RMarkov is a package for computing steady-state and transient solutions of Markov chaings. Mainly, the package offers the computation funcsions for continuous-time Markov chian (CTMC).
You can install Rsrat from github with:
install.packages("devtools")
devtools::install_github("okamumu/RMarkov")Load the library. This package requires Matrix and deformula packages.
library(RMarkov)
#> Loading required package: MatrixLet’s consider a CTMC with the following infinitesimal generator:
Q <- rbind(
c(-5,2,3,0,0),
c(1,-4,3,0,0),
c(0,2,-3,1,0),
c(0,0,1,-2,1),
c(0,0,1,4,-5)
)Get the steady-state solution with GTH algorithm. This is efficient to obtain the steady-state solution of a small-sized CTMC.
ctmc.st.gth(Q)
#> [1] 0.04166667 0.20833333 0.37500000 0.31250000 0.06250000Get the steady-state solution with GS algorithm. This is used when the
CTMC is larege. Note that the matrix Q should be CsparseMatrix (dgCMatrix) or dgeMatrix when we apply ctmc.st.gs.
ctmc.st.gs(Q)
#> $x
#> [1] 0.04166667 0.20833333 0.37500000 0.31250000 0.06250000
#>
#> $convergence
#> [1] TRUE
#>
#> $iter
#> [1] 100
#>
#> $rerror
#> [1] 0The sensitivity function or high-ordered sensitivity function are defined by the first or higher derivatives of the steady-state solution. Now we have the first derivative of Q with respect to a parameter:
dQ <- rbind(
c(-1,1,0,0,0),
c(0,-1,1,0,0),
c(0,0,-1,1,0),
c(0,0,0,-1,1),
c(0,0,0,0,0)
)Get the sensitivity function (the first derivative of steady-state probability)
pis <- ctmc.st.gth(Q)
ctmc.stsen.gs(Q, pis, pis %*% dQ)
#> $x
#> [1] -0.03125000 -0.11458333 -0.11458333 0.16493056 0.09548611
#>
#> $convergence
#> [1] TRUE
#>
#> $iter
#> [1] 30
#>
#> $rerror
#> [1] 6.231162e-10If the underlying CTMC has absorbing states, limiting probabilities go to any of absorbing states. The quasi stationary solution is a probability vector provided that the state does not go to absorbing states even if time goes to infinity.
Let T be the infinitesimal generator over transient states for the CTMC with absorbing states.
T <- rbind(
c(-5,2,0,0,0),
c(1,-4,3,0,0),
c(0,2,-3,1,0),
c(0,0,1,-2,1),
c(0,0,0,4,-5)
)The quasi stationary can be obtained by power method
res <- ctmc.st.power(T)
print(res)
#> $x
#> [1] 0.04155506 0.19958042 0.33792657 0.34839739 0.07254056
#>
#> $convergence
#> [1] TRUE
#>
#> $iter
#> [1] 100
#>
#> $rerror
#> [1] 1.714568e-09Let’s consider the same CTMC.
Q <- rbind(
c(-5,2,3,0,0),
c(1,-4,3,0,0),
c(0,2,-3,1,0),
c(0,0,1,-2,1),
c(0,0,1,4,-5)
)The initial probability vector is given by
x0 <- c(1,0,0,0,0)Get the transient solution with uniformization.
ctmc.tran.unif(Q=Q, x=x0, t=seq(0,1,length.out = 10))
#> $t
#> [1] 0.0000000 0.1111111 0.2222222 0.3333333 0.4444444 0.5555556 0.6666667
#> [8] 0.7777778 0.8888889 1.0000000
#>
#> $x
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 1 0.5820320616 0.351767424 0.224120901 0.15267248 0.11209328
#> [2,] 0 0.1612303811 0.239958487 0.274718187 0.28654935 0.28685532
#> [3,] 0 0.2423275279 0.362692298 0.418828974 0.44163544 0.44757632
#> [4,] 0 0.0139280733 0.042827224 0.075580246 0.10733132 0.13618174
#> [5,] 0 0.0004819562 0.002754567 0.006751692 0.01181142 0.01729334
#> [,7] [,8] [,9] [,10]
#> [1,] 0.08855444 0.07449601 0.06577544 0.06011389
#> [2,] 0.28175017 0.27445765 0.26663160 0.25908038
#> [3,] 0.44536423 0.43955706 0.43249361 0.42533392
#> [4,] 0.16159219 0.18361870 0.20255314 0.21875846
#> [5,] 0.02273898 0.02787058 0.03254621 0.03671335
#>
#> $cx
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 0 8.558877e-02 0.136208683 0.1675142740 0.188069810 0.202571204
#> [2,] 0 9.975905e-03 0.032810841 0.0616922710 0.093021529 0.124949303
#> [3,] 0 1.497823e-02 0.049392243 0.0932293617 0.141247921 0.190755061
#> [4,] 0 5.538287e-04 0.003634472 0.0102064685 0.020389554 0.033948876
#> [5,] 0 1.437449e-05 0.000175981 0.0006909552 0.001715627 0.003331107
#> [,7] [,8] [,9] [,10]
#> [1,] 0.21360314 0.222597057 0.23035331 0.23732592
#> [2,] 0.15657016 0.187481292 0.21754199 0.24674347
#> [3,] 0.24041226 0.289594353 0.33804647 0.38570121
#> [4,] 0.05052408 0.069732655 0.09121363 0.11464339
#> [5,] 0.00555702 0.008372415 0.01173349 0.01558601The reward vector is
r <- c(0,1,2,3,4)res <- ctmc.tran.rwd.unif(Q=Q, x=x0, t=seq(0,20,length.out = 1000), r=r)
data <- data.frame(time=res$t, instantaneous=res$irwd, average=res$crwd / res$t)
head(data, 10)
#> time instantaneous average
#> 1 0.00000000 0.0000000 NaN
#> 2 0.02002002 0.1526178 0.07754355
#> 3 0.04004004 0.2911853 0.15027698
#> 4 0.06006006 0.4171316 0.21856957
#> 5 0.08008008 0.5317360 0.28275950
#> 6 0.10010010 0.6361436 0.34315661
#> 7 0.12012012 0.7313799 0.40004488
#> 8 0.14014014 0.8183633 0.45368471
#> 9 0.16016016 0.8979160 0.50431499
#> 10 0.18018018 0.9707750 0.55215496library(ggplot2)
data_long <- tidyr::gather(data, key="type", value=reward, -time)
ggplot(data=data_long, aes(x=time, y=reward, colour=type)) + geom_line()
#> Warning: Removed 1 rows containing missing values (geom_path).
As similar to the previous section, we have the first derivative of Q with respect to a parameter:
dQ <- rbind(
c(-1,1,0,0,0),
c(0,-1,1,0,0),
c(0,0,-1,1,0),
c(0,0,0,-1,1),
c(0,0,0,0,0)
)The sensitivity function of transient solution can be obtained from the following expanded infinitesimal generator.
Qdash <- rbind2(cbind2(Q, dQ), cbind2(RMarkov::zeroM(5), Q))
x0dash <- c(x0, rep(0,5))
rdash <- c(rep(0,5), r)
res <- ctmc.tran.rwd.unif(Q=Qdash, x=x0dash, t=seq(0,20,length.out = 1000), r=rdash)
print(res)
#> $t
#> [1] 0.00000000 0.02002002 0.04004004 0.06006006 0.08008008
#> [6] 0.10010010 0.12012012 0.14014014 0.16016016 0.18018018
#> [11] 0.20020020 0.22022022 0.24024024 0.26026026 0.28028028
#> [16] 0.30030030 0.32032032 0.34034034 0.36036036 0.38038038
#> [21] 0.40040040 0.42042042 0.44044044 0.46046046 0.48048048
#> [26] 0.50050050 0.52052052 0.54054054 0.56056056 0.58058058
#> [31] 0.60060060 0.62062062 0.64064064 0.66066066 0.68068068
#> [36] 0.70070070 0.72072072 0.74074074 0.76076076 0.78078078
#> [41] 0.80080080 0.82082082 0.84084084 0.86086086 0.88088088
#> [46] 0.90090090 0.92092092 0.94094094 0.96096096 0.98098098
#> [51] 1.00100100 1.02102102 1.04104104 1.06106106 1.08108108
#> [56] 1.10110110 1.12112112 1.14114114 1.16116116 1.18118118
#> [61] 1.20120120 1.22122122 1.24124124 1.26126126 1.28128128
#> [66] 1.30130130 1.32132132 1.34134134 1.36136136 1.38138138
#> [71] 1.40140140 1.42142142 1.44144144 1.46146146 1.48148148
#> [76] 1.50150150 1.52152152 1.54154154 1.56156156 1.58158158
#> [81] 1.60160160 1.62162162 1.64164164 1.66166166 1.68168168
#> [86] 1.70170170 1.72172172 1.74174174 1.76176176 1.78178178
#> [91] 1.80180180 1.82182182 1.84184184 1.86186186 1.88188188
#> [96] 1.90190190 1.92192192 1.94194194 1.96196196 1.98198198
#> [101] 2.00200200 2.02202202 2.04204204 2.06206206 2.08208208
#> [106] 2.10210210 2.12212212 2.14214214 2.16216216 2.18218218
#> [111] 2.20220220 2.22222222 2.24224224 2.26226226 2.28228228
#> [116] 2.30230230 2.32232232 2.34234234 2.36236236 2.38238238
#> [121] 2.40240240 2.42242242 2.44244244 2.46246246 2.48248248
#> [126] 2.50250250 2.52252252 2.54254254 2.56256256 2.58258258
#> [131] 2.60260260 2.62262262 2.64264264 2.66266266 2.68268268
#> [136] 2.70270270 2.72272272 2.74274274 2.76276276 2.78278278
#> [141] 2.80280280 2.82282282 2.84284284 2.86286286 2.88288288
#> [146] 2.90290290 2.92292292 2.94294294 2.96296296 2.98298298
#> [151] 3.00300300 3.02302302 3.04304304 3.06306306 3.08308308
#> [156] 3.10310310 3.12312312 3.14314314 3.16316316 3.18318318
#> [161] 3.20320320 3.22322322 3.24324324 3.26326326 3.28328328
#> [166] 3.30330330 3.32332332 3.34334334 3.36336336 3.38338338
#> [171] 3.40340340 3.42342342 3.44344344 3.46346346 3.48348348
#> [176] 3.50350350 3.52352352 3.54354354 3.56356356 3.58358358
#> [181] 3.60360360 3.62362362 3.64364364 3.66366366 3.68368368
#> [186] 3.70370370 3.72372372 3.74374374 3.76376376 3.78378378
#> [191] 3.80380380 3.82382382 3.84384384 3.86386386 3.88388388
#> [196] 3.90390390 3.92392392 3.94394394 3.96396396 3.98398398
#> [201] 4.00400400 4.02402402 4.04404404 4.06406406 4.08408408
#> [206] 4.10410410 4.12412412 4.14414414 4.16416416 4.18418418
#> [211] 4.20420420 4.22422422 4.24424424 4.26426426 4.28428428
#> [216] 4.30430430 4.32432432 4.34434434 4.36436436 4.38438438
#> [221] 4.40440440 4.42442442 4.44444444 4.46446446 4.48448448
#> [226] 4.50450450 4.52452452 4.54454454 4.56456456 4.58458458
#> [231] 4.60460460 4.62462462 4.64464464 4.66466466 4.68468468
#> [236] 4.70470470 4.72472472 4.74474474 4.76476476 4.78478478
#> [241] 4.80480480 4.82482482 4.84484484 4.86486486 4.88488488
#> [246] 4.90490490 4.92492492 4.94494494 4.96496496 4.98498498
#> [251] 5.00500501 5.02502503 5.04504505 5.06506507 5.08508509
#> [256] 5.10510511 5.12512513 5.14514515 5.16516517 5.18518519
#> [261] 5.20520521 5.22522523 5.24524525 5.26526527 5.28528529
#> [266] 5.30530531 5.32532533 5.34534535 5.36536537 5.38538539
#> [271] 5.40540541 5.42542543 5.44544545 5.46546547 5.48548549
#> [276] 5.50550551 5.52552553 5.54554555 5.56556557 5.58558559
#> [281] 5.60560561 5.62562563 5.64564565 5.66566567 5.68568569
#> [286] 5.70570571 5.72572573 5.74574575 5.76576577 5.78578579
#> [291] 5.80580581 5.82582583 5.84584585 5.86586587 5.88588589
#> [296] 5.90590591 5.92592593 5.94594595 5.96596597 5.98598599
#> [301] 6.00600601 6.02602603 6.04604605 6.06606607 6.08608609
#> [306] 6.10610611 6.12612613 6.14614615 6.16616617 6.18618619
#> [311] 6.20620621 6.22622623 6.24624625 6.26626627 6.28628629
#> [316] 6.30630631 6.32632633 6.34634635 6.36636637 6.38638639
#> [321] 6.40640641 6.42642643 6.44644645 6.46646647 6.48648649
#> [326] 6.50650651 6.52652653 6.54654655 6.56656657 6.58658659
#> [331] 6.60660661 6.62662663 6.64664665 6.66666667 6.68668669
#> [336] 6.70670671 6.72672673 6.74674675 6.76676677 6.78678679
#> [341] 6.80680681 6.82682683 6.84684685 6.86686687 6.88688689
#> [346] 6.90690691 6.92692693 6.94694695 6.96696697 6.98698699
#> [351] 7.00700701 7.02702703 7.04704705 7.06706707 7.08708709
#> [356] 7.10710711 7.12712713 7.14714715 7.16716717 7.18718719
#> [361] 7.20720721 7.22722723 7.24724725 7.26726727 7.28728729
#> [366] 7.30730731 7.32732733 7.34734735 7.36736737 7.38738739
#> [371] 7.40740741 7.42742743 7.44744745 7.46746747 7.48748749
#> [376] 7.50750751 7.52752753 7.54754755 7.56756757 7.58758759
#> [381] 7.60760761 7.62762763 7.64764765 7.66766767 7.68768769
#> [386] 7.70770771 7.72772773 7.74774775 7.76776777 7.78778779
#> [391] 7.80780781 7.82782783 7.84784785 7.86786787 7.88788789
#> [396] 7.90790791 7.92792793 7.94794795 7.96796797 7.98798799
#> [401] 8.00800801 8.02802803 8.04804805 8.06806807 8.08808809
#> [406] 8.10810811 8.12812813 8.14814815 8.16816817 8.18818819
#> [411] 8.20820821 8.22822823 8.24824825 8.26826827 8.28828829
#> [416] 8.30830831 8.32832833 8.34834835 8.36836837 8.38838839
#> [421] 8.40840841 8.42842843 8.44844845 8.46846847 8.48848849
#> [426] 8.50850851 8.52852853 8.54854855 8.56856857 8.58858859
#> [431] 8.60860861 8.62862863 8.64864865 8.66866867 8.68868869
#> [436] 8.70870871 8.72872873 8.74874875 8.76876877 8.78878879
#> [441] 8.80880881 8.82882883 8.84884885 8.86886887 8.88888889
#> [446] 8.90890891 8.92892893 8.94894895 8.96896897 8.98898899
#> [451] 9.00900901 9.02902903 9.04904905 9.06906907 9.08908909
#> [456] 9.10910911 9.12912913 9.14914915 9.16916917 9.18918919
#> [461] 9.20920921 9.22922923 9.24924925 9.26926927 9.28928929
#> [466] 9.30930931 9.32932933 9.34934935 9.36936937 9.38938939
#> [471] 9.40940941 9.42942943 9.44944945 9.46946947 9.48948949
#> [476] 9.50950951 9.52952953 9.54954955 9.56956957 9.58958959
#> [481] 9.60960961 9.62962963 9.64964965 9.66966967 9.68968969
#> [486] 9.70970971 9.72972973 9.74974975 9.76976977 9.78978979
#> [491] 9.80980981 9.82982983 9.84984985 9.86986987 9.88988989
#> [496] 9.90990991 9.92992993 9.94994995 9.96996997 9.98998999
#> [501] 10.01001001 10.03003003 10.05005005 10.07007007 10.09009009
#> [506] 10.11011011 10.13013013 10.15015015 10.17017017 10.19019019
#> [511] 10.21021021 10.23023023 10.25025025 10.27027027 10.29029029
#> [516] 10.31031031 10.33033033 10.35035035 10.37037037 10.39039039
#> [521] 10.41041041 10.43043043 10.45045045 10.47047047 10.49049049
#> [526] 10.51051051 10.53053053 10.55055055 10.57057057 10.59059059
#> [531] 10.61061061 10.63063063 10.65065065 10.67067067 10.69069069
#> [536] 10.71071071 10.73073073 10.75075075 10.77077077 10.79079079
#> [541] 10.81081081 10.83083083 10.85085085 10.87087087 10.89089089
#> [546] 10.91091091 10.93093093 10.95095095 10.97097097 10.99099099
#> [551] 11.01101101 11.03103103 11.05105105 11.07107107 11.09109109
#> [556] 11.11111111 11.13113113 11.15115115 11.17117117 11.19119119
#> [561] 11.21121121 11.23123123 11.25125125 11.27127127 11.29129129
#> [566] 11.31131131 11.33133133 11.35135135 11.37137137 11.39139139
#> [571] 11.41141141 11.43143143 11.45145145 11.47147147 11.49149149
#> [576] 11.51151151 11.53153153 11.55155155 11.57157157 11.59159159
#> [581] 11.61161161 11.63163163 11.65165165 11.67167167 11.69169169
#> [586] 11.71171171 11.73173173 11.75175175 11.77177177 11.79179179
#> [591] 11.81181181 11.83183183 11.85185185 11.87187187 11.89189189
#> [596] 11.91191191 11.93193193 11.95195195 11.97197197 11.99199199
#> [601] 12.01201201 12.03203203 12.05205205 12.07207207 12.09209209
#> [606] 12.11211211 12.13213213 12.15215215 12.17217217 12.19219219
#> [611] 12.21221221 12.23223223 12.25225225 12.27227227 12.29229229
#> [616] 12.31231231 12.33233233 12.35235235 12.37237237 12.39239239
#> [621] 12.41241241 12.43243243 12.45245245 12.47247247 12.49249249
#> [626] 12.51251251 12.53253253 12.55255255 12.57257257 12.59259259
#> [631] 12.61261261 12.63263263 12.65265265 12.67267267 12.69269269
#> [636] 12.71271271 12.73273273 12.75275275 12.77277277 12.79279279
#> [641] 12.81281281 12.83283283 12.85285285 12.87287287 12.89289289
#> [646] 12.91291291 12.93293293 12.95295295 12.97297297 12.99299299
#> [651] 13.01301301 13.03303303 13.05305305 13.07307307 13.09309309
#> [656] 13.11311311 13.13313313 13.15315315 13.17317317 13.19319319
#> [661] 13.21321321 13.23323323 13.25325325 13.27327327 13.29329329
#> [666] 13.31331331 13.33333333 13.35335335 13.37337337 13.39339339
#> [671] 13.41341341 13.43343343 13.45345345 13.47347347 13.49349349
#> [676] 13.51351351 13.53353353 13.55355355 13.57357357 13.59359359
#> [681] 13.61361361 13.63363363 13.65365365 13.67367367 13.69369369
#> [686] 13.71371371 13.73373373 13.75375375 13.77377377 13.79379379
#> [691] 13.81381381 13.83383383 13.85385385 13.87387387 13.89389389
#> [696] 13.91391391 13.93393393 13.95395395 13.97397397 13.99399399
#> [701] 14.01401401 14.03403403 14.05405405 14.07407407 14.09409409
#> [706] 14.11411411 14.13413413 14.15415415 14.17417417 14.19419419
#> [711] 14.21421421 14.23423423 14.25425425 14.27427427 14.29429429
#> [716] 14.31431431 14.33433433 14.35435435 14.37437437 14.39439439
#> [721] 14.41441441 14.43443443 14.45445445 14.47447447 14.49449449
#> [726] 14.51451451 14.53453453 14.55455455 14.57457457 14.59459459
#> [731] 14.61461461 14.63463463 14.65465465 14.67467467 14.69469469
#> [736] 14.71471471 14.73473473 14.75475475 14.77477477 14.79479479
#> [741] 14.81481481 14.83483483 14.85485485 14.87487487 14.89489489
#> [746] 14.91491491 14.93493493 14.95495495 14.97497497 14.99499499
#> [751] 15.01501502 15.03503504 15.05505506 15.07507508 15.09509510
#> [756] 15.11511512 15.13513514 15.15515516 15.17517518 15.19519520
#> [761] 15.21521522 15.23523524 15.25525526 15.27527528 15.29529530
#> [766] 15.31531532 15.33533534 15.35535536 15.37537538 15.39539540
#> [771] 15.41541542 15.43543544 15.45545546 15.47547548 15.49549550
#> [776] 15.51551552 15.53553554 15.55555556 15.57557558 15.59559560
#> [781] 15.61561562 15.63563564 15.65565566 15.67567568 15.69569570
#> [786] 15.71571572 15.73573574 15.75575576 15.77577578 15.79579580
#> [791] 15.81581582 15.83583584 15.85585586 15.87587588 15.89589590
#> [796] 15.91591592 15.93593594 15.95595596 15.97597598 15.99599600
#> [801] 16.01601602 16.03603604 16.05605606 16.07607608 16.09609610
#> [806] 16.11611612 16.13613614 16.15615616 16.17617618 16.19619620
#> [811] 16.21621622 16.23623624 16.25625626 16.27627628 16.29629630
#> [816] 16.31631632 16.33633634 16.35635636 16.37637638 16.39639640
#> [821] 16.41641642 16.43643644 16.45645646 16.47647648 16.49649650
#> [826] 16.51651652 16.53653654 16.55655656 16.57657658 16.59659660
#> [831] 16.61661662 16.63663664 16.65665666 16.67667668 16.69669670
#> [836] 16.71671672 16.73673674 16.75675676 16.77677678 16.79679680
#> [841] 16.81681682 16.83683684 16.85685686 16.87687688 16.89689690
#> [846] 16.91691692 16.93693694 16.95695696 16.97697698 16.99699700
#> [851] 17.01701702 17.03703704 17.05705706 17.07707708 17.09709710
#> [856] 17.11711712 17.13713714 17.15715716 17.17717718 17.19719720
#> [861] 17.21721722 17.23723724 17.25725726 17.27727728 17.29729730
#> [866] 17.31731732 17.33733734 17.35735736 17.37737738 17.39739740
#> [871] 17.41741742 17.43743744 17.45745746 17.47747748 17.49749750
#> [876] 17.51751752 17.53753754 17.55755756 17.57757758 17.59759760
#> [881] 17.61761762 17.63763764 17.65765766 17.67767768 17.69769770
#> [886] 17.71771772 17.73773774 17.75775776 17.77777778 17.79779780
#> [891] 17.81781782 17.83783784 17.85785786 17.87787788 17.89789790
#> [896] 17.91791792 17.93793794 17.95795796 17.97797798 17.99799800
#> [901] 18.01801802 18.03803804 18.05805806 18.07807808 18.09809810
#> [906] 18.11811812 18.13813814 18.15815816 18.17817818 18.19819820
#> [911] 18.21821822 18.23823824 18.25825826 18.27827828 18.29829830
#> [916] 18.31831832 18.33833834 18.35835836 18.37837838 18.39839840
#> [921] 18.41841842 18.43843844 18.45845846 18.47847848 18.49849850
#> [926] 18.51851852 18.53853854 18.55855856 18.57857858 18.59859860
#> [931] 18.61861862 18.63863864 18.65865866 18.67867868 18.69869870
#> [936] 18.71871872 18.73873874 18.75875876 18.77877878 18.79879880
#> [941] 18.81881882 18.83883884 18.85885886 18.87887888 18.89889890
#> [946] 18.91891892 18.93893894 18.95895896 18.97897898 18.99899900
#> [951] 19.01901902 19.03903904 19.05905906 19.07907908 19.09909910
#> [956] 19.11911912 19.13913914 19.15915916 19.17917918 19.19919920
#> [961] 19.21921922 19.23923924 19.25925926 19.27927928 19.29929930
#> [966] 19.31931932 19.33933934 19.35935936 19.37937938 19.39939940
#> [971] 19.41941942 19.43943944 19.45945946 19.47947948 19.49949950
#> [976] 19.51951952 19.53953954 19.55955956 19.57957958 19.59959960
#> [981] 19.61961962 19.63963964 19.65965966 19.67967968 19.69969970
#> [986] 19.71971972 19.73973974 19.75975976 19.77977978 19.79979980
#> [991] 19.81981982 19.83983984 19.85985986 19.87987988 19.89989990
#> [996] 19.91991992 19.93993994 19.95995996 19.97997998 20.00000000
#>
#> $x
#> [1] 0.04166667 0.20833333 0.37500000 0.31250000 0.06250000
#> [6] -0.03125000 -0.11458333 -0.11458333 0.16493056 0.09548611
#>
#> $cx
#> [1] 1.0399306 4.2413194 7.5468750 5.9869792 1.1848958 -0.6423611
#> [7] -2.2031250 -2.2204861 3.2397280 1.8262442
#>
#> $irwd
#> [1] 0.00000000 0.01889809 0.03584589 0.05122592 0.06535314 0.07848544
#> [7] 0.09083263 0.10256414 0.11381561 0.12469446 0.13528472 0.14565104
#> [13] 0.15584217 0.16589384 0.17583122 0.18567102 0.19542318 0.20509233
#> [19] 0.21467901 0.22418068 0.23359247 0.24290797 0.25211966 0.26121943
#> [25] 0.27019892 0.27904977 0.28776385 0.29633345 0.30475136 0.31301097
#> [31] 0.32110634 0.32903221 0.33678405 0.34435802 0.35175096 0.35896042
#> [37] 0.36598457 0.37282220 0.37947266 0.38593587 0.39221221 0.39830256
#> [43] 0.40420817 0.40993070 0.41547216 0.42083483 0.42602130 0.43103436
#> [49] 0.43587705 0.44055255 0.44506421 0.44941552 0.45361004 0.45765145
#> [55] 0.46154347 0.46528987 0.46889447 0.47236107 0.47569350 0.47889558
#> [61] 0.48197109 0.48492381 0.48775746 0.49047573 0.49308225 0.49558060
#> [67] 0.49797429 0.50026679 0.50246146 0.50456164 0.50657055 0.50849136
#> [73] 0.51032717 0.51208099 0.51375574 0.51535430 0.51687944 0.51833386
#> [79] 0.51972018 0.52104096 0.52229866 0.52349568 0.52463435 0.52571692
#> [85] 0.52674557 0.52772240 0.52864946 0.52952871 0.53036208 0.53115140
#> [91] 0.53189846 0.53260496 0.53327259 0.53390293 0.53449754 0.53505791
#> [97] 0.53558548 0.53608163 0.53654771 0.53698499 0.53739473 0.53777813
#> [103] 0.53813632 0.53847043 0.53878152 0.53907061 0.53933870 0.53958674
#> [109] 0.53981564 0.54002628 0.54021952 0.54039615 0.54055698 0.54070273
#> [115] 0.54083415 0.54095191 0.54105670 0.54114914 0.54122985 0.54129942
#> [121] 0.54135841 0.54140737 0.54144682 0.54147726 0.54149916 0.54151299
#> [127] 0.54151918 0.54151815 0.54151032 0.54149606 0.54147575 0.54144974
#> [133] 0.54141837 0.54138196 0.54134083 0.54129527 0.54124558 0.54119200
#> [139] 0.54113482 0.54107428 0.54101060 0.54094403 0.54087477 0.54080303
#> [145] 0.54072901 0.54065289 0.54057486 0.54049508 0.54041371 0.54033092
#> [151] 0.54024684 0.54016161 0.54007538 0.53998826 0.53990037 0.53981183
#> [157] 0.53972274 0.53963322 0.53954334 0.53945322 0.53936292 0.53927255
#> [163] 0.53918216 0.53909184 0.53900166 0.53891167 0.53882195 0.53873255
#> [169] 0.53864351 0.53855491 0.53846677 0.53837915 0.53829209 0.53820563
#> [175] 0.53811980 0.53803464 0.53795017 0.53786644 0.53778345 0.53770125
#> [181] 0.53761984 0.53753926 0.53745952 0.53738064 0.53730263 0.53722551
#> [187] 0.53714929 0.53707398 0.53699959 0.53692613 0.53685361 0.53678203
#> [193] 0.53671139 0.53664171 0.53657298 0.53650521 0.53643840 0.53637254
#> [199] 0.53630764 0.53624370 0.53618071 0.53611867 0.53605758 0.53599743
#> [205] 0.53593822 0.53587995 0.53582261 0.53576619 0.53571069 0.53565610
#> [211] 0.53560241 0.53554962 0.53549772 0.53544669 0.53539655 0.53534726
#> [217] 0.53529883 0.53525125 0.53520451 0.53515859 0.53511350 0.53506921
#> [223] 0.53502573 0.53498303 0.53494111 0.53489997 0.53485959 0.53481995
#> [229] 0.53478106 0.53474289 0.53470545 0.53466872 0.53463269 0.53459734
#> [235] 0.53456268 0.53452869 0.53449535 0.53446267 0.53443062 0.53439921
#> [241] 0.53436841 0.53433823 0.53430864 0.53427964 0.53425123 0.53422338
#> [247] 0.53419610 0.53416937 0.53414318 0.53411752 0.53409239 0.53406778
#> [253] 0.53404367 0.53402005 0.53399693 0.53397428 0.53395211 0.53393040
#> [259] 0.53390914 0.53388833 0.53386796 0.53384801 0.53382849 0.53380938
#> [265] 0.53379068 0.53377238 0.53375447 0.53373694 0.53371978 0.53370300
#> [271] 0.53368657 0.53367051 0.53365479 0.53363941 0.53362436 0.53360964
#> [277] 0.53359525 0.53358117 0.53356739 0.53355392 0.53354075 0.53352787
#> [283] 0.53351527 0.53350295 0.53349090 0.53347912 0.53346760 0.53345634
#> [289] 0.53344533 0.53343457 0.53342405 0.53341376 0.53340370 0.53339388
#> [295] 0.53338427 0.53337488 0.53336570 0.53335673 0.53334796 0.53333939
#> [301] 0.53333102 0.53332284 0.53331484 0.53330703 0.53329939 0.53329193
#> [307] 0.53328464 0.53327752 0.53327057 0.53326377 0.53325713 0.53325064
#> [313] 0.53324430 0.53323811 0.53323206 0.53322616 0.53322039 0.53321475
#> [319] 0.53320925 0.53320387 0.53319862 0.53319349 0.53318848 0.53318359
#> [325] 0.53317882 0.53317415 0.53316960 0.53316515 0.53316080 0.53315656
#> [331] 0.53315242 0.53314838 0.53314443 0.53314058 0.53313681 0.53313314
#> [337] 0.53312955 0.53312605 0.53312263 0.53311930 0.53311604 0.53311286
#> [343] 0.53310976 0.53310673 0.53310377 0.53310088 0.53309806 0.53309531
#> [349] 0.53309263 0.53309001 0.53308745 0.53308496 0.53308252 0.53308014
#> [355] 0.53307782 0.53307556 0.53307335 0.53307119 0.53306909 0.53306704
#> [361] 0.53306503 0.53306308 0.53306117 0.53305931 0.53305750 0.53305572
#> [367] 0.53305400 0.53305231 0.53305066 0.53304906 0.53304749 0.53304596
#> [373] 0.53304447 0.53304302 0.53304160 0.53304022 0.53303886 0.53303755
#> [379] 0.53303626 0.53303501 0.53303379 0.53303259 0.53303143 0.53303030
#> [385] 0.53302919 0.53302811 0.53302706 0.53302603 0.53302503 0.53302405
#> [391] 0.53302310 0.53302217 0.53302126 0.53302038 0.53301951 0.53301867
#> [397] 0.53301785 0.53301705 0.53301627 0.53301551 0.53301477 0.53301405
#> [403] 0.53301334 0.53301266 0.53301198 0.53301133 0.53301069 0.53301007
#> [409] 0.53300946 0.53300887 0.53300830 0.53300773 0.53300719 0.53300665
#> [415] 0.53300613 0.53300562 0.53300513 0.53300465 0.53300417 0.53300372
#> [421] 0.53300327 0.53300283 0.53300241 0.53300199 0.53300159 0.53300119
#> [427] 0.53300081 0.53300043 0.53300007 0.53299971 0.53299937 0.53299903
#> [433] 0.53299870 0.53299838 0.53299806 0.53299776 0.53299746 0.53299717
#> [439] 0.53299689 0.53299661 0.53299634 0.53299608 0.53299583 0.53299558
#> [445] 0.53299534 0.53299510 0.53299487 0.53299464 0.53299442 0.53299421
#> [451] 0.53299400 0.53299380 0.53299360 0.53299341 0.53299322 0.53299304
#> [457] 0.53299286 0.53299269 0.53299252 0.53299235 0.53299219 0.53299204
#> [463] 0.53299188 0.53299174 0.53299159 0.53299145 0.53299131 0.53299118
#> [469] 0.53299105 0.53299092 0.53299080 0.53299067 0.53299056 0.53299044
#> [475] 0.53299033 0.53299022 0.53299011 0.53299001 0.53298991 0.53298981
#> [481] 0.53298972 0.53298962 0.53298953 0.53298944 0.53298936 0.53298927
#> [487] 0.53298919 0.53298911 0.53298903 0.53298896 0.53298888 0.53298881
#> [493] 0.53298874 0.53298867 0.53298861 0.53298854 0.53298848 0.53298842
#> [499] 0.53298836 0.53298830 0.53298824 0.53298819 0.53298813 0.53298808
#> [505] 0.53298803 0.53298798 0.53298793 0.53298788 0.53298784 0.53298779
#> [511] 0.53298775 0.53298771 0.53298766 0.53298762 0.53298758 0.53298755
#> [517] 0.53298751 0.53298747 0.53298744 0.53298740 0.53298737 0.53298733
#> [523] 0.53298730 0.53298727 0.53298724 0.53298721 0.53298718 0.53298716
#> [529] 0.53298713 0.53298710 0.53298708 0.53298705 0.53298703 0.53298700
#> [535] 0.53298698 0.53298696 0.53298693 0.53298691 0.53298689 0.53298687
#> [541] 0.53298685 0.53298683 0.53298681 0.53298679 0.53298678 0.53298676
#> [547] 0.53298674 0.53298673 0.53298671 0.53298669 0.53298668 0.53298666
#> [553] 0.53298665 0.53298663 0.53298662 0.53298661 0.53298659 0.53298658
#> [559] 0.53298657 0.53298656 0.53298655 0.53298653 0.53298652 0.53298651
#> [565] 0.53298650 0.53298649 0.53298648 0.53298647 0.53298646 0.53298645
#> [571] 0.53298644 0.53298643 0.53298643 0.53298642 0.53298641 0.53298640
#> [577] 0.53298639 0.53298639 0.53298638 0.53298637 0.53298637 0.53298636
#> [583] 0.53298635 0.53298635 0.53298634 0.53298633 0.53298633 0.53298632
#> [589] 0.53298632 0.53298631 0.53298631 0.53298630 0.53298630 0.53298629
#> [595] 0.53298629 0.53298628 0.53298628 0.53298627 0.53298627 0.53298626
#> [601] 0.53298626 0.53298626 0.53298625 0.53298625 0.53298624 0.53298624
#> [607] 0.53298624 0.53298623 0.53298623 0.53298623 0.53298622 0.53298622
#> [613] 0.53298622 0.53298622 0.53298621 0.53298621 0.53298621 0.53298621
#> [619] 0.53298620 0.53298620 0.53298620 0.53298620 0.53298619 0.53298619
#> [625] 0.53298619 0.53298619 0.53298619 0.53298618 0.53298618 0.53298618
#> [631] 0.53298618 0.53298618 0.53298617 0.53298617 0.53298617 0.53298617
#> [637] 0.53298617 0.53298617 0.53298616 0.53298616 0.53298616 0.53298616
#> [643] 0.53298616 0.53298616 0.53298616 0.53298616 0.53298615 0.53298615
#> [649] 0.53298615 0.53298615 0.53298615 0.53298615 0.53298615 0.53298615
#> [655] 0.53298615 0.53298614 0.53298614 0.53298614 0.53298614 0.53298614
#> [661] 0.53298614 0.53298614 0.53298614 0.53298614 0.53298614 0.53298614
#> [667] 0.53298614 0.53298614 0.53298613 0.53298613 0.53298613 0.53298613
#> [673] 0.53298613 0.53298613 0.53298613 0.53298613 0.53298613 0.53298613
#> [679] 0.53298613 0.53298613 0.53298613 0.53298613 0.53298613 0.53298613
#> [685] 0.53298613 0.53298613 0.53298613 0.53298613 0.53298612 0.53298612
#> [691] 0.53298612 0.53298612 0.53298612 0.53298612 0.53298612 0.53298612
#> [697] 0.53298612 0.53298612 0.53298612 0.53298612 0.53298612 0.53298612
#> [703] 0.53298612 0.53298612 0.53298612 0.53298612 0.53298612 0.53298612
#> [709] 0.53298612 0.53298612 0.53298612 0.53298612 0.53298612 0.53298612
#> [715] 0.53298612 0.53298612 0.53298612 0.53298612 0.53298612 0.53298612
#> [721] 0.53298612 0.53298612 0.53298612 0.53298612 0.53298612 0.53298612
#> [727] 0.53298612 0.53298612 0.53298612 0.53298612 0.53298612 0.53298612
#> [733] 0.53298612 0.53298612 0.53298612 0.53298611 0.53298611 0.53298611
#> [739] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [745] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [751] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [757] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [763] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [769] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [775] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [781] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [787] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [793] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [799] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [805] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [811] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [817] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [823] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [829] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [835] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [841] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [847] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [853] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [859] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [865] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [871] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [877] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [883] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [889] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [895] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [901] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [907] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [913] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [919] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [925] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [931] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [937] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [943] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [949] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [955] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [961] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [967] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [973] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [979] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [985] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [991] 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611 0.53298611
#> [997] 0.53298611 0.53298611 0.53298611 0.53298611
#>
#> $crwd
#> [1] 0.000000000 0.000192783 0.000743683 0.001617607 0.002786423
#> [6] 0.004227720 0.005923753 0.007860561 0.010027228 0.012415261
#> [11] 0.015018078 0.017830576 0.020848785 0.024069574 0.027490420
#> [16] 0.031109214 0.034924111 0.038933413 0.043135467 0.047528602
#> [21] 0.052111071 0.056881011 0.061836423 0.066975146 0.072294856
#> [26] 0.077793063 0.083467108 0.089314175 0.095331298 0.101515375
#> [31] 0.107863174 0.114371354 0.121036475 0.127855014 0.134823376
#> [36] 0.141937912 0.149194929 0.156590704 0.164121496 0.171783555
#> [41] 0.179573136 0.187486506 0.195519954 0.203669796 0.211932387
#> [46] 0.220304124 0.228781454 0.237360877 0.246038950 0.254812295
#> [51] 0.263677599 0.272631614 0.281671168 0.290793157 0.299994553
#> [56] 0.309272404 0.318623832 0.328046036 0.337536293 0.347091953
#> [61] 0.356710446 0.366389275 0.376126020 0.385918334 0.395763942
#> [66] 0.405660645 0.415606311 0.425598880 0.435636361 0.445716827
#> [71] 0.455838420 0.465999344 0.476197867 0.486432317 0.496701082
#> [76] 0.507002609 0.517335399 0.527698010 0.538089053 0.548507190
#> [81] 0.558951133 0.569419644 0.579911532 0.590425650 0.600960898
#> [86] 0.611516218 0.622090592 0.632683044 0.643292637 0.653918469
#> [91] 0.664559678 0.675215434 0.685884943 0.696567441 0.707262199
#> [96] 0.717968516 0.728685720 0.739413170 0.750150250 0.760896370
#> [101] 0.771650966 0.782413500 0.793183456 0.803960340 0.814743680
#> [106] 0.825533026 0.836327948 0.847128036 0.857932895 0.868742153
#> [111] 0.879555453 0.890372454 0.901192831 0.912016276 0.922842495
#> [116] 0.933671206 0.944502144 0.955335056 0.966169699 0.977005846
#> [121] 0.987843279 0.998681791 1.009521188 1.020361283 1.031201902
#> [126] 1.042042878 1.052884053 1.063725279 1.074566416 1.085407331
#> [131] 1.096247899 1.107088003 1.117927533 1.128766383 1.139604456
#> [136] 1.150441662 1.161277913 1.172113130 1.182947239 1.193780168
#> [141] 1.204611853 1.215442235 1.226271256 1.237098865 1.247925016
#> [146] 1.258749663 1.269572766 1.280394290 1.291214200 1.302032467
#> [151] 1.312849063 1.323663965 1.334477149 1.345288598 1.356098295
#> [156] 1.366906226 1.377712379 1.388516744 1.399319312 1.410120079
#> [161] 1.420919039 1.431716191 1.442511534 1.453305067 1.464096793
#> [166] 1.474886716 1.485674840 1.496461171 1.507245715 1.518028481
#> [171] 1.528809478 1.539588716 1.550366205 1.561141957 1.571915984
#> [176] 1.582688300 1.593458917 1.604227851 1.614995116 1.625760727
#> [181] 1.636524701 1.647287053 1.658047800 1.668806959 1.679564548
#> [186] 1.690320584 1.701075085 1.711828069 1.722579555 1.733329560
#> [191] 1.744078105 1.754825206 1.765570885 1.776315158 1.787058047
#> [196] 1.797799568 1.808539743 1.819278590 1.830016127 1.840752375
#> [201] 1.851487353 1.862221079 1.872953572 1.883684852 1.894414937
#> [206] 1.905143846 1.915871598 1.926598211 1.937323703 1.948048094
#> [211] 1.958771401 1.969493642 1.980214835 1.990934998 2.001654148
#> [216] 2.012372303 2.023089479 2.033805695 2.044520966 2.055235310
#> [221] 2.065948743 2.076661282 2.087372941 2.098083739 2.108793689
#> [226] 2.119502807 2.130211110 2.140918611 2.151625327 2.162331271
#> [231] 2.173036459 2.183740904 2.194444620 2.205147622 2.215849924
#> [236] 2.226551538 2.237252478 2.247952758 2.258652389 2.269351385
#> [241] 2.280049759 2.290747522 2.301444687 2.312141265 2.322837269
#> [246] 2.333532710 2.344227598 2.354921947 2.365615765 2.376309064
#> [251] 2.387001855 2.397694149 2.408385954 2.419077282 2.429768141
#> [256] 2.440458543 2.451148496 2.461838010 2.472527094 2.483215756
#> [261] 2.493904007 2.504591854 2.515279306 2.525966371 2.536653057
#> [266] 2.547339374 2.558025327 2.568710926 2.579396178 2.590081090
#> [271] 2.600765670 2.611449924 2.622133861 2.632817486 2.643500806
#> [276] 2.654183828 2.664866560 2.675549006 2.686231173 2.696913067
#> [281] 2.707594695 2.718276062 2.728957174 2.739638037 2.750318656
#> [286] 2.760999036 2.771679183 2.782359102 2.793038798 2.803718276
#> [291] 2.814397541 2.825076597 2.835755450 2.846434105 2.857112564
#> [296] 2.867790833 2.878468917 2.889146819 2.899824543 2.910502094
#> [301] 2.921179475 2.931856690 2.942533744 2.953210639 2.963887380
#> [306] 2.974563969 2.985240411 2.995916709 3.006592866 3.017268885
#> [311] 3.027944769 3.038620522 3.049296147 3.059971647 3.070647024
#> [316] 3.081322281 3.091997421 3.102672447 3.113347362 3.124022168
#> [321] 3.134696867 3.145371463 3.156045957 3.166720352 3.177394650
#> [326] 3.188068853 3.198742965 3.209416986 3.220090919 3.230764767
#> [331] 3.241438530 3.252112212 3.262785813 3.273459337 3.284132784
#> [336] 3.294806157 3.305479457 3.316152686 3.326825846 3.337498938
#> [341] 3.348171964 3.358844926 3.369517825 3.380190663 3.390863440
#> [346] 3.401536159 3.412208821 3.422881427 3.433553979 3.444226478
#> [351] 3.454898925 3.465571321 3.476243668 3.486915967 3.497588219
#> [356] 3.508260425 3.518932586 3.529604703 3.540276778 3.550948812
#> [361] 3.561620804 3.572292757 3.582964672 3.593636548 3.604308388
#> [366] 3.614980192 3.625651961 3.636323695 3.646995397 3.657667066
#> [371] 3.668338703 3.679010309 3.689681885 3.700353431 3.711024949
#> [376] 3.721696438 3.732367900 3.743039336 3.753710745 3.764382129
#> [381] 3.775053489 3.785724824 3.796396135 3.807067424 3.817738690
#> [386] 3.828409934 3.839081157 3.849752359 3.860423540 3.871094702
#> [391] 3.881765845 3.892436968 3.903108074 3.913779161 3.924450231
#> [396] 3.935121284 3.945792320 3.956463340 3.967134345 3.977805333
#> [401] 3.988476307 3.999147266 4.009818211 4.020489142 4.031160059
#> [406] 4.041830963 4.052501854 4.063172733 4.073843599 4.084514453
#> [411] 4.095185296 4.105856127 4.116526947 4.127197756 4.137868554
#> [416] 4.148539343 4.159210121 4.169880890 4.180551648 4.191222398
#> [421] 4.201893139 4.212563870 4.223234593 4.233905308 4.244576015
#> [426] 4.255246713 4.265917404 4.276588087 4.287258762 4.297929431
#> [431] 4.308600092 4.319270747 4.329941395 4.340612036 4.351282671
#> [436] 4.361953300 4.372623923 4.383294540 4.393965151 4.404635757
#> [441] 4.415306357 4.425976952 4.436647541 4.447318126 4.457988705
#> [446] 4.468659280 4.479329851 4.490000416 4.500670977 4.511341534
#> [451] 4.522012087 4.532682636 4.543353180 4.554023721 4.564694258
#> [456] 4.575364791 4.586035320 4.596705846 4.607376369 4.618046888
#> [461] 4.628717404 4.639387917 4.650058427 4.660728933 4.671399437
#> [466] 4.682069938 4.692740436 4.703410931 4.714081424 4.724751914
#> [471] 4.735422402 4.746092887 4.756763370 4.767433850 4.778104328
#> [476] 4.788774804 4.799445278 4.810115750 4.820786220 4.831456687
#> [481] 4.842127153 4.852797617 4.863468079 4.874138539 4.884808998
#> [486] 4.895479454 4.906149909 4.916820363 4.927490815 4.938161265
#> [491] 4.948831714 4.959502161 4.970172607 4.980843052 4.991513495
#> [496] 5.002183937 5.012854378 5.023524817 5.034195255 5.044865692
#> [501] 5.055536128 5.066206563 5.076876996 5.087547429 5.098217860
#> [506] 5.108888291 5.119558720 5.130229149 5.140899576 5.151570003
#> [511] 5.162240429 5.172910854 5.183581278 5.194251701 5.204922124
#> [516] 5.215592545 5.226262966 5.236933387 5.247603806 5.258274225
#> [521] 5.268944643 5.279615060 5.290285477 5.300955893 5.311626309
#> [526] 5.322296724 5.332967138 5.343637552 5.354307965 5.364978378
#> [531] 5.375648790 5.386319202 5.396989613 5.407660023 5.418330434
#> [536] 5.429000843 5.439671253 5.450341662 5.461012070 5.471682478
#> [541] 5.482352886 5.493023293 5.503693700 5.514364106 5.525034512
#> [546] 5.535704918 5.546375323 5.557045728 5.567716133 5.578386537
#> [551] 5.589056942 5.599727345 5.610397749 5.621068152 5.631738555
#> [556] 5.642408958 5.653079360 5.663749762 5.674420164 5.685090566
#> [561] 5.695760967 5.706431368 5.717101769 5.727772170 5.738442571
#> [566] 5.749112971 5.759783371 5.770453771 5.781124171 5.791794570
#> [571] 5.802464970 5.813135369 5.823805768 5.834476167 5.845146565
#> [576] 5.855816964 5.866487362 5.877157760 5.887828158 5.898498556
#> [581] 5.909168954 5.919839352 5.930509749 5.941180146 5.951850544
#> [586] 5.962520941 5.973191338 5.983861735 5.994532131 6.005202528
#> [591] 6.015872925 6.026543321 6.037213717 6.047884114 6.058554510
#> [596] 6.069224906 6.079895302 6.090565698 6.101236094 6.111906489
#> [601] 6.122576885 6.133247280 6.143917676 6.154588071 6.165258467
#> [606] 6.175928862 6.186599257 6.197269652 6.207940047 6.218610442
#> [611] 6.229280837 6.239951232 6.250621627 6.261292021 6.271962416
#> [616] 6.282632811 6.293303205 6.303973600 6.314643994 6.325314389
#> [621] 6.335984783 6.346655177 6.357325572 6.367995966 6.378666360
#> [626] 6.389336754 6.400007148 6.410677542 6.421347936 6.432018330
#> [631] 6.442688724 6.453359118 6.464029512 6.474699906 6.485370300
#> [636] 6.496040694 6.506711087 6.517381481 6.528051875 6.538722268
#> [641] 6.549392662 6.560063056 6.570733449 6.581403843 6.592074236
#> [646] 6.602744630 6.613415023 6.624085417 6.634755810 6.645426204
#> [651] 6.656096597 6.666766990 6.677437384 6.688107777 6.698778170
#> [656] 6.709448564 6.720118957 6.730789350 6.741459743 6.752130137
#> [661] 6.762800530 6.773470923 6.784141316 6.794811709 6.805482102
#> [666] 6.816152495 6.826822889 6.837493282 6.848163675 6.858834068
#> [671] 6.869504461 6.880174854 6.890845247 6.901515640 6.912186033
#> [676] 6.922856426 6.933526819 6.944197212 6.954867605 6.965537998
#> [681] 6.976208391 6.986878784 6.997549177 7.008219570 7.018889963
#> [686] 7.029560355 7.040230748 7.050901141 7.061571534 7.072241927
#> [691] 7.082912320 7.093582713 7.104253106 7.114923498 7.125593891
#> [696] 7.136264284 7.146934677 7.157605070 7.168275463 7.178945855
#> [701] 7.189616248 7.200286641 7.210957034 7.221627427 7.232297819
#> [706] 7.242968212 7.253638605 7.264308998 7.274979390 7.285649783
#> [711] 7.296320176 7.306990569 7.317660961 7.328331354 7.339001747
#> [716] 7.349672140 7.360342532 7.371012925 7.381683318 7.392353711
#> [721] 7.403024103 7.413694496 7.424364889 7.435035281 7.445705674
#> [726] 7.456376067 7.467046460 7.477716852 7.488387245 7.499057638
#> [731] 7.509728030 7.520398423 7.531068816 7.541739208 7.552409601
#> [736] 7.563079994 7.573750386 7.584420779 7.595091172 7.605761564
#> [741] 7.616431957 7.627102350 7.637772742 7.648443135 7.659113528
#> [746] 7.669783920 7.680454313 7.691124706 7.701795098 7.712465491
#> [751] 7.723135884 7.733806276 7.744476669 7.755147062 7.765817454
#> [756] 7.776487847 7.787158240 7.797828632 7.808499025 7.819169418
#> [761] 7.829839810 7.840510203 7.851180595 7.861850988 7.872521381
#> [766] 7.883191773 7.893862166 7.904532559 7.915202951 7.925873344
#> [771] 7.936543737 7.947214129 7.957884522 7.968554914 7.979225307
#> [776] 7.989895700 8.000566092 8.011236485 8.021906878 8.032577270
#> [781] 8.043247663 8.053918055 8.064588448 8.075258841 8.085929233
#> [786] 8.096599626 8.107270019 8.117940411 8.128610804 8.139281196
#> [791] 8.149951589 8.160621982 8.171292374 8.181962767 8.192633159
#> [796] 8.203303552 8.213973945 8.224644337 8.235314730 8.245985123
#> [801] 8.256655515 8.267325908 8.277996300 8.288666693 8.299337086
#> [806] 8.310007478 8.320677871 8.331348263 8.342018656 8.352689049
#> [811] 8.363359441 8.374029834 8.384700226 8.395370619 8.406041012
#> [816] 8.416711404 8.427381797 8.438052190 8.448722582 8.459392975
#> [821] 8.470063367 8.480733760 8.491404153 8.502074545 8.512744938
#> [826] 8.523415330 8.534085723 8.544756116 8.555426508 8.566096901
#> [831] 8.576767293 8.587437686 8.598108079 8.608778471 8.619448864
#> [836] 8.630119256 8.640789649 8.651460042 8.662130434 8.672800827
#> [841] 8.683471219 8.694141612 8.704812005 8.715482397 8.726152790
#> [846] 8.736823182 8.747493575 8.758163968 8.768834360 8.779504753
#> [851] 8.790175146 8.800845538 8.811515931 8.822186323 8.832856716
#> [856] 8.843527109 8.854197501 8.864867894 8.875538286 8.886208679
#> [861] 8.896879072 8.907549464 8.918219857 8.928890249 8.939560642
#> [866] 8.950231035 8.960901427 8.971571820 8.982242212 8.992912605
#> [871] 9.003582998 9.014253390 9.024923783 9.035594175 9.046264568
#> [876] 9.056934961 9.067605353 9.078275746 9.088946138 9.099616531
#> [881] 9.110286924 9.120957316 9.131627709 9.142298101 9.152968494
#> [886] 9.163638887 9.174309279 9.184979672 9.195650064 9.206320457
#> [891] 9.216990850 9.227661242 9.238331635 9.249002027 9.259672420
#> [896] 9.270342813 9.281013205 9.291683598 9.302353990 9.313024383
#> [901] 9.323694776 9.334365168 9.345035561 9.355705953 9.366376346
#> [906] 9.377046739 9.387717131 9.398387524 9.409057916 9.419728309
#> [911] 9.430398702 9.441069094 9.451739487 9.462409879 9.473080272
#> [916] 9.483750665 9.494421057 9.505091450 9.515761842 9.526432235
#> [921] 9.537102628 9.547773020 9.558443413 9.569113805 9.579784198
#> [926] 9.590454591 9.601124983 9.611795376 9.622465768 9.633136161
#> [931] 9.643806554 9.654476946 9.665147339 9.675817731 9.686488124
#> [936] 9.697158517 9.707828909 9.718499302 9.729169694 9.739840087
#> [941] 9.750510480 9.761180872 9.771851265 9.782521657 9.793192050
#> [946] 9.803862443 9.814532835 9.825203228 9.835873620 9.846544013
#> [951] 9.857214406 9.867884798 9.878555191 9.889225583 9.899895976
#> [956] 9.910566369 9.921236761 9.931907154 9.942577546 9.953247939
#> [961] 9.963918332 9.974588724 9.985259117 9.995929509 10.006599902
#> [966] 10.017270295 10.027940687 10.038611080 10.049281472 10.059951865
#> [971] 10.070622258 10.081292650 10.091963043 10.102633435 10.113303828
#> [976] 10.123974221 10.134644613 10.145315006 10.155985399 10.166655791
#> [981] 10.177326184 10.187996576 10.198666969 10.209337362 10.220007754
#> [986] 10.230678147 10.241348539 10.252018932 10.262689325 10.273359717
#> [991] 10.284030110 10.294700502 10.305370895 10.316041288 10.326711680
#> [996] 10.337382073 10.348052465 10.358722858 10.369393251 10.380063643