_sparse_n20000_LM_k6, RSpectra: returned the wrong or an incomplete target set (value error 8.2e-02, backward error 2.6e-09).
  • banded_lap1d_n20000_SA_k8, RSpectra: returned the wrong or an incomplete target set (value error -, backward error -).
  • laplacian2d_g150_SA_k10, RSpectra: returned the wrong or an incomplete target set (value error 3.1e-13, backward error 3.1e-13).

A residual certificate checks that each returned pair is accurate; it does not by itself prove that no wanted eigenvalue was skipped, which is what the target-set column measures.

Scaling

The scaling profile sweeps one variable at a time: the dimension n of a random sparse symmetric matrix (5 nonzeros per row), its density at n = 20000, the number of wanted eigenvalues k, the long side of a sparse m x 2000 SVD, and the dimension of the 1-D Laplacian, whose smallest eigenvalues have gaps shrinking like 1/n^2. Solid lines are 1 thread, dashed lines 4 threads; open symbols mark results that missed the wanted set.

Fitted log-log slope of median time against the swept variable.
group method threads slope
scaling_n eigencore 1 0.97
scaling_n RSpectra 1 1.17
scaling_n eigencore 4 0.68
scaling_n RSpectra 4 1.14
scaling_nnz eigencore 1 0.64
scaling_nnz RSpectra 1 0.60
scaling_nnz eigencore 4 0.44
scaling_nnz RSpectra 4 0.60
scaling_k eigencore 1 0.99
scaling_k RSpectra 1 0.92
scaling_k eigencore 4 0.84
scaling_k RSpectra 4 0.93
scaling_svd_m eigencore 1 1.13
scaling_svd_m RSpectra 1 1.36
scaling_svd_m irlba 1 1.34
scaling_svd_m eigencore 4 0.82
scaling_svd_m RSpectra 4 1.32
scaling_svd_m irlba 4 1.29
scaling_banded_n eigencore 1 0.59
scaling_banded_n RSpectra 1 1.13
scaling_banded_n eigencore 4 0.56
scaling_banded_n RSpectra 4 1.15

How to reproduce or add your machine

Install eigencore from the commit you want to measure into a private library and run the suite from the repository root:

R CMD INSTALL --preclean --no-docs -l .rlib .
Rscript inst/benchmarks/run-suite.R --lib=.rlib --profile=quick              # ~2 min smoke run
Rscript inst/benchmarks/run-suite.R --lib=.rlib --profile=standard --threads=1,4
Rscript inst/benchmarks/run-suite.R --lib=.rlib --profile=scaling  --threads=1,4

Useful options: --families=sym_sparse,svd_sparse (or a sweep such as scaling_k), --cases=<substring>, --methods=eigencore,RSpectra,irlba,base,PRIMME, --reps=N, --suitesparse (adds a few SuiteSparse Matrix Collection matrices, downloaded once into the cache and skipped when offline), and --list to print the selected cases. Each run writes inst/benchmarks/results/<YYYYMMDD>-<machine-id>-<profile>/ containing results.csv (one row per case, method and thread setting), cases.csv (generators, reference sources and ||A||_2), and environment.json/.rds (CPU, cores, R, BLAS/LAPACK library and threads, package versions, eigencore git SHA, load average). The machine id is a hash; no hostname is stored.

To add your machine, commit that directory; this article and the README table pick up the newest run per machine and profile. Summaries are available without rebuilding the site:

Rscript inst/benchmarks/report.R inst/benchmarks/results/<run-dir>

The benchmarks GitHub workflow (manual dispatch, inputs profile and threads) runs the same command on ubuntu-latest with OpenBLAS and uploads the result directory as an artifact; it never commits results.

Limits

  • Shared, noisy hardware. Stored results may come from a shared container whose load varies while the suite runs; the environment table records the load average at the start and end, and results.csv keeps every repetition (times) and the per-row load. Medians absorb short spikes, not sustained contention. Multithreaded rows are the most sensitive: eigencore’s OpenMP governor shrinks the team when regions stop paying off, but a multithreaded BLAS can still spin on an oversubscribed machine (set OPENBLAS_NUM_THREADS=1 there), which is why matvec counts are reported alongside times.
  • BLAS dependence. Dense cases and eigencore’s blocked kernels depend on the BLAS/LAPACK library and its threads; RSpectra uses Eigen’s own kernels. Compare rows only within one result set, and check the BLAS line of its environment.
  • Random matrices are not real matrices. Most cases are synthetic with known structure (random sparsity, Laplacians, prescribed spectra). Real matrices have different sparsity patterns, spectra and conditioning; the optional SuiteSparse cases are a first step, and your own matrices are the only benchmark that settles a choice.
  • Different stopping rules. All methods receive tol = 1e-8, but each package interprets it differently (RSpectra relative to the Ritz value, irlba relative to the largest singular value, eigencore as a backward error with a certificate). The independent accuracy columns are there so that a fast but inaccurate or incomplete answer is visible.
  • What the times include. eigencore’s time includes certification and result construction; dense base R rows include as.matrix(); the RSpectra generalized row includes the sparse Cholesky factorisation.