Numpy库学习

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Learning Numpy, a lib of python with matrix and other calculation.

Prepare

Generate matrices A, with random Gaussian entries, B, a Toeplitz matrix, where A ∈Rn×m and B ∈Rm×m, for n = 200, m = 500.

Code

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import numpy 
from scipy.linalg import toeplitz

numpy.random.seed()
n = 200
m = 500

A = numpy.random.normal(size = [n, m])
B = toeplitz(numpy.random.normal(size=m))

print("------------A-----------\n"+str(A))
print("------------B-----------\n"+str(B))

Result

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[[-0.49837536 -0.3247838   0.50678674 ..., -1.2703934   0.76603943
   0.03755445]
 [-0.03462894 -0.21418493  1.22520135 ...,  0.84866266  0.40158282
  -0.65401731]
 [ 1.1291174   0.49403126 -0.28698245 ...,  0.61324651  0.73156374
  -1.58197964]
 ...,
 [-0.68737602 -0.89741706  0.12970931 ..., -0.82061635  0.09854131
   1.00242481]
 [-0.31053207 -0.05350655 -1.51289055 ..., -2.26626492  0.79240834
  -0.73548947]
 [ 0.29646267  0.44302565  1.34552379 ...,  0.12526137 -0.74942753
  -1.40240914]]
------------B-----------
[[ 1.44784231 -0.33696399 -0.1432767  ...,  0.26606438 -0.09475546
  -1.50552307]
 [-0.33696399  1.44784231 -0.33696399 ..., -1.16119122  0.26606438
  -0.09475546]
 [-0.1432767  -0.33696399  1.44784231 ..., -1.39562368 -1.16119122
   0.26606438]
 ...,
 [ 0.26606438 -1.16119122 -1.39562368 ...,  1.44784231 -0.33696399
  -0.1432767 ]
 [-0.09475546  0.26606438 -1.16119122 ..., -0.33696399  1.44784231
  -0.33696399]
 [-1.50552307 -0.09475546  0.26606438 ..., -0.1432767  -0.33696399
   1.44784231]]

Knowledge

  1. random.normal() to create a matrix with Gaussian entries.
  2. toepltitz() to create a toepltitz matrix.
  3. random.seed() , without argument, set random seed by system time.

Exercise 9.1: Matrix operations

Calculate A + A, AA’, A’A and AB. Write a function that computes A(B − λI) for any λ.

Code

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A_add_A = A + A
A_mul_AT = numpy.matmul(A, A.transpose())
AT_mul_A = numpy.matmul(A.transpose(), A)
A_mul_B = numpy.matmul(A, B)
print("---------- A+A -----------\n"+str(A_add_A))
print("---------- AA' -----------\n"+str(A_mul_AT))
print("---------- A'A -----------\n"+str(AT_mul_A))
print("---------- AB ------------\n"+str(A_mul_B))

def E_9_1(A, B, N):
    return numpy.matmul(A, B-N*numpy.eye([len(B), len(B)]))

print("--------- E_9_1 ----------\n"+str(E_9_1(A, B, 3)))

Result

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---------- A+A -----------
[[-0.99675071 -0.6495676   1.01357349 ..., -2.5407868   1.53207886
   0.0751089 ]
 [-0.06925787 -0.42836986  2.4504027  ...,  1.69732532  0.80316565
  -1.30803462]
 [ 2.25823479  0.98806253 -0.5739649  ...,  1.22649302  1.46312748
  -3.16395929]
 ...,
 [-1.37475203 -1.79483411  0.25941863 ..., -1.64123271  0.19708262
   2.00484962]
 [-0.62106414 -0.1070131  -3.0257811  ..., -4.53252983  1.58481669
  -1.47097894]
 [ 0.59292533  0.8860513   2.69104758 ...,  0.25052273 -1.49885506
  -2.80481829]]
---------- AA' -----------
[[ 524.50900633  -23.41422424   -8.4109328  ...,  -17.09509132
   -36.58226516   22.36854305]
 [ -23.41422424  474.25665538  -20.03371708 ...,  -15.65752606
    14.95693007   -6.58265367]
 [  -8.4109328   -20.03371708  534.41749229 ...,  -10.27103728
    19.68694583   30.70304678]
 ...,
 [ -17.09509132  -15.65752606  -10.27103728 ...,  440.6393497    -3.86846872
    24.44312808]
 [ -36.58226516   14.95693007   19.68694583 ...,   -3.86846872
   460.44551208  -24.6430389 ]
 [  22.36854305   -6.58265367   30.70304678 ...,   24.44312808  -24.6430389
   533.38452907]]
---------- A'A -----------
[[ 207.71499004   -8.83528023   17.29332948 ...,    1.35094389
     9.97232568   18.87072786]
 [  -8.83528023  151.22418385   18.14374619 ...,    0.76927965
     4.16070767  -16.40379885]
 [  17.29332948   18.14374619  205.02249127 ...,   -8.18432486
    -9.26707911    2.07063859]
 ...,
 [   1.35094389    0.76927965   -8.18432486 ...,  208.97270412
    -5.71621365    8.51065013]
 [   9.97232568    4.16070767   -9.26707911 ...,   -5.71621365
   174.85487874   -6.98119382]
 [  18.87072786  -16.40379885    2.07063859 ...,    8.51065013
    -6.98119382  186.52479968]]
---------- AB ------------
[[-14.87853525   8.70049763   0.66427479 ..., -40.15235048  38.96819378
   36.98761191]
 [  8.62765483 -38.65652814   1.54867974 ...,  51.0022929    2.29161362
    2.36645638]
 [-28.67531663 -23.03994764   5.9337154  ...,  -1.57998333  35.43202865
   -1.02604062]
 ...,
 [ 21.1088727  -24.70137887   2.29477385 ...,  20.56806809 -33.32895855
  -10.71314031]
 [ 16.33190225 -17.93547546  -9.57141851 ..., -40.75340698  65.49378203
  -19.09691261]
 [-29.11460503 -43.80853239  35.2469557  ...,  15.52438326  20.15859321
  -22.71350542]]
--------- E_9_1 ----------
[[-13.38340918   9.67484904  -0.85608544 ..., -36.34117028  36.67007549
   36.87494856]
 [  8.73154164 -38.01397334  -2.12692431 ...,  48.45630492   1.08686515
    4.32850831]
 [-32.06266882 -24.52204143   6.79466275 ...,  -3.41972285  33.23733743
    3.71989831]
 ...,
 [ 23.17100075 -22.0091277    1.90564591 ...,  23.02991715 -33.62458247
  -13.72041474]
 [ 17.26349845 -17.77495581  -5.03274686 ..., -33.95461223  63.116557
  -16.89044419]
 [-30.00399303 -45.13760934  31.21038433 ...,  15.14859916  22.40687581
  -18.50627799]]

Knowledge

  1. matmul() to calculate the mul of 2 matrics.
  2. eye(k) to calculate a diagonal matrix, k*k
  3. transpose() to return matrix’s Inverse matrix

Exercise 9.2: Solving a linear system

Generate a vector b with m entries and solve Bx = b

Code

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b = numpy.random.normal(size=m)
x = numpy.linalg.solve(B, b)

print("--------- E_9_2 ----------\n"+str(x))

Result

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--------- E_9_2 ----------
[ 0.77346147 -0.80520425 -0.35482528 -0.56178735 -0.21607332  0.09578284
  0.45176672 -0.349336   -0.07061673 -0.67937045  0.77608967  0.02769846
 -0.00266615 -0.78878366  0.32858871 -0.22730774  0.46410093  0.08025457
 -0.39018536  0.31449226 -0.15399999 -0.90204851  0.36424562  0.61096627
  0.33907227 -0.78779662 -0.48162797 -0.64274742  0.12995262  0.99430534
 -0.14856798 -1.15273747  0.09436944  0.40647302  0.65276383 -0.10382041
  0.41243053 -0.29856156  0.63452485 -0.22499859 -0.30249207 -0.14497932
  0.85522387  0.21110914 -0.23258855 -0.9897657  -0.31338026 -0.26002534
  0.924013   -0.47043625 -0.45807153 -0.75775808  1.12480813  0.30321897
 -0.29125859 -0.33289894 -0.64971968  0.50094724  0.36235713 -0.00269059
 -0.15268968 -0.01259865  0.41392597 -0.83904654 -0.47292343 -0.18366766
  0.15149868  1.16077591 -0.53859254 -0.98168055 -0.42284948  1.11918415
  0.47384948 -0.21576477 -0.57785042  0.16373249  0.58937659  0.05069881
 -0.60646457 -0.07515867 -0.25892701  0.60705024 -0.40504958 -0.41706291
  0.02971281  0.69272723 -0.27455502 -0.26949703 -0.88870169 -0.1796936
  0.3022728   0.32450019 -0.40309466 -0.28997186  0.28757543  0.83201615
  0.18273413  0.19503945 -0.83802164  0.13698447  0.35706851  0.41057604
  0.15437131 -0.12593118 -0.78262502  0.46311827  0.18733747 -0.12280002
 -0.17000113  0.72736406  0.04638688 -0.0446584  -0.3740762   0.10985653
  0.55118338  0.58155485 -0.31415809  0.02940531  0.26057846  1.10319678
  0.32003339 -0.66484382 -0.24373552  0.85396941  0.25111252 -0.07437428
 -0.42373725 -0.00220312  0.49085239  0.58744762 -0.73623476  0.00606373
  0.30887715  0.40351642  0.24192468 -0.60608552 -0.06592915  0.57245086
  1.18607234 -0.05819671 -0.63773722 -0.8275492   0.47117888  0.8240523
  0.09995859 -1.1189529  -0.68472816  0.08958122  0.94135285 -0.21713449
 -0.22415887 -0.73924185 -0.00985684  0.53434199 -0.22782094 -0.00677473
  0.18406202  0.25589949  0.02174665 -0.47176496 -1.00158691  0.29509947
  0.46664914 -0.65511163 -0.15173256 -0.22610517  0.71416397  0.42033317
 -0.8626862  -0.95496022  0.41023983  0.69597171  0.79422447 -0.10630244
 -0.90460346  0.04098153  0.52159034  0.65757128 -0.40015286 -0.10895269
 -0.21034717  0.56477992  0.35828024  0.08569485 -0.29118951  0.26209546
 -0.08259761 -0.49767682  0.00403437  0.15095307  0.35145479  0.5900982
  0.02764191 -0.2800535   0.37674186  0.24215259  0.19028278 -0.22832022
 -0.58452005  0.23195453  1.01708531  0.21018793 -0.07069491 -0.20749496
 -0.30999703  0.98823655  0.88781881  0.01026305 -0.31819752 -0.45149265
  0.23417009  0.20983107  0.22389019 -0.00310436  0.33739053 -0.02782074
  0.05340627 -0.24885277  0.25400433  0.64797729 -0.03559467 -0.05009246
 -0.08380507  0.36138515  0.5550696   0.28391703 -0.39964763  0.33188527
 -0.15623489  0.53671244 -0.07852636 -0.33260849 -0.26049435  0.16521133
 -0.29556138 -0.14604016  0.1189506  -0.72699557  0.75795078 -0.72868493
  0.60271873 -0.62082569  0.20283552  0.06472603  0.31866239 -0.90097767
 -0.15603714 -0.53582783  0.88605567 -0.36223599 -0.59827917 -1.29951569
  0.1108652   0.66520771  0.01635112 -0.47215453 -1.24440155 -0.08236616
  0.77307018  0.08110942  0.20703049 -1.09960752  0.53298015 -0.1027147
  0.57313429 -0.26114314  0.36161462 -0.63713623  0.24713223 -0.12035843
  0.11065281  0.09846977 -0.43356553 -0.02745542 -0.03750936  0.15417482
  0.02681122  0.35293284 -0.22082268  0.52545969 -0.52711138  0.94215319
  0.59478862  0.44168817 -0.27060941 -0.43870625  0.34660181  1.0296679
  0.96253921  0.09362239  0.12398285 -0.32504824  0.59667371 -0.29713152
 -0.15533326 -0.08547569  0.24128572 -0.25817327  0.12638511 -0.32930891
  0.34818106  0.62209695  0.00579373 -0.09340648  0.29222168  0.50828812
  0.03484351  0.3983519  -0.65951852  0.07763202  0.09992687  0.31966399
  0.19685432 -0.37118144 -0.13390334  0.54808429  0.50920353 -0.074975
 -0.76687343 -0.43923484 -0.09119431  0.32772465  0.34325338  0.01731712
 -0.28073798  0.39537496  0.09365181 -0.24057869 -0.5238085  -0.3035102
  0.53520429  0.94732342 -0.40593518 -0.37112931 -0.0610225   0.26211594
  0.51096725 -0.74657876 -0.4668039   0.34383364  0.65990638 -0.67455365
 -0.20302455 -0.51223175  0.20865335  0.31091778 -0.3855612  -0.58707168
  0.21290637  0.28386365  0.25445908 -0.42757652 -0.34554999  0.57824806
  0.56947678 -0.44043856 -0.59497373 -0.19046726  0.78359797  0.125395
  0.01909045 -1.25379069 -0.09729798  0.52626197  0.78733424 -0.54835847
 -0.47638112 -0.07003016  0.55939806  1.16409273 -0.0810496  -0.18461299
  0.41337285  0.72340257  0.47537153  0.11145316 -0.41716581  0.15447172
  0.87950718  0.32799822 -0.72525093 -0.25424443 -0.33243891  0.89624186
 -0.24327463 -0.377505   -0.01318703  1.15073861  0.50827694  0.14629782
 -0.60745337  0.35594258  0.77265696  0.51980829 -0.43604754  0.19361863
  0.0974694   0.12374675  0.22393652 -0.25111504 -0.16791473  0.66888136
  0.08905014 -0.24185122 -0.82934214  0.17298365  0.32383379  0.38730725
 -0.55691312 -0.14960641 -0.3161664   1.00423863 -0.13530144  0.2758329
 -0.59409022 -0.72711962  0.18561096  0.29493483 -0.49462417 -0.02535275
 -0.32179269  0.63802358  0.06450911 -0.36244535 -0.55088169  0.47010579
 -0.03932424 -0.05139748 -0.72489824  0.43677258  0.85363747  0.59513256
 -1.1427693  -0.75490234 -0.05380666  0.84642801  0.97061214 -0.82256622
 -1.12268162 -0.11974293  0.25058075 -0.14730279 -0.41490229  0.47094733
  0.0695535  -0.06111872 -0.00325791 -0.69918284  0.97462969  0.10417971
 -0.06695869 -1.23309132 -0.17049135  0.64043296  0.82727541 -0.20828416
 -0.59323465 -0.65802288  0.32621394  0.5368197   0.37921229 -1.08605397
  0.16858299  0.03444726  0.11315069  0.05051093  0.03860219 -0.35531796
  0.3553949   0.39874088 -0.52344943 -0.13202249  0.91263459  0.32986441
  0.26566883 -0.62952324 -0.54520924 -0.43928257  1.0950506   0.1665704
 -1.02249691 -0.05817481 -0.30783025  0.09503854  0.06272448 -0.35042346
  0.06067587  0.08451991  0.3850205  -0.77288649  0.17066585  0.10336318
  0.37866533 -0.63292289 -0.15179168  0.06741788  0.41384271 -0.29224392
  0.39719698 -1.22197233]

Knowledge

  1. numpy.linalg.solve(a, b) to calculate the result of ax = b

Exercise 9.3: Norms

Compute the Frobenius norm of A: |A|F and the infinity norm of B: |B|∞. Also find the largest and smallest singular values of B.

Code

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A_FN = numpy.linalg.norm(A, 'fro')
B_IN = numpy.linalg.norm(B, numpy.inf)
print("---------- A_FN ------------\n"+str(A_FN))
print("---------- B_IN ------------\n"+str(B_IN))

U, sigma, V = numpy.linalg.svd(B)
B_max_singular = numpy.max(sigma)
B_min_singular = numpy.min(sigma)
print("---------- B_max_singular ------------\n"+str(B_max_singular))
print("---------- B_min_singular ------------\n"+str(B_min_singular))

Result

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---------- A_FN ------------
316.813378033
---------- B_IN ------------
387.19726512
---------- B_max_singular ------------
51.6474766426
---------- B_min_singular ------------
0.0562452006537

Knowledge

  1. 矩阵范数：norm, numpy.linalg.norm()
  2. 奇异值分解：singular values， numpy.linalg.svd()

Exercise 9.4: Power iteration

Generate a matrix Z, n × n, with Gaussian entries, and use the power iteration to find the largest eigenvalue and corresponding eigenvector of Z. How many iterations are needed till convergence?

Optional: use the time.clock() method to compare computation time when varying n.

Code

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import time

Z = numpy.random.normal(size=[n, n])

def power_iteration_eigenvalue( X, err=0.01):
    x = numpy.random.normal(size=len(X)).transpose()
    step = numpy.inf
    times = 0
    time_begin = time.clock()
    
    while step > err:
        old = numpy.max(x)
        x = x/old
        x = numpy.matmul(X, x)
        new = numpy.max(x)
        step = abs(old-new)
        times += 1
        # print("step "+str(times)+" : "+str(new))

    time_end = time.clock()
    return x, new, times, time_end-time_begin

[eigenvector, eigenvalue, times, running_time] = power_iteration_eigenvalue(Z)

print("------------ E_9_4 ----------------")
print("eigenvector: "+str(eigenvector))
print(" eigenvalue: "+str(eigenvalue))
print(" step count: "+str(times))
print("  time used: "+str(running_time))

Result

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------------ E_9_4 ----------------
eigenvector: [  6.19366521e+00  -7.19111305e+00   7.47822455e+00   1.76596816e+00
  -1.22108278e+01  -6.68997718e+00  -1.07959431e+01   7.38413171e+00
   9.60996846e+00  -3.39415864e+00  -1.23836889e+01   1.60311426e+01
  -3.55845284e-01  -1.69182976e+01   6.50582275e+00  -3.93769884e+00
   7.23395058e+00  -6.30992140e+00   3.77506490e+00  -3.23313667e+00
   6.64974007e+00  -7.24150321e+00   9.88020162e+00   2.08854412e-03
   3.07409841e+00   3.42129651e+00  -5.92382223e+00  -1.35043015e+00
  -1.40770229e+00   1.24531978e+01  -6.32025548e+00  -1.26441627e+01
   3.26831772e-01  -1.42739520e+00   1.32925887e+00   1.06598272e+01
  -8.48197449e+00   4.82646327e+00   3.67864359e+00  -2.92456117e+00
   1.49065221e+01  -3.89251672e+00   1.24128342e+01   1.27677631e+01
  -1.11263113e+01  -1.46350312e+01   9.36651121e+00  -3.84124488e-01
  -5.67795245e+00  -8.75638867e+00   6.02814460e+00  -9.83530958e-01
   6.50873016e+00   9.18944025e+00  -1.08015910e+01  -2.27073094e+00
   2.05116066e+00   9.74180888e+00   4.34640807e+00  -1.45355560e+00
   1.72229371e+00   1.60129994e+00  -9.49922896e+00  -9.26528224e+00
   1.41245323e+01   1.29324148e+00   1.15170129e+01  -2.78286401e+00
  -2.41103992e+00  -6.04664999e+00   1.60510520e+01   9.06866735e+00
  -2.39722406e+00  -7.74249726e+00  -6.22329381e+00   5.66999289e+00
   7.07244637e-01  -1.42750525e+01   1.00944549e+01  -1.56414851e-01
  -6.77685962e+00  -1.31324847e+00   1.40962962e+01  -2.82314517e+00
   1.15847925e+00  -1.45499963e+01  -1.47587123e+00  -5.45102517e+00
   2.07919993e+00   1.48515566e+00  -3.34405736e+00  -4.93831256e-01
   3.52542969e+00  -4.49030531e+00  -4.14344570e+00  -6.73085059e+00
  -1.96366164e+00  -4.73358636e+00  -1.42491876e+01   7.14478669e-01
  -4.03345293e+00  -8.64438839e-01  -1.91710524e+00   4.02300501e-01
  -3.12336156e+00   3.25908960e+00   8.73984942e+00   1.80387754e+00
   5.37126471e-01   2.33427637e-01   7.20753102e+00   8.38898454e+00
   6.73706722e+00   2.25368920e+00   4.03456123e+00  -5.26635000e-01
  -1.09389778e+01  -1.69632014e+01  -5.04630146e+00   2.99535217e+00
  -4.97695971e+00   7.08832858e+00  -1.42509689e+00   5.44444010e-01
   1.14700391e+00  -3.34913269e+00  -1.68841359e+00  -1.85218648e+00
   6.17064696e+00  -1.17127043e+00  -7.25388347e+00  -2.70293145e+00
  -8.28104371e+00   1.64055792e+00   4.94486141e+00   4.18690047e+00
  -9.94856800e+00  -8.29269461e+00   5.81317629e+00  -1.31414167e+01
   8.10423657e-01  -1.30089443e+01  -7.10088385e+00  -7.62918634e+00
   5.14150493e+00   4.43537243e+00  -3.52960575e-01  -1.35215797e+00
  -5.68239918e+00  -5.78998103e+00  -4.91632266e-01  -2.68038511e+00
  -1.03668811e+01   5.49950249e+00  -2.26436209e+00   1.23865979e+01
   6.14732355e+00   4.31623361e+00  -8.40648179e+00  -3.72063488e+00
   2.01815136e+00  -5.14877954e-01  -1.16816147e+00  -2.24455122e+01
  -5.57878300e+00   3.10970510e-01   1.29690655e+00  -1.69972653e+01
   1.58677609e+01  -5.77551543e+00  -4.36274195e+00  -5.63981586e+00
  -4.54383986e+00  -7.16379461e-01  -4.07212591e+00  -5.45036929e+00
  -9.33501142e+00  -1.46760018e+01  -4.31132630e+00   1.16448053e+01
   6.21216923e+00  -6.78022246e+00   4.13536082e+00   3.69218241e+00
   1.97466803e+00  -1.08666389e+00   8.80475201e+00   2.46159249e+00
   2.21402112e+00  -2.46765512e+00   3.57973968e+00  -1.11716056e-01
  -8.24346370e+00  -2.90508233e+00   4.13721878e+00  -4.10881873e+00
   5.15073997e+00  -1.22542384e+01   3.96149388e+00  -1.13627510e+01]
 eigenvalue: 16.0510519646
 step count: 68
  time used: 0.007216

Knowledge

  1. eigenvalue: 特征值
  2. 幂迭代法：x(k) = x(k-1)*A

Exercise 9.5: Singular values

Generate an n × n matrix, denoted by C, where each entry is 1 with probability p and 0 otherwise. Use
the linear algebra library of Scipy to compute the singular values of C. What can you say about the
relationship between n, p and the largest singular value?

Code

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p = 0.1
print("-------------- E_9_5 ------------------")
while p<1:
    C = numpy.random.binomial(1, p, [n, n])
    U, singular_value, V = numpy.linalg.svd(C)
    max_singular = numpy.max(singular_value)
    print("n= "+str(n)+"    p= "+str(p)+"   max_singular= "+ str(max_singular))
    p += 0.1

Result

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-------------- E_9_5 ------------------
n= 200    p= 0.1   max_singular= 20.6567657213
n= 200    p= 0.2   max_singular= 40.4920520455
n= 200    p= 0.3   max_singular= 60.6782510601
n= 200    p= 0.4   max_singular= 81.0362830637
n= 200    p= 0.5   max_singular= 101.14764207
n= 200    p= 0.6   max_singular= 120.373897161
n= 200    p= 0.7   max_singular= 139.768077408
n= 200    p= 0.8   max_singular= 160.37471516
n= 200    p= 0.9   max_singular= 179.191890776
n= 200    p= 1.0   max_singular= 200.0

Knowledge

  1. random.binomial(n, p, size), n is 0~n, p is probability, size is size
  2. we can find that max_singular = n * p

Exercise 9.6: Nearest neighbor

Write a function that takes a value z and an array A and finds the element in A that is closest to z. The
function should return the closest value, not index.

Hint: Use the built-in functionality of Numpy rather than writing code to find this value manually. In
particular, use brackets and argmin.

Code

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def closest( X, z):
    ind = numpy.argmin(numpy.abs(X - z * numpy.ones((len(X), len(X[0])))))
    return X[ind//len(X[0])][ind%len(X[0])]

print("-------------- Closest ---------------")
z = -1
while z <= 1:
    print("z= "+str(z)+"     closest= "+str(closest(A, z)))
    z += 0.1

Result

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-------------- Closest ---------------
z= -1     closest= -1.00000868018
z= -0.9     closest= -0.899991177575
z= -0.8     closest= -0.799982994952
z= -0.7     closest= -0.699995902891
z= -0.6     closest= -0.59999964816
z= -0.5     closest= -0.499995702358
z= -0.4     closest= -0.399997886746
z= -0.3     closest= -0.299999122356
z= -0.2     closest= -0.200002337307
z= -0.1     closest= -0.0999872558528
z= -1.38777878078e-16     closest= -1.45838021333e-05
z= 0.1     closest= 0.10001924864
z= 0.2     closest= 0.200009578588
z= 0.3     closest= 0.30001507798
z= 0.4     closest= 0.399992197442
z= 0.5     closest= 0.499998597393
z= 0.6     closest= 0.599983955592
z= 0.7     closest= 0.700043443047
z= 0.8     closest= 0.800006259776
z= 0.9     closest= 0.900013425609
z= 1.0     closest= 1.00000424317

Knowledge

  1. argmin(X) to return X’s minimum’s index

Reference