Implement Gradient Descent in Python | by Rohan Joseph | Towards Data Science

It is an optimization algorithm to find the minimum of a function. We start with a random point on the function and move in the negative direction of the gradient of the function to reach the local/global minima.
Question : Find the local minima of the function y=(x+5)2 starting from the point x=3
Solution : We know the answer just by looking at the graph. y = (x+5)2 reaches it’s minimum value when x = -5 (i.e when x=-5, y=0). Hence x=-5 is the local and global minima of the function.
Now, let’s see how to obtain the same numerically using gradient descent.
Step 1 : Initialize x =3. Then, find the gradient of the function, dy/dx = 2*(x+5).
Step 2 : Move in the direction of the negative of the gradient (Why?). But wait, how much to move? For that, we require a learning rate. Let us assume the learning rate → 0.01
Step 3 : Let’s perform 2 iterations of gradient descent
Step 4 : We can observe that the X value is slowly decreasing and should converge to -5 (the local minima). However, how many iterations should we perform?
Let us set a precision variable in our algorithm which calculates the difference between two consecutive “x” values . If the difference between x values from 2 consecutive iterations is lesser than the precision we set, stop the algorithm !
Step 1 : Initialize parameters
cur_x = 3 # The algorithm starts at x=3rate = 0.01 # Learning rateprecision = 0.000001 #This tells us when to stop the algorithmprevious_step_size = 1 #max_iters = 10000 # maximum number of iterationsiters = 0 #iteration counterdf = lambda x: 2*(x+5) #Gradient of our function 
Step 2 : Run a loop to perform gradient descent :
i. Stop loop when difference between x values from 2 consecutive iterations is less than 0.000001 or when number of iterations exceeds 10,000
while previous_step_size > precision and iters < max_iters:    prev_x = cur_x #Store current x value in prev_x    cur_x = cur_x - rate * df(prev_x) #Grad descent    previous_step_size = abs(cur_x - prev_x) #Change in x    iters = iters+1 #iteration count    print("Iteration",iters,"\nX value is",cur_x) #Print iterations    print("The local minimum occurs at", cur_x)
Output : From the output below, we can observe the x values for the first 10 iterations- which can be cross checked with our calculation above. The algorithm runs for 595 iterations before it terminates. The code and solution is embedded below for reference.
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cur_x = 3 # The algorithm starts at x=3
rate = 0.01 # Learning rate
precision = 0.000001 #This tells us when to stop the algorithm
previous_step_size = 1 #
max_iters = 10000 # maximum number of iterations
iters = 0 #iteration counter
df = lambda x: 2*(x+5) #Gradient of our function

while previous_step_size > precision and iters < max_iters:
    prev_x = cur_x #Store current x value in prev_x
    cur_x = cur_x - rate * df(prev_x) #Grad descent
    previous_step_size = abs(cur_x - prev_x) #Change in x
    iters = iters+1 #iteration count
    print("Iteration",iters,"\nX value is",cur_x) #Print iterations
    
print("The local minimum occurs at", cur_x)

Iteration 1 
X value is 2.84
Iteration 2 
X value is 2.6832
Iteration 3 
X value is 2.529536
Iteration 4 
X value is 2.37894528
Iteration 5 
X value is 2.2313663744
Iteration 6 
X value is 2.0867390469119997
Iteration 7 
X value is 1.9450042659737599
Iteration 8 
X value is 1.8061041806542846
Iteration 9 
X value is 1.669982097041199
Iteration 10 
X value is 1.5365824551003748
Iteration 11 
X value is 1.4058508059983674
Iteration 12 
X value is 1.2777337898784
Iteration 13 
X value is 1.152179114080832
Iteration 14 
X value is 1.0291355317992152
Iteration 15 
X value is 0.9085528211632309
Iteration 16 
X value is 0.7903817647399662
Iteration 17 
X value is 0.6745741294451669
Iteration 18 
X value is 0.5610826468562635
Iteration 19 
X value is 0.44986099391913825
Iteration 20 
X value is 0.3408637740407555
Iteration 21 
X value is 0.23404649855994042
Iteration 22 
X value is 0.1293655685887416
Iteration 23 
X value is 0.026778257216966764
Iteration 24 
X value is -0.07375730792737258
Iteration 25 
X value is -0.1722821617688251
Iteration 26 
X value is -0.2688365185334486
Iteration 27 
X value is -0.36345978816277963
Iteration 28 
X value is -0.45619059239952403
Iteration 29 
X value is -0.5470667805515336
Iteration 30 
X value is -0.6361254449405029
Iteration 31 
X value is -0.7234029360416929
Iteration 32 
X value is -0.8089348773208591
Iteration 33 
X value is -0.8927561797744419
Iteration 34 
X value is -0.9749010561789531
Iteration 35 
X value is -1.055403035055374
Iteration 36 
X value is -1.1342949743542665
Iteration 37 
X value is -1.2116090748671813
Iteration 38 
X value is -1.2873768933698377
Iteration 39 
X value is -1.361629355502441
Iteration 40 
X value is -1.4343967683923922
Iteration 41 
X value is -1.5057088330245443
Iteration 42 
X value is -1.5755946563640535
Iteration 43 
X value is -1.6440827632367725
Iteration 44 
X value is -1.711201107972037
Iteration 45 
X value is -1.7769770858125964
Iteration 46 
X value is -1.8414375440963444
Iteration 47 
X value is -1.9046087932144176
Iteration 48 
X value is -1.9665166173501292
Iteration 49 
X value is -2.0271862850031264
Iteration 50 
X value is -2.0866425593030637
Iteration 51 
X value is -2.1449097081170025
Iteration 52 
X value is -2.2020115139546625
Iteration 53 
X value is -2.257971283675569
Iteration 54 
X value is -2.312811858002058
Iteration 55 
X value is -2.3665556208420164
Iteration 56 
X value is -2.419224508425176
Iteration 57 
X value is -2.4708400182566725
Iteration 58 
X value is -2.521423217891539
Iteration 59 
X value is -2.570994753533708
Iteration 60 
X value is -2.619574858463034
Iteration 61 
X value is -2.667183361293773
Iteration 62 
X value is -2.713839694067898
Iteration 63 
X value is -2.75956290018654
Iteration 64 
X value is -2.804371642182809
Iteration 65 
X value is -2.8482842093391527
Iteration 66 
X value is -2.8913185251523696
Iteration 67 
X value is -2.9334921546493224
Iteration 68 
X value is -2.974822311556336
Iteration 69 
X value is -3.015325865325209
Iteration 70 
X value is -3.055019348018705
Iteration 71 
X value is -3.093918961058331
Iteration 72 
X value is -3.1320405818371646
Iteration 73 
X value is -3.1693997702004215
Iteration 74 
X value is -3.206011774796413
Iteration 75 
X value is -3.2418915393004846
Iteration 76 
X value is -3.277053708514475
Iteration 77 
X value is -3.3115126343441856
Iteration 78 
X value is -3.345282381657302
Iteration 79 
X value is -3.378376734024156
Iteration 80 
X value is -3.4108091993436727
Iteration 81 
X value is -3.4425930153567994
Iteration 82 
X value is -3.4737411550496633
Iteration 83 
X value is -3.50426633194867
Iteration 84 
X value is -3.534181005309697
Iteration 85 
X value is -3.563497385203503
Iteration 86 
X value is -3.5922274374994325
Iteration 87 
X value is -3.620382888749444
Iteration 88 
X value is -3.6479752309744553
Iteration 89 
X value is -3.675015726354966
Iteration 90 
X value is -3.7015154118278666
Iteration 91 
X value is -3.7274851035913095
Iteration 92 
X value is -3.7529354015194833
Iteration 93 
X value is -3.7778766934890937
Iteration 94 
X value is -3.8023191596193118
Iteration 95 
X value is -3.8262727764269258
Iteration 96 
X value is -3.8497473208983872
Iteration 97 
X value is -3.8727523744804193
Iteration 98 
X value is -3.895297326990811
Iteration 99 
X value is -3.917391380450995
Iteration 100 
X value is -3.939043552841975
Iteration 101 
X value is -3.9602626817851356
Iteration 102 
X value is -3.981057428149433
Iteration 103 
X value is -4.001436279586445
Iteration 104 
X value is -4.021407553994716
Iteration 105 
X value is -4.040979402914822
Iteration 106 
X value is -4.060159814856525
Iteration 107 
X value is -4.078956618559395
Iteration 108 
X value is -4.097377486188207
Iteration 109 
X value is -4.115429936464443
Iteration 110 
X value is -4.133121337735154
Iteration 111 
X value is -4.150458910980451
Iteration 112 
X value is -4.167449732760842
Iteration 113 
X value is -4.1841007381056246
Iteration 114 
X value is -4.200418723343512
Iteration 115 
X value is -4.216410348876642
Iteration 116 
X value is -4.2320821418991095
Iteration 117 
X value is -4.247440499061128
Iteration 118 
X value is -4.262491689079905
Iteration 119 
X value is -4.277241855298307
Iteration 120 
X value is -4.291697018192341
Iteration 121 
X value is -4.305863077828494
Iteration 122 
X value is -4.319745816271924
Iteration 123 
X value is -4.333350899946486
Iteration 124 
X value is -4.3466838819475555
Iteration 125 
X value is -4.359750204308605
Iteration 126 
X value is -4.372555200222433
Iteration 127 
X value is -4.385104096217984
Iteration 128 
X value is -4.3974020142936245
Iteration 129 
X value is -4.409453974007752
Iteration 130 
X value is -4.421264894527597
Iteration 131 
X value is -4.432839596637045
Iteration 132 
X value is -4.444182804704305
Iteration 133 
X value is -4.4552991486102185
Iteration 134 
X value is -4.466193165638014
Iteration 135 
X value is -4.4768693023252535
Iteration 136 
X value is -4.487331916278748
Iteration 137 
X value is -4.497585277953173
Iteration 138 
X value is -4.50763357239411
Iteration 139 
X value is -4.517480900946228
Iteration 140 
X value is -4.527131282927304
Iteration 141 
X value is -4.536588657268758
Iteration 142 
X value is -4.545856884123382
Iteration 143 
X value is -4.5549397464409145
Iteration 144 
X value is -4.563840951512097
Iteration 145 
X value is -4.572564132481855
Iteration 146 
X value is -4.581112849832218
Iteration 147 
X value is -4.589490592835574
Iteration 148 
X value is -4.597700780978863
Iteration 149 
X value is -4.605746765359285
Iteration 150 
X value is -4.6136318300521
Iteration 151 
X value is -4.621359193451058
Iteration 152 
X value is -4.628932009582036
Iteration 153 
X value is -4.636353369390395
Iteration 154 
X value is -4.643626302002588
Iteration 155 
X value is -4.650753775962536
Iteration 156 
X value is -4.657738700443285
Iteration 157 
X value is -4.664583926434419
Iteration 158 
X value is -4.671292247905731
Iteration 159 
X value is -4.6778664029476165
Iteration 160 
X value is -4.684309074888664
Iteration 161 
X value is -4.6906228933908904
Iteration 162 
X value is -4.696810435523073
Iteration 163 
X value is -4.702874226812612
Iteration 164 
X value is -4.708816742276359
Iteration 165 
X value is -4.714640407430832
Iteration 166 
X value is -4.720347599282215
Iteration 167 
X value is -4.725940647296571
Iteration 168 
X value is -4.731421834350639
Iteration 169 
X value is -4.736793397663627
Iteration 170 
X value is -4.742057529710355
Iteration 171 
X value is -4.747216379116147
Iteration 172 
X value is -4.752272051533824
Iteration 173 
X value is -4.757226610503148
Iteration 174 
X value is -4.762082078293084
Iteration 175 
X value is -4.766840436727223
Iteration 176 
X value is -4.771503627992678
Iteration 177 
X value is -4.776073555432824
Iteration 178 
X value is -4.780552084324168
Iteration 179 
X value is -4.784941042637685
Iteration 180 
X value is -4.7892422217849315
Iteration 181 
X value is -4.793457377349233
Iteration 182 
X value is -4.7975882298022485
Iteration 183 
X value is -4.801636465206204
Iteration 184 
X value is -4.805603735902079
Iteration 185 
X value is -4.809491661184038
Iteration 186 
X value is -4.813301827960357
Iteration 187 
X value is -4.81703579140115
Iteration 188 
X value is -4.820695075573127
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X value is -4.824281174061665
Iteration 190 
X value is -4.827795550580431
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X value is -4.831239639568823
Iteration 192 
X value is -4.834614846777447
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X value is -4.837922549841898
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X value is -4.84116409884506
Iteration 195 
X value is -4.844340816868159
Iteration 196 
X value is -4.847454000530796
Iteration 197 
X value is -4.85050492052018
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X value is -4.853494822109776
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X value is -4.85642492566758
Iteration 200 
X value is -4.859296427154229
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X value is -4.862110498611145
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X value is -4.864868288638922
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X value is -4.867570922866143
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X value is -4.87021950440882
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X value is -4.872815114320644
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X value is -4.875358812034231
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X value is -4.877851635793546
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X value is -4.880294603077676
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X value is -4.882688711016122
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X value is -4.8850349367958
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X value is -4.887334238059884
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X value is -4.8895875532986866
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X value is -4.891795802232712
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X value is -4.893959886188058
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X value is -4.896080688464297
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X value is -4.898159074695011
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X value is -4.9001958932011105
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X value is -4.902191975337089
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X value is -4.904148135830347
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X value is -4.90606517311374
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X value is -4.907943869651465
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X value is -4.909784992258436
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X value is -4.911589292413267
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X value is -4.913357506565002
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X value is -4.915090356433702
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X value is -4.9167885493050285
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X value is -4.918452778318928
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X value is -4.920083722752549
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X value is -4.921682048297498
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X value is -4.923248407331548
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X value is -4.9247834391849175
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X value is -4.926287770401219
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X value is -4.927762014993195
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X value is -4.929206774693331
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X value is -4.930622639199464
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X value is -4.932010186415474
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X value is -4.933369982687164
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X value is -4.934702583033421
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X value is -4.936008531372753
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X value is -4.937288360745298
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X value is -4.938542593530392
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X value is -4.939771741659784
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X value is -4.940976306826588
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X value is -4.942156780690056
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X value is -4.943313645076255
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X value is -4.94444737217473
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X value is -4.945558424731236
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X value is -4.946647256236611
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X value is -4.947714311111879
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X value is -4.9487600248896415
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X value is -4.949784824391848
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X value is -4.950789127904011
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X value is -4.951773345345931
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X value is -4.952737878439012
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X value is -4.953683120870232
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X value is -4.954609458452827
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X value is -4.955517269283771
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X value is -4.956406923898095
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X value is -4.957278785420133
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X value is -4.958133209711731
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X value is -4.958970545517496
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X value is -4.959791134607146
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X value is -4.960595311915003
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X value is -4.9613834056767026
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X value is -4.962155737563169
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X value is -4.962912622811905
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X value is -4.963654370355667
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X value is -4.964381282948554
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X value is -4.965093657289583
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X value is -4.965791784143791
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X value is -4.966475948460915
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X value is -4.967146429491697
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X value is -4.967803500901863
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X value is -4.968447430883826
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X value is -4.969078482266149
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X value is -4.969696912620826
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X value is -4.970302974368409
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X value is -4.970896914881041
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X value is -4.97147897658342
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X value is -4.972049397051752
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X value is -4.972608409110717
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X value is -4.973156240928502
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X value is -4.973693116109932
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X value is -4.974219253787734
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X value is -4.974734868711979
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X value is -4.975240171337739
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X value is -4.975735367910985
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X value is -4.976220660552765
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X value is -4.976696247341709
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X value is -4.977162322394875
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X value is -4.977619075946977
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X value is -4.978066694428038
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X value is -4.978505360539477
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X value is -4.978935253328687
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X value is -4.979356548262113
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X value is -4.979769417296871
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X value is -4.980174028950934
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X value is -4.980570548371915
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X value is -4.980959137404477
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X value is -4.981339954656387
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X value is -4.981713155563259
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X value is -4.982078892451994
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X value is -4.9824373146029535
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X value is -4.982788568310895
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X value is -4.983132796944677
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X value is -4.983470141005784
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X value is -4.983800738185668
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X value is -4.984124723421955
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X value is -4.984442228953515
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X value is -4.984753384374445
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X value is -4.985058316686956
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X value is -4.9853571503532175
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X value is -4.985650007346153
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X value is -4.9859370071992295
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X value is -4.986218267055245
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X value is -4.98702874320626
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X value is -4.987288168342134
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X value is -4.988739320779054
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X value is -4.995642974069633
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X value is -4.99573011458824
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X value is -4.995899202050547
Iteration 376 
X value is -4.995981218009535
Iteration 377 
X value is -4.996061593649345
Iteration 378 
X value is -4.996140361776358
Iteration 379 
X value is -4.996217554540831
Iteration 380 
X value is -4.996293203450014
Iteration 381 
X value is -4.996367339381013
Iteration 382 
X value is -4.996439992593393
Iteration 383 
X value is -4.996511192741525
Iteration 384 
X value is -4.996580968886694
Iteration 385 
X value is -4.99664934950896
Iteration 386 
X value is -4.9967163625187805
Iteration 387 
X value is -4.996782035268405
Iteration 388 
X value is -4.996846394563037
Iteration 389 
X value is -4.996909466671776
Iteration 390 
X value is -4.996971277338341
Iteration 391 
X value is -4.997031851791574
Iteration 392 
X value is -4.997091214755742
Iteration 393 
X value is -4.997149390460628
Iteration 394 
X value is -4.997206402651415
Iteration 395 
X value is -4.997262274598387
Iteration 396 
X value is -4.997317029106419
Iteration 397 
X value is -4.997370688524291
Iteration 398 
X value is -4.997423274753805
Iteration 399 
X value is -4.997474809258729
Iteration 400 
X value is -4.997525313073554
Iteration 401 
X value is -4.997574806812083
Iteration 402 
X value is -4.997623310675841
Iteration 403 
X value is -4.997670844462324
Iteration 404 
X value is -4.997717427573078
Iteration 405 
X value is -4.997763079021617
Iteration 406 
X value is -4.997807817441185
Iteration 407 
X value is -4.997851661092361
Iteration 408 
X value is -4.997894627870514
Iteration 409 
X value is -4.997936735313104
Iteration 410 
X value is -4.9979780006068415
Iteration 411 
X value is -4.998018440594705
Iteration 412 
X value is -4.998058071782811
Iteration 413 
X value is -4.998096910347155
Iteration 414 
X value is -4.998134972140212
Iteration 415 
X value is -4.998172272697408
Iteration 416 
X value is -4.9982088272434595
Iteration 417 
X value is -4.998244650698591
Iteration 418 
X value is -4.998279757684619
Iteration 419 
X value is -4.998314162530927
Iteration 420 
X value is -4.998347879280309
Iteration 421 
X value is -4.998380921694703
Iteration 422 
X value is -4.998413303260809
Iteration 423 
X value is -4.998445037195593
Iteration 424 
X value is -4.998476136451681
Iteration 425 
X value is -4.998506613722648
Iteration 426 
X value is -4.998536481448195
Iteration 427 
X value is -4.998565751819231
Iteration 428 
X value is -4.998594436782846
Iteration 429 
X value is -4.998622548047189
Iteration 430 
X value is -4.998650097086245
Iteration 431 
X value is -4.9986770951445205
Iteration 432 
X value is -4.99870355324163
Iteration 433 
X value is -4.998729482176797
Iteration 434 
X value is -4.998754892533261
Iteration 435 
X value is -4.998779794682596
Iteration 436 
X value is -4.998804198788944
Iteration 437 
X value is -4.998828114813166
Iteration 438 
X value is -4.998851552516903
Iteration 439 
X value is -4.998874521466565
Iteration 440 
X value is -4.998897031037234
Iteration 441 
X value is -4.998919090416489
Iteration 442 
X value is -4.99894070860816
Iteration 443 
X value is -4.998961894435997
Iteration 444 
X value is -4.998982656547277
Iteration 445 
X value is -4.999003003416331
Iteration 446 
X value is -4.999022943348004
Iteration 447 
X value is -4.999042484481044
Iteration 448 
X value is -4.999061634791423
Iteration 449 
X value is -4.999080402095594
Iteration 450 
X value is -4.999098794053682
Iteration 451 
X value is -4.999116818172609
Iteration 452 
X value is -4.999134481809157
Iteration 453 
X value is -4.999151792172974
Iteration 454 
X value is -4.999168756329515
Iteration 455 
X value is -4.999185381202924
Iteration 456 
X value is -4.999201673578866
Iteration 457 
X value is -4.999217640107289
Iteration 458 
X value is -4.999233287305143
Iteration 459 
X value is -4.9992486215590395
Iteration 460 
X value is -4.999263649127859
Iteration 461 
X value is -4.999278376145302
Iteration 462 
X value is -4.999292808622396
Iteration 463 
X value is -4.999306952449948
Iteration 464 
X value is -4.999320813400949
Iteration 465 
X value is -4.99933439713293
Iteration 466 
X value is -4.999347709190272
Iteration 467 
X value is -4.9993607550064665
Iteration 468 
X value is -4.999373539906337
Iteration 469 
X value is -4.99938606910821
Iteration 470 
X value is -4.9993983477260455
Iteration 471 
X value is -4.999410380771525
Iteration 472 
X value is -4.999422173156094
Iteration 473 
X value is -4.9994337296929725
Iteration 474 
X value is -4.999445055099113
Iteration 475 
X value is -4.999456153997131
Iteration 476 
X value is -4.999467030917188
Iteration 477 
X value is -4.9994776902988445
Iteration 478 
X value is -4.999488136492867
Iteration 479 
X value is -4.99949837376301
Iteration 480 
X value is -4.99950840628775
Iteration 481 
X value is -4.999518238161995
Iteration 482 
X value is -4.999527873398756
Iteration 483 
X value is -4.99953731593078
Iteration 484 
X value is -4.999546569612165
Iteration 485 
X value is -4.999555638219921
Iteration 486 
X value is -4.999564525455523
Iteration 487 
X value is -4.999573234946412
Iteration 488 
X value is -4.9995817702474845
Iteration 489 
X value is -4.999590134842535
Iteration 490 
X value is -4.999598332145684
Iteration 491 
X value is -4.99960636550277
Iteration 492 
X value is -4.999614238192715
Iteration 493 
X value is -4.999621953428861
Iteration 494 
X value is -4.999629514360284
Iteration 495 
X value is -4.999636924073078
Iteration 496 
X value is -4.999644185591617
Iteration 497 
X value is -4.999651301879784
Iteration 498 
X value is -4.999658275842188
Iteration 499 
X value is -4.999665110325345
Iteration 500 
X value is -4.999671808118838
Iteration 501 
X value is -4.9996783719564615
Iteration 502 
X value is -4.999684804517332
Iteration 503 
X value is -4.999691108426985
Iteration 504 
X value is -4.999697286258446
Iteration 505 
X value is -4.9997033405332765
Iteration 506 
X value is -4.999709273722611
Iteration 507 
X value is -4.999715088248159
Iteration 508 
X value is -4.999720786483196
Iteration 509 
X value is -4.999726370753532
Iteration 510 
X value is -4.999731843338461
Iteration 511 
X value is -4.999737206471692
Iteration 512 
X value is -4.999742462342258
Iteration 513 
X value is -4.999747613095413
Iteration 514 
X value is -4.999752660833504
Iteration 515 
X value is -4.999757607616834
Iteration 516 
X value is -4.999762455464498
Iteration 517 
X value is -4.999767206355208
Iteration 518 
X value is -4.999771862228104
Iteration 519 
X value is -4.999776424983542
Iteration 520 
X value is -4.9997808964838715
Iteration 521 
X value is -4.999785278554194
Iteration 522 
X value is -4.9997895729831106
Iteration 523 
X value is -4.999793781523448
Iteration 524 
X value is -4.999797905892979
Iteration 525 
X value is -4.999801947775119
Iteration 526 
X value is -4.999805908819617
Iteration 527 
X value is -4.999809790643225
Iteration 528 
X value is -4.99981359483036
Iteration 529 
X value is -4.999817322933753
Iteration 530 
X value is -4.999820976475077
Iteration 531 
X value is -4.999824556945576
Iteration 532 
X value is -4.999828065806665
Iteration 533 
X value is -4.9998315044905315
Iteration 534 
X value is -4.999834874400721
Iteration 535 
X value is -4.999838176912706
Iteration 536 
X value is -4.999841413374452
Iteration 537 
X value is -4.999844585106963
Iteration 538 
X value is -4.999847693404824
Iteration 539 
X value is -4.999850739536727
Iteration 540 
X value is -4.999853724745993
Iteration 541 
X value is -4.999856650251073
Iteration 542 
X value is -4.999859517246051
Iteration 543 
X value is -4.99986232690113
Iteration 544 
X value is -4.999865080363108
Iteration 545 
X value is -4.999867778755846
Iteration 546 
X value is -4.999870423180729
Iteration 547 
X value is -4.999873014717115
Iteration 548 
X value is -4.999875554422772
Iteration 549 
X value is -4.999878043334316
Iteration 550 
X value is -4.99988048246763
Iteration 551 
X value is -4.999882872818278
Iteration 552 
X value is -4.999885215361912
Iteration 553 
X value is -4.999887511054674
Iteration 554 
X value is -4.999889760833581
Iteration 555 
X value is -4.999891965616909
Iteration 556 
X value is -4.999894126304571
Iteration 557 
X value is -4.999896243778479
Iteration 558 
X value is -4.999898318902909
Iteration 559 
X value is -4.999900352524851
Iteration 560 
X value is -4.9999023454743545
Iteration 561 
X value is -4.999904298564868
Iteration 562 
X value is -4.9999062125935705
Iteration 563 
X value is -4.999908088341699
Iteration 564 
X value is -4.9999099265748645
Iteration 565 
X value is -4.999911728043367
Iteration 566 
X value is -4.9999134934825
Iteration 567 
X value is -4.99991522361285
Iteration 568 
X value is -4.999916919140593
Iteration 569 
X value is -4.999918580757781
Iteration 570 
X value is -4.999920209142625
Iteration 571 
X value is -4.999921804959773
Iteration 572 
X value is -4.9999233688605775
Iteration 573 
X value is -4.999924901483366
Iteration 574 
X value is -4.999926403453699
Iteration 575 
X value is -4.999927875384625
Iteration 576 
X value is -4.999929317876933
Iteration 577 
X value is -4.999930731519394
Iteration 578 
X value is -4.999932116889006
Iteration 579 
X value is -4.999933474551226
Iteration 580 
X value is -4.999934805060202
Iteration 581 
X value is -4.999936108958998
Iteration 582 
X value is -4.999937386779818
Iteration 583 
X value is -4.999938639044221
Iteration 584 
X value is -4.999939866263337
Iteration 585 
X value is -4.99994106893807
Iteration 586 
X value is -4.999942247559309
Iteration 587 
X value is -4.999943402608123
Iteration 588 
X value is -4.9999445345559606
Iteration 589 
X value is -4.999945643864842
Iteration 590 
X value is -4.999946730987545
Iteration 591 
X value is -4.999947796367794
Iteration 592 
X value is -4.999948840440438
Iteration 593 
X value is -4.999949863631629
Iteration 594 
X value is -4.999950866358997
Iteration 595 
X value is -4.9999518490318176
The local minimum occurs at -4.9999518490318176