[R] MASS::mvrnorm() on MKL may produce different numbers even when the seed is the same?

[Shu Fai Cheung]

Hi All,

When addressing an error in one of my packages in the CRAN check at CRAN,
I found something strange which, I believe, is unrelated to my package.
I am not sure whether it is a bug or a known issue. Therefore, I would like
to have advice from experts here.

The error at CRAN check occurred in a test, and only on R-devel with MKL. No
issues on all other platforms. The test compares two sets of random numbers,
supposed to be identical because generated from the same seed. However, when
I tried that in R-devel (2023-08-15 r84957) on Ubuntu 22.04.3 LTS, linked to
MKL (2023.2.0), I found that this is not the case in one situation.

I don't know why but somehow only one particular set of means and
covariance matrix, among many others in the code, led to that problem.
Please find
below the code to reproduce the means and the covariance matrix (pardon me
for the long code):

mu0 <- structure(c(0.52252416853188655, 0.39883382931927774, 1.6296642535174521,
2.9045763671121816, -0.19816874840500939, 0.42610841566522556,
0.30155498531316366, 1.0339601619394503, 3.4125587827873192,
13.125481598475405, 19.275480386183503, 658.94225353462195, 1.0997193726987393,
9.9980286642877214, 6.4947188998602092, -12.952617780813679,
8.4991882784024799, 106.82209520278377, 0.19623712449219838), class =
c("numeric"))

sigma0 <- structure(c(0.0047010182215945009, 1.4688424622163808e-17,
-2.5269697696885822e-17,
-1.2451516929358655e-18, 3.6553475725253408e-19, 3.2092363914356227e-19,
-2.6341938382508503e-18, 8.6439582878287556e-19, -1.295240602054123e-17,
1.1154057497258702e-18, -8.2407621704807367e-33, -1.0891686096304908e-15,
-2.6800671450262819e-18, -0.00092251429799113333, -1.4833018148344697e-16,
2.2888892242132203e-16, -3.5128615476876035e-18, 2.8004770623734002e-33,
2.0818099301348832e-34, 1.294907062174661e-17, 0.012788608283099183,
1.5603789410838518e-15, 1.0425182851251724e-16, 2.758318745465066e-17,
-9.7047963858148572e-18, -7.4685438142667792e-17, -1.9426723638193857e-17,
-7.8775307262071738e-17, -4.0773523140359949e-17, 4.1212975037936416e-18,
-1.199934828824835e-14, 2.301740948367742e-17, -2.5410883836579325e-18,
-0.129172198695584, -1.9922957470809647e-14, 1.7077690739402761e-16,
1.663702957392817e-30, 5.5362608501514495e-32, -2.5575000656645995e-17,
7.3879960338026333e-16, 0.052246980157830372, -0.007670030643844231,
-1.5468402204507243e-17, 1.0518757786432683e-17, 2.9992131812421253e-16,
1.6588830885081527e-17, -8.3276045185697992e-16, -3.3713008849128695e-15,
1.5823980389961169e-17, 2.2243227896948194e-15, -3.2071920299461956e-17,
5.0187645877462975e-18, -7.4622951185527265e-15, -0.44701112744184102,
-1.854066605407503e-16, -5.2511356793649759e-30, -6.7993385117653912e-32,
-1.5343740968067595e-18, -3.6785921616583949e-17, -0.0076700306438433133,
0.019231143599164325, 5.7818784939170702e-18, 1.7819897158350214e-17,
-2.3927039320969442e-17, -3.9630951242056644e-18, 1.9779107704573469e-15,
7.8103367010164485e-15, -1.2351275675875924e-17, -8.6959160183013082e-15,
-1.6013333002787863e-18, 3.0110116065267407e-19, 3.7155867714997651e-16,
-0.12490087179942209, -2.2029631919898945e-16, -8.0869824211018481e-31,
-5.0586739111186695e-33, 2.0667225433033359e-19, 1.0141735013231455e-18,
3.2845977347050714e-17, -2.9662734037779067e-19, 9.9595082331331027e-05,
-1.9982466114987023e-18, 2.6420113469249318e-18, 5.2415327077690342e-19,
3.007533948014542e-18, -5.1128158961673558e-17, 6.7791246559883483e-19,
-3.042327665547861e-15, 1.9523738271941284e-19, -4.0556768902105205e-20,
-1.024372770865438e-17, -0.010638955366526863, 2.5786757042756338e-17,
1.3546130005558746e-32, 1.7947249868721723e-33, 1.9228405363682382e-19,
-4.1209014001548407e-18, 1.5228291529275058e-17, 2.0443162438349059e-17,
-1.9572693345249383e-18, 0.0012709581028842473, -0.0018515998737074948,
9.3128024867175073e-20, -5.1895788671618993e-17, 2.7373981615289174e-16,
1.2812977711597223e-17, -2.2792319486263267e-15, 4.1599503721278813e-19,
-3.7733269771394201e-20, 4.16234419478765e-17, -1.5986158133468129e-16,
-0.016037670900735834, -5.0763064708173244e-33, -1.0176066166236902e-50,
-5.9296704797665404e-18, -8.0698801402861772e-17, 2.9646619457173492e-16,
-2.8879431119718227e-17, 2.9393253663483117e-18, -0.0018515998737074959,
0.090335687736246548, 5.6849412731758135e-19, 8.8934458836007799e-17,
-4.1390858756690365e-16, 4.120323677410211e-16, 2.8000915545933503e-15,
2.8094462743052983e-17, 1.1636214841356605e-18, 8.1510367497000071e-16,
-3.004558574117108e-15, 0.0061666744014873013, -2.5381532354086619e-33,
-2.7110175619881585e-34, -3.9720857469692968e-18, -2.2722246209861298e-17,
6.5087631702810744e-18, -5.8833898464959171e-18, 6.0598974659958357e-19,
-6.4324667436451943e-19, -4.9865458549184915e-19, 0.010690736164778532,
-2.5717220776886191e-17, -2.9932234433250055e-17, -1.7586566364625091e-32,
-2.0572361612722435e-15, -4.1554892682395136e-18, 7.7947068522170098e-19,
2.2950757715435305e-16, -6.8563401956907788e-17, 8.3986032618135276e-18,
-3.0929447793865389e-45, -9.1799940583213546e-47, 1.9205112318761867e-18,
-6.309535361758424e-17, -8.5794270550008683e-16, 1.9513428795022368e-15,
6.975277535512248e-18, -6.2501857399599432e-17, 1.0008535047159303e-16,
-2.6690493301637561e-17, 0.11645557445978426, -5.0477656527863884e-17,
-1.9142598204346162e-30, -4.3865060631657549e-14, -9.2939338818469881e-18,
-3.7687560169835257e-19, 6.3729886582182645e-16, -1.2613712396302566e-14,
7.8691203779893581e-16, -7.96410737362738e-44, -5.4321486152322576e-46,
-1.7940674523534164e-17, -8.157501463813461e-17, -3.3732156955561982e-15,
7.677772572020035e-15, -4.7764170203097078e-17, 2.7500679201468897e-16,
-3.9378464227822661e-16, -1.9805344603340844e-17, -2.0590552280380157e-16,
1.7227826719189774, -2.5152187296963038e-30, 1.7551764763918126e-14,
-5.4770085336579068e-18, 3.5206263799487875e-18, 8.2395392572605337e-16,
-4.1620820803543103e-14, -3.4715380153547894e-15, -3.0529078265571622e-43,
-2.8046838240348109e-45, -2.5479250226286821e-32, 4.121297503793449e-18,
1.5823980389961397e-17, -1.235127567587614e-17, 1.0285610559852655e-18,
1.0765062746617977e-17, 4.1501588000780158e-16, 5.2738511582000153e-33,
-1.6831260511527137e-30, -3.3541828527847257e-30, 3.7154414411814374,
3.4040266595702152e-15, 5.1153310393987592e-18, 4.9999747986237238e-33,
-4.1627442819337855e-17, -1.3972969142201618e-16, -2.2035880670651683e-16,
-7.2138653746624827e-47, 2.5041819857453665e-47, -1.0640003589174214e-15,
-7.9766775252940816e-15, -1.86380928624219e-15, -2.0409565483972152e-14,
-4.5010528758464713e-16, -2.5476765314056885e-15, 5.7438239741680622e-15,
-5.1005913365528254e-15, -4.0052224901292309e-14, 2.0275682895130953e-14,
8.001473599959262e-15, 4342.0489349328445, -2.2045103993340862e-15,
2.087963708926218e-16, 8.0568968211307842e-14, 2.8167998366347968e-13,
3.1749229442581484e-14, 8.9567352491809379e-43, -2.3670298646799273e-44,
4.2781010071619158e-32, 2.5122662667827335e-17, -3.4359127754145909e-17,
-1.4706600926070093e-18, 1.876439844639638e-19, -9.1730442893545148e-20,
2.8080363645722388e-17, -4.4705925750481468e-32, -7.809262716208445e-18,
-5.8577085238749991e-19, 5.1153310393987515e-18, -6.9351338840590931e-16,
0.012093826986889086, -8.3952223993262708e-33, -2.5375314514710393e-16,
3.424781603493142e-16, -4.3266783076648891e-18, 7.3570155156909413e-45,
1.6171838713232998e-46, -0.00092251429799113343, -7.8240087156410085e-18,
2.1083287560348406e-17, -8.9775735933099227e-18, 3.2531284806658474e-20,
-2.3137400470270888e-19, 7.166222774364368e-19, -1.6962655186338848e-19,
2.5417429127255315e-18, -2.1888401697215364e-19, 7.492787666039381e-33,
2.1373531604105888e-16, 5.2592866998595688e-19, 0.0053508323628379687,
6.939324860007723e-17, -1.1345120732507663e-16, -1.3255448103671732e-18,
1.0169078705557416e-16, -3.701528155571061e-19, -1.3050383760465465e-16,
-0.129172198695584, -1.5780816956514754e-14, -1.1071526119482569e-15,
-2.7949359117065728e-16, 9.8321930809926744e-17, 7.5470671367369062e-16,
1.9622092961540385e-16, 7.9567529294094524e-16, 4.1183571472477233e-16,
-4.1627442819339772e-17, 1.2120022499676643e-13, -2.3248889366737827e-16,
1.4240632385802796e-17, 1.3217752807216008, 2.0311176273844298e-13,
-1.9570716831733758e-15, 4.7962597568986823e-16, 2.9165264143720962e-17,
2.5255547350017705e-16, -7.2487511301945062e-15, -0.44701112744185578,
-0.12490087179941219, -0.010638955366526433, -7.9593195538363592e-17,
-3.0600743713008676e-15, -1.8184344227119647e-16, -1.2723803109870239e-14,
-4.267580759582334e-14, -1.0227267431703487e-16, 3.9402946853735757e-13,
3.1994002076770255e-16, -4.2061670894524917e-17, 7.4223786548689626e-14,
7.0315216015826767, 2.2390669558248378e-15, -1.2244512793012972e-15,
-3.5253412228046429e-17, -1.2061427540284553e-18, 9.9899822637462946e-17,
-2.4550261028932595e-16, -2.5317628873304297e-16, 2.5199623506843722e-17,
-0.016037670900735834, 0.0061666744014872909, -1.3133261974683634e-18,
6.5223470430906591e-16, -3.4512393153190672e-15, -2.4620056039335759e-16,
2.8862767590185475e-14, -1.0881366439433239e-17, -3.9120910874012715e-18,
-1.2372606267786687e-15, 3.2639925872240233e-15, 0.30212469777249018,
4.3378386520217752e-16, 1.1399322374260218e-17, 2.5273795605894831e-18,
7.0372281397307745e-16, -1.4493840318281129e-15, 2.3909766400689098e-16,
-2.3621724602135697e-17, 4.1164300362570093e-18, -5.9299103903990776e-18,
-1.6549632537822852e-30, 2.0576427541669881e-29, 1.1492923590019121e-28,
-3.4572603795675457e-31, -2.0561228611852425e-29, 2.1007427249504219e-30,
9.6204477501104748e-17, -6.7166997975872844e-15, 1.3428325376228686e-14,
3.3362903423571782e-16, 3.2947112676731014, 6.6046445483993014e-18,
-4.0887487160499735e-20, -3.0968390611719402e-18, -6.0930441874738707e-19,
-3.1404422740276848e-18, -5.6584069489838701e-20, 1.6491214275041756e-19,
6.3584544371811611e-22, 4.4921015596154276e-33, -3.1054789027110318e-31,
-1.1913370661394943e-30, 3.0677051354054613e-33, 5.8136140932799355e-30,
-4.0289214710854143e-33, 6.392220812489477e-19, 6.2007829741095621e-17,
1.0190707777780152e-17, 9.6388161040420896e-18, -4.3748284667364627e-18,
0.0054985968634936964), dim = c(19L, 19L), class = c("matrix"))

If I run the following several times, given that the same seed is used,
I expect the numbers generated are the same:

set.seed(1234)
MASS::mvrnorm(n = 5, mu = mu0, Sigma = sigma0)[1, 1:5]

However, this is not the case:

> set.seed(1234)
> MASS::mvrnorm(n = 5, mu = mu0, Sigma = sigma0)[1, 1:5]
[1]  0.4851605  0.5704446  1.6873036  2.7645014 -0.2020908
> set.seed(1234)
> MASS::mvrnorm(n = 5, mu = mu0, Sigma = sigma0)[1, 1:5]
[1]  0.4851605  0.5704446  1.6872818  2.7644787 -0.2023346
> set.seed(1234)
> MASS::mvrnorm(n = 5, mu = mu0, Sigma = sigma0)[1, 1:5]
[1]  0.4851605  0.5704446  1.6872818  2.7644787 -0.2023346
> set.seed(1234)
> MASS::mvrnorm(n = 5, mu = mu0, Sigma = sigma0)[1, 1:5]
[1]  0.4851605  0.5704446  1.6873036  2.7645014 -0.2020908

It seems that the numbers in some columns alternate between
two versions.

I understand that MKL can yield random numbers different from
those generated in the "standard R", due to the output of
eigen() used in MASS::mvrnorm(). However, should the numbers
generated identical if the same seed is used?

This is the sessionInfo() output. I used a fresh installation of Ubuntu,
installed MKL, and then compiled R-devel.

> sessionInfo()
R Under development (unstable) (2023-08-15 r84957)
Platform: x86_64-pc-linux-gnu
Running under: Ubuntu 22.04.3 LTS

Matrix products: default
BLAS/LAPACK: /opt/intel/oneapi/mkl/2023.2.0/lib/intel64/libmkl_rt.so.2;
 LAPACK version 3.10.1

locale:
 [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C
 [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8
 [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8
 [7] LC_PAPER=en_US.UTF-8       LC_NAME=C
 [9] LC_ADDRESS=C               LC_TELEPHONE=C
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C

time zone: Asia/Shanghai
tzcode source: system (glibc)

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base

loaded via a namespace (and not attached):
[1] MASS_7.3-60.1  compiler_4.4.0

I cannot create the same MKL R-devel machine used in CRAN checks.
Therefore, I am not 100% certain whether the cause of the error
is the same. Nevertheless, this phenomenon, non-identical random
numbers generated even with the same seed, can explain the
error I encountered.

Regards,
Shu Fai