Let's again consider the function of two variables that we saw before:

$$f(x, y) = -0.4+(x+15)/30 + (y+15)/40 + 0.5\sin(r), r = \sqrt{x^2+y^2}$$

We can plot this function as before:

In [1]:
%matplotlib inline

from numpy import *
from numpy.linalg import norm

from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
from matplotlib.pyplot import *
from numpy import *

def f(x, y):
r = sqrt(x**2 + y**2)
return -.4 + (x+15)/30. + (y+15)/40.+.5*sin(r)

x = arange(-15, 15, 0.25)
y = arange(-15, 15, 0.25)
X, Y = meshgrid(x, y)
R = sqrt(X**2 + Y**2)
Z = -.4 + (X+15)/30. + (Y+15)/40.+.5*sin(R)

/usr/lib/python2.7/dist-packages/matplotlib/font_manager.py:273: UserWarning: Matplotlib is building the font cache using fc-list. This may take a moment.
warnings.warn('Matplotlib is building the font cache using fc-list. This may take a moment.')

In [2]:
fig = figure(3)
ax = fig.gca(projection='3d')
surf = ax.plot_surface(X, Y, Z, rstride=1, cstride=1, cmap=cm.coolwarm,
linewidth=0, antialiased=False)
ax.set_zlim(-2.01, 2.01)

ax.zaxis.set_major_locator(LinearLocator(10))
ax.zaxis.set_major_formatter(FormatStrFormatter('%.02f'))

fig.colorbar(surf, shrink=0.5, aspect=5)
show()

In [3]:
x = arange(-15, 15, 0.25)
y = arange(-15, 15, 0.25)
X, Y = meshgrid(x, y)
R = sqrt(X**2 + Y**2)
Z = -.4 + (X+15)/30. + (Y+15)/40.+.5*sin(R)

figure(1)
CS = contour(X, Y, Z, cmap=cm.coolwarm)
clabel(CS, inline=1, fontsize=10)
title('Contour plot')

show()


Suppose we want to find a local minimum of this function. In order to do that, we need to compute its gradient.

Here is a sample computation:

$$\frac{\partial f}{\partial x} = 1/30 + 0.5\cos(r)\frac{\partial r}{\partial x}$$$$\frac{\partial r}{\partial x} = (0.5(x^2+y^2)^{-1/2})(2 x)$$

We can get the following:

In [4]:
def drdx(x, y, r):
return (.5*(x**2 + y**2)**-.5)*(2*x)

def drdy(x, y, r):
return (.5*(x**2 + y**2)**-.5)*(2*y)

def dfdx(x, y):
r = sqrt(x**2 + y**2)
return 1/30. + .5*cos(r)*drdx(x, y, r)

def dfdy(x, y):
r = sqrt(x**2 + y**2)
return 1/40. + .5*cos(r)*drdy(x, y, r)

return array([dfdx(x, y), dfdy(x, y)])


Let's check that we got things right:

In [5]:
x = 0
y = 5
h = 0.001
(f(x+h, y)-f(x, y))/h

Out[5]:
0.033347516444903746
In [6]:
dfdx(x, y)

Out[6]:
0.033333333333333333
In [7]:
(f(x, y+h)-f(x, y))/h

Out[7]:
0.16707080014188858
In [8]:
dfdy(x, y)

Out[8]:
0.16683109273161312

Pretty close! We'd expect to get closer if we make $h$ smaller.

In [9]:
gradf(x, y)

Out[9]:
array([ 0.03333333,  0.16683109])

That's just the same thing. We are now ready to implement gradient descent

In [10]:
def grad_descent2(f, gradf, init_t, alpha):
EPS = 1e-5
prev_t = init_t-10*EPS
t = init_t.copy()

max_iter = 1000
iter = 0
while norm(t - prev_t) > EPS and iter < max_iter:
prev_t = t.copy()
print t, f(t[0], t[1]), gradf(t[0], t[1])
iter += 1

return t


Let's try to run it, guided by the countour plot about where to start.

In [11]:
f(10., 8.)

Out[11]:
1.1271253262964116
In [12]:
grad_descent2(f, gradf, array([10.0, 8.0]), 0.01)

[ 9.99587412  7.99671596] 1.124342902 [ 0.413072    0.32879157]
[ 9.9917434   7.99342804] 1.12155397037 [ 0.41354556  0.32917105]
[ 9.98760794  7.99013633] 1.11875865954 [ 0.41400905  0.32954248]
[ 9.98346785  7.98684091] 1.11595709897 [ 0.41446241  0.3299058 ]
[ 9.97932323  7.98354185] 1.11314941939 [ 0.41490557  0.33026098]
[ 9.97517417  7.98023924] 1.11033575278 [ 0.41533849  0.33060795]
[ 9.97102078  7.97693316] 1.10751623235 [ 0.4157611   0.33094668]
[ 9.96686317  7.97362369] 1.10469099251 [ 0.41617334  0.33127712]
[ 9.96270144  7.97031092] 1.10186016887 [ 0.41657517  0.33159923]
[ 9.95853569  7.96699493] 1.09902389818 [ 0.41696652  0.33191296]
[ 9.95436602  7.9636758 ] 1.09618231834 [ 0.41734736  0.33221828]
[ 9.95019255  7.96035362] 1.09333556834 [ 0.41771762  0.33251514]
[ 9.94601537  7.95702847] 1.09048378828 [ 0.41807726  0.3328035 ]
[ 9.9418346   7.95370043] 1.0876271193 [ 0.41842622  0.33308333]
[ 9.93765034  7.9503696 ] 1.08476570356 [ 0.41876447  0.33335458]
[ 9.93346269  7.94703605] 1.08189968424 [ 0.41909195  0.33361722]
[ 9.92927177  7.94369988] 1.0790292055 [ 0.41940863  0.33387122]
[ 9.92507769  7.94036117] 1.07615441243 [ 0.41971445  0.33411654]
[ 9.92088054  7.93702   ] 1.07327545105 [ 0.42000938  0.33435314]
[ 9.91668045  7.93367647] 1.07039246827 [ 0.42029338  0.334581  ]
[ 9.91247752  7.93033066] 1.06750561186 [ 0.42056641  0.33480008]
[ 9.90827185  7.92698266] 1.06461503043 [ 0.42082843  0.33501036]
[ 9.90406357  7.92363256] 1.06172087337 [ 0.42107941  0.3352118 ]
[ 9.89985277  7.92028044] 1.05882329085 [ 0.4213193   0.33540438]
[ 9.89563958  7.9169264 ] 1.05592243381 [ 0.42154809  0.33558808]
[ 9.8914241   7.91357051] 1.05301845385 [ 0.42176573  0.33576285]
[ 9.88720644  7.91021289] 1.05011150328 [ 0.4219722   0.33592869]
[ 9.88298672  7.9068536 ] 1.04720173505 [ 0.42216747  0.33608557]
[ 9.87876505  7.90349274] 1.04428930272 [ 0.4223515   0.33623347]
[ 9.87454153  7.90013041] 1.04137436044 [ 0.42252429  0.33637236]
[ 9.87031629  7.89676669] 1.03845706291 [ 0.42268579  0.33650223]
[ 9.86608943  7.89340166] 1.03553756533 [ 0.42283599  0.33662306]
[ 9.86186107  7.89003543] 1.03261602341 [ 0.42297488  0.33673483]
[ 9.85763132  7.88666808] 1.02969259328 [ 0.42310242  0.33683753]
[ 9.8534003   7.88329971] 1.02676743151 [ 0.4232186   0.33693114]
[ 9.84916811  7.8799304 ] 1.02384069506 [ 0.4233234   0.33701565]
[ 9.84493488  7.87656024] 1.0209125412 [ 0.42341681  0.33709105]
[ 9.84070071  7.87318933] 1.01798312756 [ 0.42349882  0.33715732]
[ 9.83646572  7.86981776] 1.01505261203 [ 0.42356942  0.33721446]
[ 9.83223003  7.86644561] 1.01212115275 [ 0.42362858  0.33726246]
[ 9.82799374  7.86307299] 1.00918890808 [ 0.42367632  0.33730131]
[ 9.82375698  7.85969997] 1.00625603654 [ 0.42371261  0.33733101]
[ 9.81951985  7.85632666] 1.0033226968 [ 0.42373745  0.33735155]
[ 9.81528248  7.85295315] 1.00038904766 [ 0.42375084  0.33736293]
[ 9.81104497  7.84957952] 0.997455247966 [ 0.42375278  0.33736515]
[ 9.80680744  7.84620587] 0.994521456619 [ 0.42374326  0.3373582 ]
[ 9.80257001  7.84283229] 0.991587832522 [ 0.4237223   0.33734209]
[ 9.79833279  7.83945887] 0.988654534544 [ 0.42368988  0.33731681]
[ 9.79409589  7.8360857 ] 0.985721721494 [ 0.42364601  0.33728238]
[ 9.78985943  7.83271287] 0.982789552077 [ 0.4235907  0.3372388]
[ 9.78562352  7.82934049] 0.979858184866 [ 0.42352396  0.33718606]
[ 9.78138828  7.82596863] 0.976927778266 [ 0.42344579  0.33712419]
[ 9.77715382  7.82259738] 0.973998490475 [ 0.42335621  0.33705318]
[ 9.77292026  7.81922685] 0.971070479457 [ 0.42325522  0.33697305]
[ 9.76868771  7.81585712] 0.968143902901 [ 0.42314284  0.3368838 ]
[ 9.76445628  7.81248828] 0.965218918189 [ 0.42301909  0.33678545]
[ 9.76022609  7.80912043] 0.962295682363 [ 0.42288398  0.33667802]
[ 9.75599725  7.80575365] 0.959374352093 [ 0.42273753  0.33656151]
[ 9.75176987  7.80238803] 0.956455083636 [ 0.42257976  0.33643594]
[ 9.74754408  7.79902367] 0.953538032809 [ 0.42241069  0.33630133]
[ 9.74331997  7.79566066] 0.950623354955 [ 0.42223034  0.3361577 ]
[ 9.73909767  7.79229908] 0.947711204907 [ 0.42203873  0.33600507]
[ 9.73487728  7.78893903] 0.944801736956 [ 0.4218359   0.33584345]
[ 9.73065892  7.7855806 ] 0.941895104819 [ 0.42162187  0.33567287]
[ 9.7264427   7.78222387] 0.938991461608 [ 0.42139667  0.33549335]
[ 9.72222873  7.77886894] 0.936090959795 [ 0.42116033  0.33530491]
[ 9.71801713  7.77551589] 0.933193751182 [ 0.42091287  0.33510759]
[ 9.713808    7.77216481] 0.930299986867 [ 0.42065434  0.3349014 ]
[ 9.70960146  7.7688158 ] 0.927409817218 [ 0.42038477  0.33468638]
[ 9.70539761  7.76546893] 0.924523391836 [ 0.42010419  0.33446255]
[ 9.70119657  7.76212431] 0.921640859528 [ 0.41981265  0.33422994]
[ 9.69699844  7.75878201] 0.918762368277 [ 0.41951017  0.33398859]
[ 9.69280334  7.75544212] 0.915888065209 [ 0.4191968   0.33373852]
[ 9.68861137  7.75210474] 0.913018096569 [ 0.41887259  0.33347978]
[ 9.68442265  7.74876994] 0.910152607686 [ 0.41853757  0.33321238]
[ 9.68023727  7.74543782] 0.907291742948 [ 0.41819179  0.33293638]
[ 9.67605535  7.74210845] 0.904435645772 [ 0.41783529  0.3326518 ]
[ 9.671877    7.73878193] 0.90158445858 [ 0.41746813  0.33235869]
[ 9.66770232  7.73545835] 0.898738322765 [ 0.41709035  0.33205708]
[ 9.66353142  7.73213778] 0.89589737867 [ 0.41670199  0.331747  ]
[ 9.6593644   7.72882031] 0.893061765559 [ 0.41630312  0.33142852]
[ 9.65520136  7.72550602] 0.890231621592 [ 0.41589379  0.33110165]
[ 9.65104243  7.722195  ] 0.887407083798 [ 0.41547404  0.33076645]
[ 9.64688769  7.71888734] 0.884588288051 [ 0.41504393  0.33042297]
[ 9.64273725  7.71558311] 0.881775369046 [ 0.41460351  0.33007124]
[ 9.63859121  7.7122824 ] 0.878968460275 [ 0.41415286  0.32971131]
[ 9.63444968  7.70898528] 0.876167694002 [ 0.41369201  0.32934322]
[ 9.63031276  7.70569185] 0.873373201242 [ 0.41322104  0.32896704]
[ 9.62618055  7.70240218] 0.870585111737 [ 0.41274001  0.32858279]
[ 9.62205315  7.69911635] 0.867803553935 [ 0.41224896  0.32819054]
[ 9.61793066  7.69583445] 0.865028654967 [ 0.41174798  0.32779034]
[ 9.61381318  7.69255655] 0.862260540632 [ 0.41123712  0.32738223]
[ 9.60970081  7.68928272] 0.859499335367 [ 0.41071645  0.32696626]
[ 9.60559365  7.68601306] 0.856745162236 [ 0.41018602  0.3265425 ]
[ 9.60149179  7.68274764] 0.853998142907 [ 0.40964592  0.32611099]
[ 9.59739533  7.67948653] 0.851258397635 [ 0.40909621  0.32567178]
[ 9.59330437  7.67622981] 0.848526045241 [ 0.40853696  0.32522494]
[ 9.589219    7.67297756] 0.845801203099 [ 0.40796823  0.32477052]
[ 9.58513931  7.66972985] 0.843083987115 [ 0.40739011  0.32430858]
[ 9.58106541  7.66648677] 0.840374511714 [ 0.40680266  0.32383917]
[ 9.57699739  7.66324838] 0.837672889823 [ 0.40620595  0.32336235]
[ 9.57293533  7.66001475] 0.834979232857 [ 0.40560005  0.32287818]
[ 9.56887933  7.65678597] 0.832293650701 [ 0.40498506  0.32238673]
[ 9.56482948  7.6535621 ] 0.829616251701 [ 0.40436103  0.32188804]
[ 9.56078587  7.65034322] 0.826947142648 [ 0.40372804  0.32138219]
[ 9.55674858  7.6471294 ] 0.824286428767 [ 0.40308618  0.32086923]
[ 9.55271772  7.64392071] 0.821634213702 [ 0.40243552  0.32034923]
[ 9.54869337  7.64071722] 0.818990599506 [ 0.40177613  0.31982225]
[ 9.54467561  7.63751899] 0.816355686633 [ 0.40110811  0.31928835]
[ 9.54066453  7.63432611] 0.813729573924 [ 0.40043152  0.3187476 ]
[ 9.53666021  7.63113863] 0.811112358598 [ 0.39974645  0.31820006]
[ 9.53266275  7.62795663] 0.808504136243 [ 0.39905299  0.31764579]
[ 9.52867222  7.62478018] 0.80590500081 [ 0.3983512   0.31708487]
[ 9.5246887   7.62160933] 0.803315044602 [ 0.39764119  0.31651736]
[ 9.52071229  7.61844415] 0.800734358266 [ 0.39692302  0.31594333]
[ 9.51674306  7.61528472] 0.798163030792 [ 0.39619679  0.31536284]
[ 9.51278109  7.61213109] 0.795601149499 [ 0.39546257  0.31477596]
[ 9.50882647  7.60898333] 0.793048800035 [ 0.39472046  0.31418275]
[ 9.50487926  7.6058415 ] 0.79050606637 [ 0.39397054  0.3135833 ]
[ 9.50093956  7.60270567] 0.787973030793 [ 0.39321289  0.31297766]
[ 9.49700743  7.59957589] 0.785449773905 [ 0.39244761  0.31236591]
[ 9.49308295  7.59645224] 0.78293637462 [ 0.39167477  0.31174811]
[ 9.48916621  7.59333475] 0.780432910157 [ 0.39089447  0.31112434]
[ 9.48525726  7.59022351] 0.777939456045 [ 0.39010679  0.31049466]
[ 9.48135619  7.58711856] 0.775456086112 [ 0.38931183  0.30985915]
[ 9.47746307  7.58401997] 0.772982872494 [ 0.38850967  0.30921787]
[ 9.47357798  7.58092779] 0.770519885627 [ 0.38770039  0.30857091]
[ 9.46970097  7.57784209] 0.768067194249 [ 0.3868841   0.30791832]
[ 9.46583213  7.5747629 ] 0.765624865403 [ 0.38606088  0.30726018]
[ 9.46197152  7.5716903 ] 0.763192964435 [ 0.38523081  0.30659657]
[ 9.45811922  7.56862433] 0.760771554995 [ 0.384394    0.30592755]
[ 9.45427528  7.56556506] 0.758360699044 [ 0.38355052  0.3052532 ]
[ 9.45043977  7.56251253] 0.755960456851 [ 0.38270048  0.30457359]
[ 9.44661277  7.55946679] 0.753570886997 [ 0.38184395  0.3038888 ]
[ 9.44279433  7.5564279 ] 0.751192046382 [ 0.38098104  0.30319888]
[ 9.43898452  7.55339591] 0.748823990226 [ 0.38011183  0.30250393]
[ 9.4351834   7.55037087] 0.746466772074 [ 0.37923642  0.30180401]
[ 9.43139103  7.54735283] 0.7441204438 [ 0.37835489  0.30109919]
[ 9.42760748  7.54434184] 0.741785055615 [ 0.37746734  0.30038955]
[ 9.42383281  7.54133795] 0.73946065607 [ 0.37657385  0.29967516]
[ 9.42006707  7.5383412 ] 0.737147292063 [ 0.37567453  0.29895609]
[ 9.41631033  7.53535164] 0.734845008849 [ 0.37476946  0.29823242]
[ 9.41256263  7.53236931] 0.732553850041 [ 0.37385873  0.29750422]
[ 9.40882405  7.52939427] 0.730273857621 [ 0.37294244  0.29677157]
[ 9.40509462  7.52642655] 0.728005071949 [ 0.37202067  0.29603453]
[ 9.40137441  7.52346621] 0.72574753177 [ 0.37109352  0.29529318]
[ 9.39766348  7.52051328] 0.72350127422 [ 0.37016109  0.29454759]
[ 9.39396187  7.5175678 ] 0.721266334841 [ 0.36922345  0.29379785]
[ 9.39026963  7.51462982] 0.719042747585 [ 0.3682807   0.29304401]
[ 9.38658683  7.51169938] 0.716830544826 [ 0.36733294  0.29228615]
[ 9.3829135   7.50877652] 0.71462975737 [ 0.36638026  0.29152435]
[ 9.37924969  7.50586128] 0.712440414466 [ 0.36542274  0.29075868]
[ 9.37559547  7.50295369] 0.710262543815 [ 0.36446047  0.28998921]
[ 9.37195086  7.5000538 ] 0.708096171582 [ 0.36349356  0.28921601]
[ 9.36831593  7.49716164] 0.70594132241 [ 0.36252208  0.28843916]
[ 9.36469071  7.49427725] 0.703798019428 [ 0.36154613  0.28765872]
[ 9.36107524  7.49140066] 0.701666284263 [ 0.3605658   0.28687478]
[ 9.35746959  7.48853191] 0.699546137056 [ 0.35958117  0.28608739]
[ 9.35387378  7.48567104] 0.69743759647 [ 0.35859234  0.28529664]
[ 9.35028785  7.48281807] 0.695340679707 [ 0.3575994  0.2845026]
[ 9.34671186  7.47997304] 0.69325540252 [ 0.35660243  0.28370533]
[ 9.34314583  7.47713599] 0.691181779221 [ 0.35560153  0.2829049 ]
[ 9.33958982  7.47430694] 0.689119822704 [ 0.35459678  0.2821014 ]
[ 9.33604385  7.47148593] 0.687069544451 [ 0.35358827  0.28129488]
[ 9.33250797  7.46867298] 0.68503095455 [ 0.35257608  0.28048542]
[ 9.32898221  7.46586813] 0.683004061707 [ 0.35156032  0.27967309]
[ 9.3254666   7.46307139] 0.680988873263 [ 0.35054105  0.27885795]
[ 9.32196119  7.46028281] 0.678985395206 [ 0.34951837  0.27804008]
[ 9.31846601  7.45750241] 0.676993632187 [ 0.34849237  0.27721954]
[ 9.31498109  7.45473022] 0.675013587535 [ 0.34746312  0.27639641]
[ 9.31150645  7.45196625] 0.673045263272 [ 0.34643072  0.27557075]
[ 9.30804215  7.44921055] 0.671088660128 [ 0.34539526  0.27474263]
[ 9.30458819  7.44646312] 0.669143777557 [ 0.3443568   0.27391212]
[ 9.30114463  7.443724  ] 0.667210613751 [ 0.34331545  0.27307929]
[ 9.29771147  7.44099321] 0.665289165659 [ 0.34227128  0.27224419]
[ 9.29428876  7.43827076] 0.663379428999 [ 0.34122437  0.2714069 ]
[ 9.29087652  7.4355567 ] 0.661481398277 [ 0.34017481  0.27056749]
[ 9.28747477  7.43285102] 0.6595950668 [ 0.33912269  0.26972602]
[ 9.28408354  7.43015376] 0.657720426695 [ 0.33806807  0.26888255]
[ 9.28070286  7.42746494] 0.655857468924 [ 0.33701105  0.26803715]
[ 9.27733275  7.42478456] 0.654006183301 [ 0.3359517   0.26718989]
[ 9.27397323  7.42211266] 0.652166558505 [ 0.3348901   0.26634082]
[ 9.27062433  7.41944926] 0.650338582102 [ 0.33382634  0.26549002]
[ 9.26728607  7.41679436] 0.648522240558 [ 0.33276049  0.26463754]
[ 9.26395846  7.41414798] 0.646717519256 [ 0.33169263  0.26378345]
[ 9.26064154  7.41151015] 0.644924402511 [ 0.33062283  0.26292781]
[ 9.25733531  7.40888087] 0.643142873591 [ 0.32955119  0.26207068]
[ 9.2540398   7.40626016] 0.64137291473 [ 0.32847776  0.26121213]
[ 9.25075502  7.40364804] 0.639614507146 [ 0.32740263  0.26035221]
[ 9.24748099  7.40104452] 0.637867631056 [ 0.32632588  0.25949098]
[ 9.24421773  7.39844961] 0.636132265697 [ 0.32524758  0.25862851]
[ 9.24096526  7.39586332] 0.634408389338 [ 0.3241678   0.25776486]
[ 9.23772358  7.39328567] 0.632695979298 [ 0.32308661  0.25690007]
[ 9.23449271  7.39071667] 0.630995011964 [ 0.3220041   0.25603422]
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[ 8.6014079   6.89118298] 0.434160702854 [ 0.04343383  0.0330922 ]
[ 8.60097356  6.89085206] 0.434130961414 [ 0.04322088  0.03292162]
[ 8.60054135  6.89052284] 0.434101516373 [ 0.04300899  0.03275188]
[ 8.60011126  6.89019532] 0.434072364778 [ 0.04279816  0.03258299]
[ 8.59968328  6.88986949] 0.434043503708 [ 0.04258839  0.03241494]
[ 8.59925739  6.88954534] 0.43401493027 [ 0.04237965  0.03224772]
[ 8.5988336   6.88922287] 0.433986641597 [ 0.04217196  0.03208134]
[ 8.59841188  6.88890205] 0.433958634855 [ 0.04196531  0.03191579]
[ 8.59799223  6.8885829 ] 0.433930907234 [ 0.04175969  0.03175107]
[ 8.59757463  6.88826539] 0.433903455955 [ 0.04155509  0.03158717]
[ 8.59715908  6.88794951] 0.433876278265 [ 0.04135151  0.03142408]
[ 8.59674556  6.88763527] 0.433849371437 [ 0.04114895  0.03126181]
[ 8.59633407  6.88732266] 0.433822732774 [ 0.0409474   0.03110034]
[ 8.5959246   6.88701165] 0.433796359605 [ 0.04074686  0.03093968]
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[ 8.59511166  6.88639446] 0.433744399189 [ 0.04034877  0.03062076]
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[ 8.59430666  6.88578362] 0.433693469343 [ 0.03995465  0.03030502]
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[ 8.59350952  6.88517909] 0.43364354963 [ 0.03956445  0.02999241]
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[ 8.59155057  6.88369491] 0.433523039188 [ 0.03860587  0.02922445]
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[ 8.59078034  6.88311193] 0.433476499763 [ 0.03822911  0.0289226 ]
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[ 8.58963906  6.88224874] 0.433408415613 [ 0.037671    0.02847545]
[ 8.58926235  6.88196398] 0.433386171761 [ 0.03748681  0.02832789]
[ 8.58888748  6.88168071] 0.433364149601 [ 0.03730355  0.02818106]
[ 8.58851445  6.88139889] 0.433342346923 [ 0.03712121  0.02803497]
[ 8.58814324  6.88111855] 0.43332076154 [ 0.03693977  0.0278896 ]
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[ 8.58740625  6.8805622 ] 0.433278234012 [ 0.03657962  0.02760105]
[ 8.58704045  6.88028619] 0.433257287598 [ 0.03640089  0.02745785]
[ 8.58667644  6.88001161] 0.433236549941 [ 0.03622306  0.02731537]
[ 8.58631421  6.87973846] 0.43321601896 [ 0.03604611  0.0271736 ]
[ 8.58595375  6.87946672] 0.433195692593 [ 0.03587005  0.02703253]
[ 8.58559505  6.8791964 ] 0.433175568799 [ 0.03569487  0.02689218]
[ 8.5852381   6.87892747] 0.43315564556 [ 0.03552056  0.02675252]
[ 8.5848829   6.87865995] 0.433135920874 [ 0.03534713  0.02661356]
[ 8.58452942  6.87839381] 0.433116392763 [ 0.03517456  0.02647529]
[ 8.58417768  6.87812906] 0.433097059264 [ 0.03500286  0.02633772]
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[ 8.58244454  6.87682579] 0.433003243715 [ 0.03415713  0.02566008]
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[ 8.58108821  6.87580737] 0.432931501424 [ 0.03349565  0.02513006]
[ 8.58075326  6.87555607] 0.432914010436 [ 0.03333234  0.0249992 ]
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[ 8.57845383  6.87383325] 0.432796357624 [ 0.03221183  0.02410135]
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[ 8.57717474  6.87287671] 0.432732738988 [ 0.03158898  0.02360225]
[ 8.57685885  6.87264069] 0.432717228518 [ 0.03143521  0.02347903]
[ 8.5765445  6.8724059] 0.432701872585 [ 0.03128221  0.02335643]
[ 8.57623168  6.87217233] 0.432686669649 [ 0.03112998  0.02323444]
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[ 8.57530232  6.87147893] 0.432641963624 [ 0.03067783  0.02287212]
[ 8.57499554  6.87125021] 0.432627357557 [ 0.03052862  0.02275255]
[ 8.57469025  6.87102269] 0.432612897004 [ 0.03038016  0.02263358]
[ 8.57438645  6.87079635] 0.432598580513 [ 0.03023244  0.0225152 ]
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[ 8.57348388  6.87012442] 0.432556481111 [ 0.0297937   0.02216361]
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[ 8.5705688   6.86795958] 0.432424940903 [ 0.02837803  0.02102911]
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[ 8.54981447  6.85286949] 0.431682853044 [ 0.01838179  0.013016  ]
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[ 8.5489041   6.85222568] 0.431658048492 [ 0.01794796  0.01266812]
[ 8.54872462  6.852099  ] 0.431653234398 [ 0.01786249  0.01259959]
[ 8.548546    6.85197301] 0.431648468107 [ 0.01777745  0.01253139]
[ 8.54836822  6.85184769] 0.431643749142 [ 0.01769284  0.01246354]
[ 8.5481913   6.85172306] 0.431639077032 [ 0.01760865  0.01239603]
[ 8.54801521  6.8515991 ] 0.431634451308 [ 0.01752489  0.01232885]
[ 8.54783996  6.85147581] 0.431629871509 [ 0.01744154  0.01226202]
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[ 8.5473192   6.85110994] 0.431616403097 [ 0.01719399  0.0120635 ]
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[ 8.54547212  6.84981845] 0.431570327924 [ 0.01631756  0.01136061]
[ 8.54530895  6.84970485] 0.431566384491 [ 0.01624025  0.01129862]
[ 8.54514654  6.84959186] 0.431562480176 [ 0.01616334  0.01123693]
[ 8.54498491  6.84947949] 0.431558614589 [ 0.01608681  0.01117555]
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[ 8.54434599  6.84903612] 0.431543531852 [ 0.0157845   0.01093308]
[ 8.54418815  6.84892679] 0.431539854187 [ 0.01570987  0.01087322]
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[ 8.54387469  6.84870992] 0.431532607894 [ 0.01556172  0.0107544 ]
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[ 8.54341004  6.84838906] 0.43152200565 [ 0.01534227  0.01057838]
[ 8.54325662  6.84828327] 0.431518541405 [ 0.01526985  0.01052029]
[ 8.54310392  6.84817807] 0.431515111499 [ 0.01519779  0.01046249]
[ 8.54295194  6.84807344] 0.431511715589 [ 0.0151261   0.01040499]
[ 8.54280068  6.84796939] 0.431508353336 [ 0.01505476  0.01034777]
[ 8.54265013  6.84786592] 0.431505024404 [ 0.01498378  0.01029083]
[ 8.5425003   6.84776301] 0.431501728462 [ 0.01491315  0.01023418]
[ 8.54235116  6.84766067] 0.43149846518 [ 0.01484288  0.01017781]
[ 8.54220274  6.84755889] 0.431495234233 [ 0.01477296  0.01012173]
[ 8.54205501  6.84745767] 0.431492035297 [ 0.0147034   0.01006592]
[ 8.54190797  6.84735701] 0.431488868055 [ 0.01463417  0.0100104 ]
[ 8.54176163  6.84725691] 0.431485732189 [ 0.0145653   0.00995515]
[ 8.54161598  6.84715736] 0.431482627387 [ 0.01449677  0.00990017]
[ 8.54147101  6.84705835] 0.431479553339 [ 0.01442858  0.00984548]
[ 8.54132672  6.8469599 ] 0.431476509738 [ 0.01436074  0.00979105]
[ 8.54118312  6.84686199] 0.431473496281 [ 0.01429323  0.0097369 ]
[ 8.54104018  6.84676462] 0.431470512667 [ 0.01422607  0.00968302]
[ 8.54089792  6.84666779] 0.431467558599 [ 0.01415923  0.0096294 ]
[ 8.54075633  6.8465715 ] 0.431464633782 [ 0.01409274  0.00957606]
[ 8.5406154   6.84647574] 0.431461737923 [ 0.01402657  0.00952298]
[ 8.54047514  6.84638051] 0.431458870736 [ 0.01396074  0.00947017]
[ 8.54033553  6.8462858 ] 0.431456031932 [ 0.01389524  0.00941762]
[ 8.54019658  6.84619163] 0.431453221231 [ 0.01383006  0.00936533]
[ 8.54005828  6.84609797] 0.43145043835 [ 0.01376521  0.00931331]
[ 8.53992063  6.84600484] 0.431447683014 [ 0.01370069  0.00926154]
[ 8.53978362  6.84591223] 0.431444954947 [ 0.01363649  0.00921004]
[ 8.53964725  6.84582013] 0.431442253877 [ 0.01357261  0.00915879]
[ 8.53951153  6.84572854] 0.431439579535 [ 0.01350905  0.00910779]
[ 8.53937644  6.84563746] 0.431436931654 [ 0.0134458   0.00905705]
[ 8.53924198  6.84554689] 0.431434309972 [ 0.01338288  0.00900657]
[ 8.53910815  6.84545682] 0.431431714226 [ 0.01332027  0.00895634]
[ 8.53897495  6.84536726] 0.431429144158 [ 0.01325797  0.00890636]
[ 8.53884237  6.8452782 ] 0.431426599512 [ 0.01319599  0.00885662]
[ 8.53871041  6.84518963] 0.431424080035 [ 0.01313431  0.00880714]
[ 8.53857906  6.84510156] 0.431421585474 [ 0.01307294  0.0087579 ]
[ 8.53844834  6.84501398] 0.431419115583 [ 0.01301188  0.00870891]
[ 8.53831822  6.84492689] 0.431416670115 [ 0.01295113  0.00866017]
[ 8.53818871  6.84484029] 0.431414248827 [ 0.01289068  0.00861167]
[ 8.5380598   6.84475417] 0.431411851477 [ 0.01283053  0.00856341]
[ 8.53793149  6.84466854] 0.431409477828 [ 0.01277069  0.00851539]
[ 8.53780379  6.84458338] 0.431407127641 [ 0.01271114  0.00846761]
[ 8.53767667  6.84449871] 0.431404800684 [ 0.01265189  0.00842007]
[ 8.53755016  6.84441451] 0.431402496726 [ 0.01259294  0.00837277]
[ 8.53742423  6.84433078] 0.431400215536 [ 0.01253428  0.0083257 ]
[ 8.53729888  6.84424752] 0.431397956888 [ 0.01247592  0.00827887]
[ 8.53717412  6.84416473] 0.431395720557 [ 0.01241785  0.00823227]
[ 8.53704995  6.84408241] 0.43139350632 [ 0.01236007  0.00818591]
[ 8.53692635  6.84400055] 0.431391313958 [ 0.01230258  0.00813977]
[ 8.53680332  6.84391915] 0.431389143252 [ 0.01224537  0.00809387]
[ 8.53668087  6.84383822] 0.431386993987 [ 0.01218845  0.0080482 ]
[ 8.53655898  6.84375773] 0.431384865948 [ 0.01213182  0.00800275]
[ 8.53643766  6.84367771] 0.431382758923 [ 0.01207547  0.00795753]
[ 8.53631691  6.84359813] 0.431380672705 [ 0.01201941  0.00791254]
[ 8.53619671  6.84351901] 0.431378607084 [ 0.01196362  0.00786777]
[ 8.53607708  6.84344033] 0.431376561856 [ 0.01190811  0.00782323]
[ 8.535958   6.8433621] 0.431374536818 [ 0.01185288  0.00777891]
[ 8.53583947  6.84328431] 0.431372531768 [ 0.01179793  0.00773481]
[ 8.53572149  6.84320696] 0.431370546507 [ 0.01174325  0.00769093]
[ 8.53560406  6.84313005] 0.431368580837 [ 0.01168884  0.00764727]
[ 8.53548717  6.84305358] 0.431366634564 [ 0.01163471  0.00760383]
[ 8.53537082  6.84297754] 0.431364707494 [ 0.01158085  0.0075606 ]
[ 8.53525501  6.84290193] 0.431362799436 [ 0.01152726  0.00751759]
[ 8.53513974  6.84282676] 0.431360910199 [ 0.01147393  0.00747479]
[ 8.535025    6.84275201] 0.431359039597 [ 0.01142088  0.00743221]
[ 8.53491079  6.84267769] 0.431357187444 [ 0.01136809  0.00738984]
[ 8.53479711  6.84260379] 0.431355353556 [ 0.01131556  0.00734769]
[ 8.53468396  6.84253031] 0.431353537751 [ 0.01126329  0.00730574]
[ 8.53457132  6.84245725] 0.431351739848 [ 0.01121129  0.007264  ]
[ 8.53445921  6.84238461] 0.431349959669 [ 0.01115955  0.00722247]
[ 8.53434761  6.84231239] 0.431348197038 [ 0.01110806  0.00718115]
[ 8.53423653  6.84224058] 0.431346451779 [ 0.01105684  0.00714004]
[ 8.53412596  6.84216918] 0.431344723719 [ 0.01100587  0.00709913]
[ 8.53401591  6.84209819] 0.431343012688 [ 0.01095515  0.00705842]
[ 8.53390635  6.8420276 ] 0.431341318515 [ 0.01090469  0.00701792]
[ 8.53379731  6.84195742] 0.431339641032 [ 0.01085448  0.00697762]
[ 8.53368876  6.84188765] 0.431337980073 [ 0.01080452  0.00693752]
[ 8.53358072  6.84181827] 0.431336335473 [ 0.01075482  0.00689762]
[ 8.53347317  6.84174929] 0.431334707069 [ 0.01070536  0.00685792]
[ 8.53336612  6.84168072] 0.4313330947 [ 0.01065615  0.00681842]
[ 8.53325955  6.84161253] 0.431331498206 [ 0.01060718  0.00677912]
[ 8.53315348  6.84154474] 0.431329917429 [ 0.01055846  0.00674001]
[ 8.5330479   6.84147734] 0.431328352212 [ 0.01050998  0.0067011 ]
[ 8.5329428   6.84141033] 0.431326802399 [ 0.01046175  0.00666238]
[ 8.53283818  6.84134371] 0.431325267838 [ 0.01041376  0.00662385]
[ 8.53273404  6.84127747] 0.431323748377 [ 0.010366    0.00658552]
[ 8.53263038  6.84121161] 0.431322243864 [ 0.01031849  0.00654738]
[ 8.5325272   6.84114614] 0.431320754152 [ 0.01027121  0.00650943]
[ 8.53242449  6.84108104] 0.431319279092 [ 0.01022418  0.00647166]
[ 8.53232224  6.84101633] 0.431317818538 [ 0.01017737  0.00643409]
[ 8.53222047  6.84095199] 0.431316372347 [ 0.0101308   0.00639671]
[ 8.53211916  6.84088802] 0.431314940374 [ 0.01008446  0.00635951]
[ 8.53201832  6.84082442] 0.431313522478 [ 0.01003836  0.00632249]
[ 8.53191793  6.8407612 ] 0.431312118519 [ 0.00999248  0.00628566]
[ 8.53181801  6.84069834] 0.431310728358 [ 0.00994684  0.00624902]
[ 8.53171854  6.84063585] 0.431309351857 [ 0.00990142  0.00621256]
[ 8.53161953  6.84057373] 0.431307988881 [ 0.00985623  0.00617628]
[ 8.53152096  6.84051196] 0.431306639294 [ 0.00981127  0.00614018]
[ 8.53142285  6.84045056] 0.431305302963 [ 0.00976653  0.00610426]
[ 8.53132519  6.84038952] 0.431303979757 [ 0.00972202  0.00606852]
[ 8.53122797  6.84032883] 0.431302669543 [ 0.00967772  0.00603296]
[ 8.53113119  6.8402685 ] 0.431301372193 [ 0.00963365  0.00599758]
[ 8.53103485  6.84020853] 0.431300087579 [ 0.0095898   0.00596237]
[ 8.53093895  6.8401489 ] 0.431298815573 [ 0.00954617  0.00592734]
[ 8.53084349  6.84008963] 0.431297556051 [ 0.00950276  0.00589248]
[ 8.53074847  6.84003071] 0.431296308887 [ 0.00945957  0.0058578 ]
[ 8.53065387  6.83997213] 0.431295073958 [ 0.00941659  0.00582329]
[ 8.5305597  6.8399139] 0.431293851143 [ 0.00937382  0.00578895]
[ 8.53046597  6.83985601] 0.431292640321 [ 0.00933127  0.00575479]
[ 8.53037265  6.83979846] 0.431291441372 [ 0.00928894  0.00572079]
[ 8.53027976  6.83974125] 0.431290254178 [ 0.00924681  0.00568697]
[ 8.5301873   6.83968438] 0.431289078623 [ 0.0092049   0.00565331]
[ 8.53009525  6.83962785] 0.431287914589 [ 0.00916319  0.00561982]
[ 8.53000361  6.83957165] 0.431286761962 [ 0.00912169  0.0055865 ]
[ 8.5299124   6.83951578] 0.431285620629 [ 0.0090804   0.00555334]
[ 8.52982159  6.83946025] 0.431284490477 [ 0.00903932  0.00552035]
[ 8.5297312   6.83940505] 0.431283371394 [ 0.00899844  0.00548753]
[ 8.52964122  6.83935017] 0.431282263271 [ 0.00895777  0.00545486]
[ 8.52955164  6.83929562] 0.431281165997 [ 0.0089173   0.00542237]
[ 8.52946247  6.8392414 ] 0.431280079466 [ 0.00887704  0.00539003]
[ 8.52937369  6.8391875 ] 0.431279003569 [ 0.00883697  0.00535785]
[ 8.52928532  6.83913392] 0.431277938202 [ 0.00879711  0.00532584]
[ 8.52919735  6.83908066] 0.431276883259 [ 0.00875744  0.00529398]
[ 8.52910978  6.83902772] 0.431275838636 [ 0.00871797  0.00526229]
[ 8.5290226  6.8389751] 0.43127480423 [ 0.0086787   0.00523075]
[ 8.52893581  6.83892279] 0.43127377994 [ 0.00863963  0.00519937]
[ 8.52884942  6.8388708 ] 0.431272765665 [ 0.00860075  0.00516814]
[ 8.52876341  6.83881912] 0.431271761304 [ 0.00856207  0.00513707]
[ 8.52867779  6.83876775] 0.43127076676 [ 0.00852358  0.00510616]
[ 8.52859255  6.83871669] 0.431269781935 [ 0.00848528  0.0050754 ]
[ 8.5285077   6.83866593] 0.431268806731 [ 0.00844718  0.0050448 ]
[ 8.52842323  6.83861548] 0.431267841053 [ 0.00840926  0.00501434]
[ 8.52833914  6.83856534] 0.431266884806 [ 0.00837154  0.00498404]
[ 8.52825542  6.8385155 ] 0.431265937896 [ 0.008334    0.00495389]
[ 8.52817208  6.83846596] 0.43126500023 [ 0.00829665  0.00492389]
[ 8.52808911  6.83841672] 0.431264071715 [ 0.00825949  0.00489404]
[ 8.52800652  6.83836778] 0.431263152262 [ 0.00822251  0.00486434]
[ 8.52792429  6.83831914] 0.431262241778 [ 0.00818572  0.00483479]
[ 8.52784244  6.83827079] 0.431261340176 [ 0.00814911  0.00480538]
[ 8.52776095  6.83822274] 0.431260447366 [ 0.00811269  0.00477612]
[ 8.52767982  6.83817497] 0.431259563261 [ 0.00807644  0.00474701]
[ 8.52759905  6.8381275 ] 0.431258687775 [ 0.00804038  0.00471804]
[ 8.52751865  6.83808032] 0.431257820821 [ 0.0080045   0.00468922]
[ 8.5274386   6.83803343] 0.431256962314 [ 0.0079688   0.00466054]
[ 8.52735892  6.83798683] 0.43125611217 [ 0.00793328  0.004632  ]
[ 8.52727958  6.83794051] 0.431255270307 [ 0.00789793  0.00460361]
[ 8.5272006   6.83789447] 0.431254436641 [ 0.00786276  0.00457535]
[ 8.52712198  6.83784872] 0.43125361109 [ 0.00782777  0.00454724]
[ 8.5270437   6.83780324] 0.431252793575 [ 0.00779295  0.00451927]
[ 8.52696577  6.83775805] 0.431251984014 [ 0.00775831  0.00449144]
[ 8.52688819  6.83771314] 0.43125118233 [ 0.00772384  0.00446374]
[ 8.52681095  6.8376685 ] 0.431250388442 [ 0.00768954  0.00443619]
[ 8.52673405  6.83762414] 0.431249602274 [ 0.00765542  0.00440877]
[ 8.5266575   6.83758005] 0.431248823748 [ 0.00762146  0.00438149]
[ 8.52658128  6.83753624] 0.43124805279 [ 0.00758768  0.00435434]
[ 8.52650541  6.83749269] 0.431247289322 [ 0.00755406  0.00432733]
[ 8.52642987  6.83744942] 0.431246533271 [ 0.00752061  0.00430046]
[ 8.52635466  6.83740641] 0.431245784562 [ 0.00748733  0.00427372]
[ 8.52627979  6.83736368] 0.431245043123 [ 0.00745421  0.00424711]
[ 8.52620525  6.83732121] 0.431244308881 [ 0.00742126  0.00422063]

Out[12]:
array([ 8.52620525,  6.83732121])