如何计算 Python 中的积分估计？
How to calculate Integral estimation in Python?
如何在 python3 中对下面列出的提供的数据集执行以下操作？
问题
知道 data.txt 有 2 列：
xValues，其中 ≤ ≤ ，并且是一些常量
gOfXValues
1计算 ′()
的二阶估计
2计算 $g'(x)$ 和 $\int_a^b g(x)dx.$
的二阶估计值
通常，我们不知道从随机数据样本中给出的值是均匀分离的。
3绘制 () 和 ′() ，打印积分。
4根据图表，你认为()是什么函数？
5通过定性比较你假设的 () 的精确导数和积分与之前获得的数值结果来验证这一点。
到目前为止我做了什么
import pandas as pd
dataFrame = pd.read_csv('/Users/Files/data.txt', sep="\s+", names=['xValue','gOfXValue'])
dataFrame.info
dataFrame.head()
dgdxArray = []
gOfXValues = []
#iterate all observations and extract the columns as the independent and dependent variable
for index, row in dataFrame.iterrows():
xValue = row['xValue']
gOfXValue = row['gOfXValue']
gOfXValues.append(gOfXValue)
if index > 0:
h = 0.05
difference = gOfXValue - gOfXValues[index-1] #check the difference between Current vs Previous value
dgdx = difference / h #get the Derivative
dgdxArray.append(dgdx) #add the derivative to an array so as to plot it
dgdxArray.insert(0,0.5) #hard code values
#plot the initial values provided
fig = plt.figure(figsize = (6,12))
ax1 = fig.add_subplot(311)
ax1.plot(dataFrame['xValue'], dataFrame['gOfXValue'])
ax1.set_title('Plot initial values x, g(x)')
ax1.set_xlabel('xValue')
ax1.set_ylabel('gOfXValue')
#Plot X value and the derivative on y axis
ax2 = fig.add_subplot(312)
ax2.plot(dataFrame['xValue'], dgdxArray)
ax2.set_title('Plot x, derivativeOfGOfx')
ax2.set_xlabel('xValue')
ax2.set_ylabel('gOfXValue')
fig.tight_layout()
plt.show()
EDIT_1
鉴于我只能访问 xValue 和 gOfXvalue，如何找到原始函数定义？
Edit_2 基于对 D Stanley 的评论
#we know that in calculus g(x) = sin(x), g'(x) = cos(x), g(x)dx = - cos(x) + C
#Therefore:
#calculate sin(x) and compare it to g(x) provided
#calculate f'(x) and compare it to g(x) provided
#calculate integral of g(x)dx and compare it to g(x) provided
constant = 2 #choose a random constant
#calculate the sin,cos and integral for existing x value considering that the original function is sin
sinXfound = np.sin(dataFrame.xValue)
cosXfound = np.cos(dataFrame.xValue)
intXfound = - np.cos(dataFrame.xValue) + constant
#create new columns in the original df with values calculate above
dataFrame['sinXfound'] = sinXfound
dataFrame['cosXfound'] = cosXfound
dataFrame['intXfound'] = intXfound
#find what is the difference between sin,cos newly found and the original xValue provided in the request
differenceSinXfound = sinXfound - dataFrame['gOfXValue']
differenceCosXfound = cosXfound - dataFrame['gOfXValue']
differenceIntXfound = intXfound - dataFrame['gOfXValue']
#add columns to df
dataFrame['differenceSinXfound'] = differenceSinXfound
dataFrame['differenceCosXfound'] = differenceCosXfound
dataFrame['differenceIntXfound'] = differenceCosXfound
print(dataFrame)
Edit_3 基于 Lutz 的回答
xValues = dataFrame.xValue
gofXValues = dataFrame.gOfXValue
firstDiffArray = []
def calculate_ALL_Divided_Differences():
for index, row in dataFrame.iterrows():
if index > 0:
xNow = row['xValue']
gNow = row['gOfXValue']
difference = (gofXValues[indexNow] - gofXValues[indexNow - 1]) / (xValues[indexNow] - xValues[indexNow -1])
firstDiffArray.append(difference)
firstDividedDifference = (gofXValues[1] - gofXValues[0]) / (xValues[1] - xValues[0])
x0 = xValues[0]
gOfXZero = gofXValues[0]
#Apply Newton's divided difference interpolation formula
for index, row in dataFrame.iterrows():
if index > 0:
xNow = row['xValue']
gNow = row['gOfXValue']
x_Minus_x0 = xNow - xValues[0]
x_Minus_x1 = xNow - xValues[1]
#Newton's divided difference interpolation formula is
#f(x) = y0+(x-x0) f [x0,x1]+ (x-x0) * (x-x1) * f [x0,x1,x2]
divided_Difference_Interpolation = gOfXZero + (xNow - x0) * firstDividedDifference + x_Minus_x0 * x_Minus_x1
数据集
0.000000000000000000e+00 1.000000000000000000e+00
3.157379551346525814e-02 1.031568549764810605e+00
6.314759102693051629e-02 1.063105631312673660e+00
9.472138654039577443e-02 1.094579807794844983e+00
1.262951820538610326e-01 1.125959705067717476e+00
1.578689775673262907e-01 1.157214042967250833e+00
1.894427730807915489e-01 1.188311666489717755e+00
2.210165685942568070e-01 1.219221576847691280e+00
2.525903641077220652e-01 1.249912962370308467e+00
2.841641596211873511e-01 1.280355229217014390e+00
3.157379551346525814e-01 1.310518031874168710e+00
3.473117506481178118e-01 1.340371303404112702e+00
3.788855461615830977e-01 1.369885285416546861e+00
4.104593416750483836e-01 1.399030557732340974e+00
4.420331371885136140e-01 1.427778067710209653e+00
4.736069327019788444e-01 1.456099159207016047e+00
5.051807282154441303e-01 1.483965601142838819e+00
5.367545237289094162e-01 1.511349615642326727e+00
5.683283192423747021e-01 1.538223905724288576e+00
5.999021147558398770e-01 1.564561682511917962e+00
6.314759102693051629e-01 1.590336691936528268e+00
6.630497057827704488e-01 1.615523240908179226e+00
6.946235012962356237e-01 1.640096222927107217e+00
7.261972968097009096e-01 1.664031143110431099e+00
7.577710923231661955e-01 1.687304142609184154e+00
7.893448878366314814e-01 1.709892022391332755e+00
8.209186833500967673e-01 1.731772266367076707e+00
8.524924788635619421e-01 1.752923063833377038e+00
8.840662743770272280e-01 1.773323331215339138e+00
9.156400698904925139e-01 1.792952733082778582e+00
9.472138654039576888e-01 1.811791702421020833e+00
9.787876609174229747e-01 1.829821460135725886e+00
1.010361456430888261e+00 1.847024033772298957e+00
1.041935251944353436e+00 1.863382275431223256e+00
1.073509047457818832e+00 1.878879878861460462e+00
1.105082842971284007e+00 1.893501395714874080e+00
1.136656638484749404e+00 1.907232250945481322e+00
1.168230433998214579e+00 1.920058757338174438e+00
1.199804229511679754e+00 1.931968129152434877e+00
1.231378025025145151e+00 1.942948494867437148e+00
1.262951820538610326e+00 1.952988909015839214e+00
1.294525616052075501e+00 1.962079363094462847e+00
1.326099411565540898e+00 1.970210795540986215e+00
1.357673207079006072e+00 1.977375100766707305e+00
1.389247002592471247e+00 1.983565137236369846e+00
1.420820798105936644e+00 1.988774734587002602e+00
1.452394593619401819e+00 1.992998699778669724e+00
1.483968389132867216e+00 1.996232822271006846e+00
1.515542184646332391e+00 1.998473878220378808e+00
1.547115980159797566e+00 1.999719633693477938e+00
1.578689775673262963e+00 1.999968846894156327e+00
1.610263571186728138e+00 1.999221269401275647e+00
1.641837366700193535e+00 1.997477646416338626e+00
1.673411162213658709e+00 1.994739716020657028e+00
1.704984957727123884e+00 1.991010207442792002e+00
1.736558753240589281e+00 1.986292838338002742e+00
1.768132548754054456e+00 1.980592311082403967e+00
1.799706344267519631e+00 1.973914308085537694e+00
1.831280139780985028e+00 1.966265486126021811e+00
1.862853935294450203e+00 1.957653469715929573e+00
1.894427730807915378e+00 1.948086843500509424e+00
1.926001526321380775e+00 1.937575143700825064e+00
1.957575321834845949e+00 1.926128848607841171e+00
1.989149117348311346e+00 1.913759368137436745e+00
2.020722912861776521e+00 1.900479032456751760e+00
2.052296708375241696e+00 1.886301079693208704e+00
2.083870503888706871e+00 1.871239642738459663e+00
2.115444299402172490e+00 1.855309735160411755e+00
2.147018094915637665e+00 1.838527236237377238e+00
2.178591890429102840e+00 1.820908875129262583e+00
2.210165685942568015e+00 1.802472214201578105e+00
2.241739481456033189e+00 1.783235631518890418e+00
2.273313276969498808e+00 1.763218302525168424e+00
2.304887072482963983e+00 1.742440180929283100e+00
2.336460867996429158e+00 1.720921978814716535e+00
2.368034663509894333e+00 1.698685145993306556e+00
2.399608459023359508e+00 1.675751848623608709e+00
2.431182254536824683e+00 1.652144947115186779e+00
2.462756050050290302e+00 1.627887973340858441e+00
2.494329845563755477e+00 1.603005107179614530e+00
2.525903641077220652e+00 1.577521152413588590e+00
2.557477436590685826e+00 1.551461512003107668e+00
2.589051232104151001e+00 1.524852162764468444e+00
2.620625027617616620e+00 1.497719629475682934e+00
2.652198823131081795e+00 1.470090958436002904e+00
2.683772618644546970e+00 1.441993690505579018e+00
2.715346414158012145e+00 1.413455833652134119e+00
2.746920209671477320e+00 1.384505835032010967e+00
2.778494005184942495e+00 1.355172552633428618e+00
2.810067800698408114e+00 1.325485226510211501e+00
2.841641596211873289e+00 1.295473449634670038e+00
2.873215391725338463e+00 1.265167138398678670e+00
2.904789187238803638e+00 1.234596502792368877e+00
2.936362982752268813e+00 1.203792016290152311e+00
2.967936778265734432e+00 1.172784385474099356e+00
2.999510573779199607e+00 1.141604519424951558e+00
3.031084369292664782e+00 1.110283498911275091e+00
3.062658164806129957e+00 1.078852545407476660e+00
3.094231960319595132e+00 1.047342989971558280e+00
3.125805755833060751e+00 1.015786242013636764e+00
3.157379551346525925e+00 9.842137579863630137e-01
3.188953346859991100e+00 9.526570100284420528e-01
3.220527142373456275e+00 9.211474545925236734e-01
3.252100937886921450e+00 8.897165010887251313e-01
3.283674733400387069e+00 8.583954805750482198e-01
3.315248528913852244e+00 8.272156145259004223e-01
3.346822324427317419e+00 7.962079837098479107e-01
3.378396119940782594e+00 7.654034972076313448e-01
3.409969915454247769e+00 7.348328616013215520e-01
3.441543710967712943e+00 7.045265503653301842e-01
3.473117506481178562e+00 6.745147734897881664e-01
3.504691301994643737e+00 6.448274473665716044e-01
3.536265097508108912e+00 6.154941649679892546e-01
3.567838893021574087e+00 5.865441663478661027e-01
3.599412688535039262e+00 5.580063094944210933e-01
3.630986484048504881e+00 5.299090415639968743e-01
3.662560279561970056e+00 5.022803705243168437e-01
3.694134075075435231e+00 4.751478372355316671e-01
3.725707870588900406e+00 4.485384879968926652e-01
3.757281666102365580e+00 4.224788475864115211e-01
3.788855461615830755e+00 3.969948928203856919e-01
3.820429257129296374e+00 3.721120266591415593e-01
3.852003052642761549e+00 3.478550528848133316e-01
3.883576848156226724e+00 3.242481513763912915e-01
3.915150643669691899e+00 3.013148540066936665e-01
3.946724439183157074e+00 2.790780211852837978e-01
3.978298234696622693e+00 2.575598190707169000e-01
4.009872030210087424e+00 2.367816974748317982e-01
4.041445825723553043e+00 2.167643684811096927e-01
4.073019621237018661e+00 1.975277857984218954e-01
4.104593416750483392e+00 1.790911248707376391e-01
4.136167212263949011e+00 1.614727637626225398e-01
4.167741007777413742e+00 1.446902648395883562e-01
4.199314803290879361e+00 1.287603572615404479e-01
4.230888598804344980e+00 1.136989203067911847e-01
4.262462394317809711e+00 9.952096754324846195e-02
4.294036189831275330e+00 8.624063186256325508e-02
4.325609985344740060e+00 7.387115139215894022e-02
4.357183780858205679e+00 6.242485629917493561e-02
4.388757576371671298e+00 5.191315649949035382e-02
4.420331371885136029e+00 4.234653028407053821e-02
4.451905167398601648e+00 3.373451387397807810e-02
4.483478962912066379e+00 2.608569191446230562e-02
4.515052758425531998e+00 1.940768891759592218e-02
4.546626553938997617e+00 1.370716166199725805e-02
4.578200349452462348e+00 8.989792557207887391e-03
4.609774144965927967e+00 5.260283979342972316e-03
4.641347940479392697e+00 2.522353583661263166e-03
4.672921735992858316e+00 7.787305987243531291e-04
4.704495531506323047e+00 3.115310584367314561e-05
4.736069327019788666e+00 2.803663065220618478e-04
4.767643122533254285e+00 1.526121779621192331e-03
4.799216918046719016e+00 3.767177728993265085e-03
4.830790713560184635e+00 7.001300221330386542e-03
4.862364509073649366e+00 1.122526541299739833e-02
4.893938304587114985e+00 1.643486276363004261e-02
4.925512100100580604e+00 2.262489923329280561e-02
4.957085895614045334e+00 2.978920445901367398e-02
4.988659691127510953e+00 3.792063690553715283e-02
5.020233486640975684e+00 4.701109098416056398e-02
5.051807282154441303e+00 5.705150513256296296e-02
5.083381077667906922e+00 6.803187084756523451e-02
5.114954873181371653e+00 7.994124266182545124e-02
5.146528668694837272e+00 9.276774905451867781e-02
5.178102464208302003e+00 1.064986042851256975e-01
5.209676259721767622e+00 1.211201211385396492e-01
5.241250055235233241e+00 1.366177245687767439e-01
5.272823850748697971e+00 1.529759662277010435e-01
5.304397646262163590e+00 1.701785398642742253e-01
5.335971441775628321e+00 1.882082975789789447e-01
5.367545237289093940e+00 2.070472669172214175e-01
5.399119032802559559e+00 2.266766687846610839e-01
5.430692828316024290e+00 2.470769361666227404e-01
5.462266623829489909e+00 2.682277336329232931e-01
5.493840419342954640e+00 2.901079776086668005e-01
5.525414214856420259e+00 3.126958573908157346e-01
5.556988010369884989e+00 3.359688568895683458e-01
5.588561805883350608e+00 3.599037770728926722e-01
5.620135601396816227e+00 3.844767590918209965e-01
5.651709396910280958e+00 4.096633080634712876e-01
5.683283192423746577e+00 4.354383174880819274e-01
5.714856987937211308e+00 4.617760942757109799e-01
5.746430783450676927e+00 4.886503843576732731e-01
5.778004578964142546e+00 5.160343988571614027e-01
5.809578374477607277e+00 5.439008407929837308e-01
5.841152169991072896e+00 5.722219322897904581e-01
5.872725965504537626e+00 6.009694422676585823e-01
5.904299761018003245e+00 6.301147145834531393e-01
5.935873556531468864e+00 6.596286965958874093e-01
5.967447352044933595e+00 6.894819681258308464e-01
5.999021147558399214e+00 7.196447707829856100e-01
6.030594943071863945e+00 7.500870376296912001e-01
6.062168738585329564e+00 7.807784231523084983e-01
6.093742534098795183e+00 8.116883335102823560e-01
6.125316329612259914e+00 8.427859570327489447e-01
6.156890125125725532e+00 8.740402949322825243e-01
6.188463920639190263e+00 9.054201922051545726e-01
6.220037716152655882e+00 9.368943686873262289e-01
6.251611511666121501e+00 9.684314502351897280e-01
6.283185307179586232e+00 9.999999999999997780e-01
你的导数是错误的，这个级别应该是
(gofx[i]-gofx[i-1]) / (x[i]-x[i-1])
但这只是导数的一阶近似，任务要求二阶误差。也就是说，对于 x[i] 处的导数，您必须通过点 x[i-1], x[i], x[i+1] 及其值
进行插值多项式
g[x[i]] + g[x[i],x[i+1]] * (x-x[i]) + g[x[i],x[i-1],x[i+1]] * (x-x[i])*(x-x[i-1])
并计算它在 x=x[i] 处的导数。或者，从泰勒展开你知道
(gofx[i]-gofx[i-1]) / (x[i]-x[i-1]) = g'(x[i]) - 0.5*g''(x[i])*(x[i]-x[i-1])+...
(gofx[i+1]-gofx[i]) / (x[i+1]-x[i]) = g'(x[i]) + 0.5*g''(x[i])*(x[i+1]-x[i])+...
结合使用 g''(x[i])。
所以如果
dx = x[1:]-x[:-1]
dg = g[1:]-g[:-1]
是单差，那么二阶误差的一阶导数是
dg_dx = dg/dx
diff_g = ( dx[:-1]*(dg_dx[1:]) + dx[1:]*(dg_dx[:-1]) ) / (dx[1:]+dx[:-1])
这样写是为了让凸组合的性质变得明显。
求积分，累加梯形求积就够了
sum( 0.5*(g[:-1]+g[1:])*(x[1:]-x[:-1]) )
如果你想要反导数作为函数，请使用累加和(table)。
您可能想直接将数据提取到 numpy 数组中，pandas 中应该有执行此操作的函数。
我总共得到了短剧本
x,g = np.loadtxt('so65602720.data').T
%matplotlib inline
plt.figure(figsize=(10,3))
plt.subplot(131)
plt.plot(x,g,x,np.sin(x)+1); plt.legend(["table g values", "+sin(x)$"]); plt.grid();
dx = x[1:]-x[:-1]
dg = g[1:]-g[:-1]
dg_dx = dg/dx
diff_g = ( dx[:-1]*(dg_dx[1:]) + dx[1:]*(dg_dx[:-1]) ) / (dx[1:]+dx[:-1])
plt.subplot(132)
plt.plot(x,g,x[1:-1],diff_g); plt.legend(["g", "g'"]); plt.grid();
int_g = np.cumsum(0.5*(g[1:]+g[:-1])*(x[1:]-x[:-1]))
plt.subplot(133)
plt.plot(x[1:],int_g,x,x); plt.legend(["integral of g","diagonal"]); plt.grid();
plt.tight_layout(); plt.show()
导致剧情合集
首先显示数据确实是函数 g(x)=1+sin(x)，导数正确地看起来像余弦，积分是 x+1-cos(x)。
我将此作为占位符来获取建议。
#Try1
import pandas as pd
from sympy import *
dataFrame = pd.read_csv('/data_2.txt', sep="\s+", names=['xValue','gOfXValue'],float_precision='round_trip', nrows=200)
pd.set_option('display.max_rows', 200)
xValues = dataFrame.xValue
gofXValues = dataFrame.gOfXValue
#Compute second order estimate for g'(x)
firstDiffArray = []
def calculate_ALL_Divided_Differences():
for index, row in dataFrame.iterrows():
if index > 0:
xNow = row['xValue']
gNow = row['gOfXValue']
difference = (gofXValues[index] - gofXValues[index - 1]) / (xValues[index] - xValues[index -1])
firstDiffArray.append(difference)
calculate_ALL_Divided_Differences()
#Plot x and g'(x) that I've found
xValuesLessOne = xValues[:-1]
fig = plt.figure(figsize = (6,12))
ax1 = fig.add_subplot(311)
ax1.plot(xValuesLessOne, firstDiffArray)
ax1.set_title('x vs g Derivated')
ax1.set_xlabel('xValue')
ax1.set_ylabel('g Derivated')
fig.tight_layout()
plt.show()
firstDiffArray.insert(0,0) #insert the first row as 0 to have same 200 rows data shape
dataFrame['derivative_For_gOfX'] = firstDiffArray #create column for the derivative of gOfX
###Find integral g(x)dx
def findIntegral():
for index, row in dataFrame.iterrows():
if index < 199:
xNow = xValues[index]
xNowPlus_1 = xValues[index + 1]
gNow = gofXValues[index]
gNowPlus_1 = gofXValues[index + 1]
intg = (gNowPlus_1 - gNow) * (xNowPlus_1 - xNow)
integralPoints.append(intg)
intgTrapez = 0.5*(gofXValues[i+1] + gofXValues[i]) * (xValues[i+1] - xValues[i])
trapezIntegralPoints.append(intgTrapez)
#integral found numerically
integralFound = findIntegral()
sumIntegralPoints = sum(integralPoints)
sumtrapezIntegralPoints = sum(trapezIntegralPoints)
print('IntegralFound', sumIntegralPoints)
print('TrapezIntegral', sumtrapezIntegralPoints)
IntegralFound -2.7538735181131813e-17
TrapezIntegral 8.328014461189756
xValue gOfXValue derivative_For_gOfX
0 0.000000 1.000000 0.000000
1 0.031574 1.031569 0.999834
2 0.063148 1.063106 0.998837
3 0.094721 1.094580 0.996845
4 0.126295 1.125960 0.993859
5 0.157869 1.157214 0.989882
6 0.189443 1.188312 0.984919
7 0.221017 1.219222 0.978974
8 0.252590 1.249913 0.972052
9 0.284164 1.280355 0.964162
10 0.315738 1.310518 0.955311
11 0.347312 1.340371 0.945508
12 0.378886 1.369885 0.934762
13 0.410459 1.399031 0.923084
14 0.442033 1.427778 0.910486
15 0.473607 1.456099 0.896981
16 0.505181 1.483966 0.882581
17 0.536755 1.511350 0.867302
18 0.568328 1.538224 0.851158
19 0.599902 1.564562 0.834166
20 0.631476 1.590337 0.816342
21 0.663050 1.615523 0.797704
22 0.694624 1.640096 0.778271
23 0.726197 1.664031 0.758063
24 0.757771 1.687304 0.737099
25 0.789345 1.709892 0.715400
26 0.820919 1.731772 0.692987
27 0.852492 1.752923 0.669885
28 0.884066 1.773323 0.646114
29 0.915640 1.792953 0.621699
30 0.947214 1.811792 0.596665
31 0.978788 1.829821 0.571035
32 1.010361 1.847024 0.544837
33 1.041935 1.863382 0.518096
34 1.073509 1.878880 0.490838
35 1.105083 1.893501 0.463090
36 1.136657 1.907232 0.434881
37 1.168230 1.920059 0.406239
38 1.199804 1.931968 0.377192
39 1.231378 1.942948 0.347768
40 1.262952 1.952989 0.317998
41 1.294526 1.962079 0.287911
42 1.326099 1.970211 0.257537
43 1.357673 1.977375 0.226907
44 1.389247 1.983565 0.196050
45 1.420821 1.988775 0.164998
46 1.452395 1.992999 0.133781
47 1.483968 1.996233 0.102431
48 1.515542 1.998474 0.070978
49 1.547116 1.999720 0.039455
50 1.578690 1.999969 0.007893
51 1.610264 1.999221 -0.023677
52 1.641837 1.997478 -0.055224
53 1.673411 1.994740 -0.086715
54 1.704985 1.991010 -0.118120
55 1.736559 1.986293 -0.149408
56 1.768133 1.980592 -0.180546
57 1.799706 1.973914 -0.211505
58 1.831280 1.966265 -0.242252
59 1.862854 1.957653 -0.272758
60 1.894428 1.948087 -0.302993
61 1.926002 1.937575 -0.332925
62 1.957575 1.926129 -0.362525
63 1.989149 1.913759 -0.391764
64 2.020723 1.900479 -0.420613
65 2.052297 1.886301 -0.449042
66 2.083871 1.871240 -0.477023
67 2.115444 1.855310 -0.504529
68 2.147018 1.838527 -0.531533
69 2.178592 1.820909 -0.558006
70 2.210166 1.802472 -0.583923
71 2.241739 1.783236 -0.609258
72 2.273313 1.763218 -0.633986
73 2.304887 1.742440 -0.658081
74 2.336461 1.720922 -0.681521
75 2.368035 1.698685 -0.704281
76 2.399608 1.675752 -0.726340
77 2.431182 1.652145 -0.747674
78 2.462756 1.627888 -0.768263
79 2.494330 1.603005 -0.788086
80 2.525904 1.577521 -0.807124
81 2.557477 1.551462 -0.825357
82 2.589051 1.524852 -0.842767
83 2.620625 1.497720 -0.859337
84 2.652199 1.470091 -0.875051
85 2.683773 1.441994 -0.889892
86 2.715346 1.413456 -0.903846
87 2.746920 1.384506 -0.916900
88 2.778494 1.355173 -0.929039
89 2.810068 1.325485 -0.940252
90 2.841642 1.295473 -0.950528
91 2.873215 1.265167 -0.959856
92 2.904789 1.234597 -0.968228
93 2.936363 1.203792 -0.975635
94 2.967937 1.172784 -0.982069
95 2.999511 1.141605 -0.987524
96 3.031084 1.110283 -0.991994
97 3.062658 1.078853 -0.995476
98 3.094232 1.047343 -0.997965
99 3.125806 1.015786 -0.999460
100 3.157380 0.984214 -0.999958
101 3.188953 0.952657 -0.999460
102 3.220527 0.921147 -0.997965
103 3.252101 0.889717 -0.995476
104 3.283675 0.858395 -0.991994
105 3.315249 0.827216 -0.987524
106 3.346822 0.796208 -0.982069
107 3.378396 0.765403 -0.975635
108 3.409970 0.734833 -0.968228
109 3.441544 0.704527 -0.959856
110 3.473118 0.674515 -0.950528
111 3.504691 0.644827 -0.940252
112 3.536265 0.615494 -0.929039
113 3.567839 0.586544 -0.916900
114 3.599413 0.558006 -0.903846
115 3.630986 0.529909 -0.889892
116 3.662560 0.502280 -0.875051
117 3.694134 0.475148 -0.859337
118 3.725708 0.448538 -0.842767
119 3.757282 0.422479 -0.825357
120 3.788855 0.396995 -0.807124
121 3.820429 0.372112 -0.788086
122 3.852003 0.347855 -0.768263
123 3.883577 0.324248 -0.747674
124 3.915151 0.301315 -0.726340
125 3.946724 0.279078 -0.704281
126 3.978298 0.257560 -0.681521
127 4.009872 0.236782 -0.658081
128 4.041446 0.216764 -0.633986
129 4.073020 0.197528 -0.609258
130 4.104593 0.179091 -0.583923
131 4.136167 0.161473 -0.558006
132 4.167741 0.144690 -0.531533
133 4.199315 0.128760 -0.504529
134 4.230889 0.113699 -0.477023
135 4.262462 0.099521 -0.449042
136 4.294036 0.086241 -0.420613
137 4.325610 0.073871 -0.391764
138 4.357184 0.062425 -0.362525
139 4.388758 0.051913 -0.332925
140 4.420331 0.042347 -0.302993
141 4.451905 0.033735 -0.272758
142 4.483479 0.026086 -0.242252
143 4.515053 0.019408 -0.211505
144 4.546627 0.013707 -0.180546
145 4.578200 0.008990 -0.149408
146 4.609774 0.005260 -0.118120
147 4.641348 0.002522 -0.086715
148 4.672922 0.000779 -0.055224
149 4.704496 0.000031 -0.023677
150 4.736069 0.000280 0.007893
151 4.767643 0.001526 0.039455
152 4.799217 0.003767 0.070978
153 4.830791 0.007001 0.102431
154 4.862365 0.011225 0.133781
155 4.893938 0.016435 0.164998
156 4.925512 0.022625 0.196050
157 4.957086 0.029789 0.226907
158 4.988660 0.037921 0.257537
159 5.020233 0.047011 0.287911
160 5.051807 0.057052 0.317998
161 5.083381 0.068032 0.347768
162 5.114955 0.079941 0.377192
163 5.146529 0.092768 0.406239
164 5.178102 0.106499 0.434881
165 5.209676 0.121120 0.463090
166 5.241250 0.136618 0.490838
167 5.272824 0.152976 0.518096
168 5.304398 0.170179 0.544837
169 5.335971 0.188208 0.571035
170 5.367545 0.207047 0.596665
171 5.399119 0.226677 0.621699
172 5.430693 0.247077 0.646114
173 5.462267 0.268228 0.669885
174 5.493840 0.290108 0.692987
175 5.525414 0.312696 0.715400
176 5.556988 0.335969 0.737099
177 5.588562 0.359904 0.758063
178 5.620136 0.384477 0.778271
179 5.651709 0.409663 0.797704
180 5.683283 0.435438 0.816342
181 5.714857 0.461776 0.834166
182 5.746431 0.488650 0.851158
183 5.778005 0.516034 0.867302
184 5.809578 0.543901 0.882581
185 5.841152 0.572222 0.896981
186 5.872726 0.600969 0.910486
187 5.904300 0.630115 0.923084
188 5.935874 0.659629 0.934762
189 5.967447 0.689482 0.945508
190 5.999021 0.719645 0.955311
191 6.030595 0.750087 0.964162
192 6.062169 0.780778 0.972052
193 6.093743 0.811688 0.978974
194 6.125316 0.842786 0.984919
195 6.156890 0.874040 0.989882
196 6.188464 0.905420 0.993859
197 6.220038 0.936894 0.996845
198 6.251612 0.968431 0.998837
199 6.283185 1.000000 0.999834
如何在 python3 中对下面列出的提供的数据集执行以下操作？
问题
知道 data.txt 有 2 列：
xValues，其中 ≤ ≤ ，并且是一些常量
gOfXValues
1计算 ′()
的二阶估计
2计算 $g'(x)$ 和 $\int_a^b g(x)dx.$
的二阶估计值
通常，我们不知道从随机数据样本中给出的值是均匀分离的。
3绘制 () 和 ′() ，打印积分。
4根据图表，你认为()是什么函数？
5通过定性比较你假设的 () 的精确导数和积分与之前获得的数值结果来验证这一点。
到目前为止我做了什么
import pandas as pd
dataFrame = pd.read_csv('/Users/Files/data.txt', sep="\s+", names=['xValue','gOfXValue'])
dataFrame.info
dataFrame.head()
dgdxArray = []
gOfXValues = []
#iterate all observations and extract the columns as the independent and dependent variable
for index, row in dataFrame.iterrows():
xValue = row['xValue']
gOfXValue = row['gOfXValue']
gOfXValues.append(gOfXValue)
if index > 0:
h = 0.05
difference = gOfXValue - gOfXValues[index-1] #check the difference between Current vs Previous value
dgdx = difference / h #get the Derivative
dgdxArray.append(dgdx) #add the derivative to an array so as to plot it
dgdxArray.insert(0,0.5) #hard code values
#plot the initial values provided
fig = plt.figure(figsize = (6,12))
ax1 = fig.add_subplot(311)
ax1.plot(dataFrame['xValue'], dataFrame['gOfXValue'])
ax1.set_title('Plot initial values x, g(x)')
ax1.set_xlabel('xValue')
ax1.set_ylabel('gOfXValue')
#Plot X value and the derivative on y axis
ax2 = fig.add_subplot(312)
ax2.plot(dataFrame['xValue'], dgdxArray)
ax2.set_title('Plot x, derivativeOfGOfx')
ax2.set_xlabel('xValue')
ax2.set_ylabel('gOfXValue')
fig.tight_layout()
plt.show()
EDIT_1 鉴于我只能访问 xValue 和 gOfXvalue，如何找到原始函数定义？
Edit_2 基于对 D Stanley 的评论
#we know that in calculus g(x) = sin(x), g'(x) = cos(x), g(x)dx = - cos(x) + C
#Therefore:
#calculate sin(x) and compare it to g(x) provided
#calculate f'(x) and compare it to g(x) provided
#calculate integral of g(x)dx and compare it to g(x) provided
constant = 2 #choose a random constant
#calculate the sin,cos and integral for existing x value considering that the original function is sin
sinXfound = np.sin(dataFrame.xValue)
cosXfound = np.cos(dataFrame.xValue)
intXfound = - np.cos(dataFrame.xValue) + constant
#create new columns in the original df with values calculate above
dataFrame['sinXfound'] = sinXfound
dataFrame['cosXfound'] = cosXfound
dataFrame['intXfound'] = intXfound
#find what is the difference between sin,cos newly found and the original xValue provided in the request
differenceSinXfound = sinXfound - dataFrame['gOfXValue']
differenceCosXfound = cosXfound - dataFrame['gOfXValue']
differenceIntXfound = intXfound - dataFrame['gOfXValue']
#add columns to df
dataFrame['differenceSinXfound'] = differenceSinXfound
dataFrame['differenceCosXfound'] = differenceCosXfound
dataFrame['differenceIntXfound'] = differenceCosXfound
print(dataFrame)
Edit_3 基于 Lutz 的回答
xValues = dataFrame.xValue
gofXValues = dataFrame.gOfXValue
firstDiffArray = []
def calculate_ALL_Divided_Differences():
for index, row in dataFrame.iterrows():
if index > 0:
xNow = row['xValue']
gNow = row['gOfXValue']
difference = (gofXValues[indexNow] - gofXValues[indexNow - 1]) / (xValues[indexNow] - xValues[indexNow -1])
firstDiffArray.append(difference)
firstDividedDifference = (gofXValues[1] - gofXValues[0]) / (xValues[1] - xValues[0])
x0 = xValues[0]
gOfXZero = gofXValues[0]
#Apply Newton's divided difference interpolation formula
for index, row in dataFrame.iterrows():
if index > 0:
xNow = row['xValue']
gNow = row['gOfXValue']
x_Minus_x0 = xNow - xValues[0]
x_Minus_x1 = xNow - xValues[1]
#Newton's divided difference interpolation formula is
#f(x) = y0+(x-x0) f [x0,x1]+ (x-x0) * (x-x1) * f [x0,x1,x2]
divided_Difference_Interpolation = gOfXZero + (xNow - x0) * firstDividedDifference + x_Minus_x0 * x_Minus_x1
数据集
0.000000000000000000e+00 1.000000000000000000e+00
3.157379551346525814e-02 1.031568549764810605e+00
6.314759102693051629e-02 1.063105631312673660e+00
9.472138654039577443e-02 1.094579807794844983e+00
1.262951820538610326e-01 1.125959705067717476e+00
1.578689775673262907e-01 1.157214042967250833e+00
1.894427730807915489e-01 1.188311666489717755e+00
2.210165685942568070e-01 1.219221576847691280e+00
2.525903641077220652e-01 1.249912962370308467e+00
2.841641596211873511e-01 1.280355229217014390e+00
3.157379551346525814e-01 1.310518031874168710e+00
3.473117506481178118e-01 1.340371303404112702e+00
3.788855461615830977e-01 1.369885285416546861e+00
4.104593416750483836e-01 1.399030557732340974e+00
4.420331371885136140e-01 1.427778067710209653e+00
4.736069327019788444e-01 1.456099159207016047e+00
5.051807282154441303e-01 1.483965601142838819e+00
5.367545237289094162e-01 1.511349615642326727e+00
5.683283192423747021e-01 1.538223905724288576e+00
5.999021147558398770e-01 1.564561682511917962e+00
6.314759102693051629e-01 1.590336691936528268e+00
6.630497057827704488e-01 1.615523240908179226e+00
6.946235012962356237e-01 1.640096222927107217e+00
7.261972968097009096e-01 1.664031143110431099e+00
7.577710923231661955e-01 1.687304142609184154e+00
7.893448878366314814e-01 1.709892022391332755e+00
8.209186833500967673e-01 1.731772266367076707e+00
8.524924788635619421e-01 1.752923063833377038e+00
8.840662743770272280e-01 1.773323331215339138e+00
9.156400698904925139e-01 1.792952733082778582e+00
9.472138654039576888e-01 1.811791702421020833e+00
9.787876609174229747e-01 1.829821460135725886e+00
1.010361456430888261e+00 1.847024033772298957e+00
1.041935251944353436e+00 1.863382275431223256e+00
1.073509047457818832e+00 1.878879878861460462e+00
1.105082842971284007e+00 1.893501395714874080e+00
1.136656638484749404e+00 1.907232250945481322e+00
1.168230433998214579e+00 1.920058757338174438e+00
1.199804229511679754e+00 1.931968129152434877e+00
1.231378025025145151e+00 1.942948494867437148e+00
1.262951820538610326e+00 1.952988909015839214e+00
1.294525616052075501e+00 1.962079363094462847e+00
1.326099411565540898e+00 1.970210795540986215e+00
1.357673207079006072e+00 1.977375100766707305e+00
1.389247002592471247e+00 1.983565137236369846e+00
1.420820798105936644e+00 1.988774734587002602e+00
1.452394593619401819e+00 1.992998699778669724e+00
1.483968389132867216e+00 1.996232822271006846e+00
1.515542184646332391e+00 1.998473878220378808e+00
1.547115980159797566e+00 1.999719633693477938e+00
1.578689775673262963e+00 1.999968846894156327e+00
1.610263571186728138e+00 1.999221269401275647e+00
1.641837366700193535e+00 1.997477646416338626e+00
1.673411162213658709e+00 1.994739716020657028e+00
1.704984957727123884e+00 1.991010207442792002e+00
1.736558753240589281e+00 1.986292838338002742e+00
1.768132548754054456e+00 1.980592311082403967e+00
1.799706344267519631e+00 1.973914308085537694e+00
1.831280139780985028e+00 1.966265486126021811e+00
1.862853935294450203e+00 1.957653469715929573e+00
1.894427730807915378e+00 1.948086843500509424e+00
1.926001526321380775e+00 1.937575143700825064e+00
1.957575321834845949e+00 1.926128848607841171e+00
1.989149117348311346e+00 1.913759368137436745e+00
2.020722912861776521e+00 1.900479032456751760e+00
2.052296708375241696e+00 1.886301079693208704e+00
2.083870503888706871e+00 1.871239642738459663e+00
2.115444299402172490e+00 1.855309735160411755e+00
2.147018094915637665e+00 1.838527236237377238e+00
2.178591890429102840e+00 1.820908875129262583e+00
2.210165685942568015e+00 1.802472214201578105e+00
2.241739481456033189e+00 1.783235631518890418e+00
2.273313276969498808e+00 1.763218302525168424e+00
2.304887072482963983e+00 1.742440180929283100e+00
2.336460867996429158e+00 1.720921978814716535e+00
2.368034663509894333e+00 1.698685145993306556e+00
2.399608459023359508e+00 1.675751848623608709e+00
2.431182254536824683e+00 1.652144947115186779e+00
2.462756050050290302e+00 1.627887973340858441e+00
2.494329845563755477e+00 1.603005107179614530e+00
2.525903641077220652e+00 1.577521152413588590e+00
2.557477436590685826e+00 1.551461512003107668e+00
2.589051232104151001e+00 1.524852162764468444e+00
2.620625027617616620e+00 1.497719629475682934e+00
2.652198823131081795e+00 1.470090958436002904e+00
2.683772618644546970e+00 1.441993690505579018e+00
2.715346414158012145e+00 1.413455833652134119e+00
2.746920209671477320e+00 1.384505835032010967e+00
2.778494005184942495e+00 1.355172552633428618e+00
2.810067800698408114e+00 1.325485226510211501e+00
2.841641596211873289e+00 1.295473449634670038e+00
2.873215391725338463e+00 1.265167138398678670e+00
2.904789187238803638e+00 1.234596502792368877e+00
2.936362982752268813e+00 1.203792016290152311e+00
2.967936778265734432e+00 1.172784385474099356e+00
2.999510573779199607e+00 1.141604519424951558e+00
3.031084369292664782e+00 1.110283498911275091e+00
3.062658164806129957e+00 1.078852545407476660e+00
3.094231960319595132e+00 1.047342989971558280e+00
3.125805755833060751e+00 1.015786242013636764e+00
3.157379551346525925e+00 9.842137579863630137e-01
3.188953346859991100e+00 9.526570100284420528e-01
3.220527142373456275e+00 9.211474545925236734e-01
3.252100937886921450e+00 8.897165010887251313e-01
3.283674733400387069e+00 8.583954805750482198e-01
3.315248528913852244e+00 8.272156145259004223e-01
3.346822324427317419e+00 7.962079837098479107e-01
3.378396119940782594e+00 7.654034972076313448e-01
3.409969915454247769e+00 7.348328616013215520e-01
3.441543710967712943e+00 7.045265503653301842e-01
3.473117506481178562e+00 6.745147734897881664e-01
3.504691301994643737e+00 6.448274473665716044e-01
3.536265097508108912e+00 6.154941649679892546e-01
3.567838893021574087e+00 5.865441663478661027e-01
3.599412688535039262e+00 5.580063094944210933e-01
3.630986484048504881e+00 5.299090415639968743e-01
3.662560279561970056e+00 5.022803705243168437e-01
3.694134075075435231e+00 4.751478372355316671e-01
3.725707870588900406e+00 4.485384879968926652e-01
3.757281666102365580e+00 4.224788475864115211e-01
3.788855461615830755e+00 3.969948928203856919e-01
3.820429257129296374e+00 3.721120266591415593e-01
3.852003052642761549e+00 3.478550528848133316e-01
3.883576848156226724e+00 3.242481513763912915e-01
3.915150643669691899e+00 3.013148540066936665e-01
3.946724439183157074e+00 2.790780211852837978e-01
3.978298234696622693e+00 2.575598190707169000e-01
4.009872030210087424e+00 2.367816974748317982e-01
4.041445825723553043e+00 2.167643684811096927e-01
4.073019621237018661e+00 1.975277857984218954e-01
4.104593416750483392e+00 1.790911248707376391e-01
4.136167212263949011e+00 1.614727637626225398e-01
4.167741007777413742e+00 1.446902648395883562e-01
4.199314803290879361e+00 1.287603572615404479e-01
4.230888598804344980e+00 1.136989203067911847e-01
4.262462394317809711e+00 9.952096754324846195e-02
4.294036189831275330e+00 8.624063186256325508e-02
4.325609985344740060e+00 7.387115139215894022e-02
4.357183780858205679e+00 6.242485629917493561e-02
4.388757576371671298e+00 5.191315649949035382e-02
4.420331371885136029e+00 4.234653028407053821e-02
4.451905167398601648e+00 3.373451387397807810e-02
4.483478962912066379e+00 2.608569191446230562e-02
4.515052758425531998e+00 1.940768891759592218e-02
4.546626553938997617e+00 1.370716166199725805e-02
4.578200349452462348e+00 8.989792557207887391e-03
4.609774144965927967e+00 5.260283979342972316e-03
4.641347940479392697e+00 2.522353583661263166e-03
4.672921735992858316e+00 7.787305987243531291e-04
4.704495531506323047e+00 3.115310584367314561e-05
4.736069327019788666e+00 2.803663065220618478e-04
4.767643122533254285e+00 1.526121779621192331e-03
4.799216918046719016e+00 3.767177728993265085e-03
4.830790713560184635e+00 7.001300221330386542e-03
4.862364509073649366e+00 1.122526541299739833e-02
4.893938304587114985e+00 1.643486276363004261e-02
4.925512100100580604e+00 2.262489923329280561e-02
4.957085895614045334e+00 2.978920445901367398e-02
4.988659691127510953e+00 3.792063690553715283e-02
5.020233486640975684e+00 4.701109098416056398e-02
5.051807282154441303e+00 5.705150513256296296e-02
5.083381077667906922e+00 6.803187084756523451e-02
5.114954873181371653e+00 7.994124266182545124e-02
5.146528668694837272e+00 9.276774905451867781e-02
5.178102464208302003e+00 1.064986042851256975e-01
5.209676259721767622e+00 1.211201211385396492e-01
5.241250055235233241e+00 1.366177245687767439e-01
5.272823850748697971e+00 1.529759662277010435e-01
5.304397646262163590e+00 1.701785398642742253e-01
5.335971441775628321e+00 1.882082975789789447e-01
5.367545237289093940e+00 2.070472669172214175e-01
5.399119032802559559e+00 2.266766687846610839e-01
5.430692828316024290e+00 2.470769361666227404e-01
5.462266623829489909e+00 2.682277336329232931e-01
5.493840419342954640e+00 2.901079776086668005e-01
5.525414214856420259e+00 3.126958573908157346e-01
5.556988010369884989e+00 3.359688568895683458e-01
5.588561805883350608e+00 3.599037770728926722e-01
5.620135601396816227e+00 3.844767590918209965e-01
5.651709396910280958e+00 4.096633080634712876e-01
5.683283192423746577e+00 4.354383174880819274e-01
5.714856987937211308e+00 4.617760942757109799e-01
5.746430783450676927e+00 4.886503843576732731e-01
5.778004578964142546e+00 5.160343988571614027e-01
5.809578374477607277e+00 5.439008407929837308e-01
5.841152169991072896e+00 5.722219322897904581e-01
5.872725965504537626e+00 6.009694422676585823e-01
5.904299761018003245e+00 6.301147145834531393e-01
5.935873556531468864e+00 6.596286965958874093e-01
5.967447352044933595e+00 6.894819681258308464e-01
5.999021147558399214e+00 7.196447707829856100e-01
6.030594943071863945e+00 7.500870376296912001e-01
6.062168738585329564e+00 7.807784231523084983e-01
6.093742534098795183e+00 8.116883335102823560e-01
6.125316329612259914e+00 8.427859570327489447e-01
6.156890125125725532e+00 8.740402949322825243e-01
6.188463920639190263e+00 9.054201922051545726e-01
6.220037716152655882e+00 9.368943686873262289e-01
6.251611511666121501e+00 9.684314502351897280e-01
6.283185307179586232e+00 9.999999999999997780e-01
你的导数是错误的，这个级别应该是
(gofx[i]-gofx[i-1]) / (x[i]-x[i-1])
但这只是导数的一阶近似，任务要求二阶误差。也就是说，对于 x[i] 处的导数，您必须通过点 x[i-1], x[i], x[i+1] 及其值
g[x[i]] + g[x[i],x[i+1]] * (x-x[i]) + g[x[i],x[i-1],x[i+1]] * (x-x[i])*(x-x[i-1])
并计算它在 x=x[i] 处的导数。或者，从泰勒展开你知道
(gofx[i]-gofx[i-1]) / (x[i]-x[i-1]) = g'(x[i]) - 0.5*g''(x[i])*(x[i]-x[i-1])+...
(gofx[i+1]-gofx[i]) / (x[i+1]-x[i]) = g'(x[i]) + 0.5*g''(x[i])*(x[i+1]-x[i])+...
结合使用 g''(x[i])。
所以如果
dx = x[1:]-x[:-1]
dg = g[1:]-g[:-1]
是单差，那么二阶误差的一阶导数是
dg_dx = dg/dx
diff_g = ( dx[:-1]*(dg_dx[1:]) + dx[1:]*(dg_dx[:-1]) ) / (dx[1:]+dx[:-1])
这样写是为了让凸组合的性质变得明显。
求积分，累加梯形求积就够了
sum( 0.5*(g[:-1]+g[1:])*(x[1:]-x[:-1]) )
如果你想要反导数作为函数，请使用累加和(table)。
您可能想直接将数据提取到 numpy 数组中，pandas 中应该有执行此操作的函数。
我总共得到了短剧本
x,g = np.loadtxt('so65602720.data').T
%matplotlib inline
plt.figure(figsize=(10,3))
plt.subplot(131)
plt.plot(x,g,x,np.sin(x)+1); plt.legend(["table g values", "+sin(x)$"]); plt.grid();
dx = x[1:]-x[:-1]
dg = g[1:]-g[:-1]
dg_dx = dg/dx
diff_g = ( dx[:-1]*(dg_dx[1:]) + dx[1:]*(dg_dx[:-1]) ) / (dx[1:]+dx[:-1])
plt.subplot(132)
plt.plot(x,g,x[1:-1],diff_g); plt.legend(["g", "g'"]); plt.grid();
int_g = np.cumsum(0.5*(g[1:]+g[:-1])*(x[1:]-x[:-1]))
plt.subplot(133)
plt.plot(x[1:],int_g,x,x); plt.legend(["integral of g","diagonal"]); plt.grid();
plt.tight_layout(); plt.show()
导致剧情合集
首先显示数据确实是函数 g(x)=1+sin(x)，导数正确地看起来像余弦，积分是 x+1-cos(x)。
我将此作为占位符来获取建议。
#Try1
import pandas as pd
from sympy import *
dataFrame = pd.read_csv('/data_2.txt', sep="\s+", names=['xValue','gOfXValue'],float_precision='round_trip', nrows=200)
pd.set_option('display.max_rows', 200)
xValues = dataFrame.xValue
gofXValues = dataFrame.gOfXValue
#Compute second order estimate for g'(x)
firstDiffArray = []
def calculate_ALL_Divided_Differences():
for index, row in dataFrame.iterrows():
if index > 0:
xNow = row['xValue']
gNow = row['gOfXValue']
difference = (gofXValues[index] - gofXValues[index - 1]) / (xValues[index] - xValues[index -1])
firstDiffArray.append(difference)
calculate_ALL_Divided_Differences()
#Plot x and g'(x) that I've found
xValuesLessOne = xValues[:-1]
fig = plt.figure(figsize = (6,12))
ax1 = fig.add_subplot(311)
ax1.plot(xValuesLessOne, firstDiffArray)
ax1.set_title('x vs g Derivated')
ax1.set_xlabel('xValue')
ax1.set_ylabel('g Derivated')
fig.tight_layout()
plt.show()
firstDiffArray.insert(0,0) #insert the first row as 0 to have same 200 rows data shape
dataFrame['derivative_For_gOfX'] = firstDiffArray #create column for the derivative of gOfX
###Find integral g(x)dx
def findIntegral():
for index, row in dataFrame.iterrows():
if index < 199:
xNow = xValues[index]
xNowPlus_1 = xValues[index + 1]
gNow = gofXValues[index]
gNowPlus_1 = gofXValues[index + 1]
intg = (gNowPlus_1 - gNow) * (xNowPlus_1 - xNow)
integralPoints.append(intg)
intgTrapez = 0.5*(gofXValues[i+1] + gofXValues[i]) * (xValues[i+1] - xValues[i])
trapezIntegralPoints.append(intgTrapez)
#integral found numerically
integralFound = findIntegral()
sumIntegralPoints = sum(integralPoints)
sumtrapezIntegralPoints = sum(trapezIntegralPoints)
print('IntegralFound', sumIntegralPoints)
print('TrapezIntegral', sumtrapezIntegralPoints)
IntegralFound -2.7538735181131813e-17
TrapezIntegral 8.328014461189756
xValue gOfXValue derivative_For_gOfX
0 0.000000 1.000000 0.000000
1 0.031574 1.031569 0.999834
2 0.063148 1.063106 0.998837
3 0.094721 1.094580 0.996845
4 0.126295 1.125960 0.993859
5 0.157869 1.157214 0.989882
6 0.189443 1.188312 0.984919
7 0.221017 1.219222 0.978974
8 0.252590 1.249913 0.972052
9 0.284164 1.280355 0.964162
10 0.315738 1.310518 0.955311
11 0.347312 1.340371 0.945508
12 0.378886 1.369885 0.934762
13 0.410459 1.399031 0.923084
14 0.442033 1.427778 0.910486
15 0.473607 1.456099 0.896981
16 0.505181 1.483966 0.882581
17 0.536755 1.511350 0.867302
18 0.568328 1.538224 0.851158
19 0.599902 1.564562 0.834166
20 0.631476 1.590337 0.816342
21 0.663050 1.615523 0.797704
22 0.694624 1.640096 0.778271
23 0.726197 1.664031 0.758063
24 0.757771 1.687304 0.737099
25 0.789345 1.709892 0.715400
26 0.820919 1.731772 0.692987
27 0.852492 1.752923 0.669885
28 0.884066 1.773323 0.646114
29 0.915640 1.792953 0.621699
30 0.947214 1.811792 0.596665
31 0.978788 1.829821 0.571035
32 1.010361 1.847024 0.544837
33 1.041935 1.863382 0.518096
34 1.073509 1.878880 0.490838
35 1.105083 1.893501 0.463090
36 1.136657 1.907232 0.434881
37 1.168230 1.920059 0.406239
38 1.199804 1.931968 0.377192
39 1.231378 1.942948 0.347768
40 1.262952 1.952989 0.317998
41 1.294526 1.962079 0.287911
42 1.326099 1.970211 0.257537
43 1.357673 1.977375 0.226907
44 1.389247 1.983565 0.196050
45 1.420821 1.988775 0.164998
46 1.452395 1.992999 0.133781
47 1.483968 1.996233 0.102431
48 1.515542 1.998474 0.070978
49 1.547116 1.999720 0.039455
50 1.578690 1.999969 0.007893
51 1.610264 1.999221 -0.023677
52 1.641837 1.997478 -0.055224
53 1.673411 1.994740 -0.086715
54 1.704985 1.991010 -0.118120
55 1.736559 1.986293 -0.149408
56 1.768133 1.980592 -0.180546
57 1.799706 1.973914 -0.211505
58 1.831280 1.966265 -0.242252
59 1.862854 1.957653 -0.272758
60 1.894428 1.948087 -0.302993
61 1.926002 1.937575 -0.332925
62 1.957575 1.926129 -0.362525
63 1.989149 1.913759 -0.391764
64 2.020723 1.900479 -0.420613
65 2.052297 1.886301 -0.449042
66 2.083871 1.871240 -0.477023
67 2.115444 1.855310 -0.504529
68 2.147018 1.838527 -0.531533
69 2.178592 1.820909 -0.558006
70 2.210166 1.802472 -0.583923
71 2.241739 1.783236 -0.609258
72 2.273313 1.763218 -0.633986
73 2.304887 1.742440 -0.658081
74 2.336461 1.720922 -0.681521
75 2.368035 1.698685 -0.704281
76 2.399608 1.675752 -0.726340
77 2.431182 1.652145 -0.747674
78 2.462756 1.627888 -0.768263
79 2.494330 1.603005 -0.788086
80 2.525904 1.577521 -0.807124
81 2.557477 1.551462 -0.825357
82 2.589051 1.524852 -0.842767
83 2.620625 1.497720 -0.859337
84 2.652199 1.470091 -0.875051
85 2.683773 1.441994 -0.889892
86 2.715346 1.413456 -0.903846
87 2.746920 1.384506 -0.916900
88 2.778494 1.355173 -0.929039
89 2.810068 1.325485 -0.940252
90 2.841642 1.295473 -0.950528
91 2.873215 1.265167 -0.959856
92 2.904789 1.234597 -0.968228
93 2.936363 1.203792 -0.975635
94 2.967937 1.172784 -0.982069
95 2.999511 1.141605 -0.987524
96 3.031084 1.110283 -0.991994
97 3.062658 1.078853 -0.995476
98 3.094232 1.047343 -0.997965
99 3.125806 1.015786 -0.999460
100 3.157380 0.984214 -0.999958
101 3.188953 0.952657 -0.999460
102 3.220527 0.921147 -0.997965
103 3.252101 0.889717 -0.995476
104 3.283675 0.858395 -0.991994
105 3.315249 0.827216 -0.987524
106 3.346822 0.796208 -0.982069
107 3.378396 0.765403 -0.975635
108 3.409970 0.734833 -0.968228
109 3.441544 0.704527 -0.959856
110 3.473118 0.674515 -0.950528
111 3.504691 0.644827 -0.940252
112 3.536265 0.615494 -0.929039
113 3.567839 0.586544 -0.916900
114 3.599413 0.558006 -0.903846
115 3.630986 0.529909 -0.889892
116 3.662560 0.502280 -0.875051
117 3.694134 0.475148 -0.859337
118 3.725708 0.448538 -0.842767
119 3.757282 0.422479 -0.825357
120 3.788855 0.396995 -0.807124
121 3.820429 0.372112 -0.788086
122 3.852003 0.347855 -0.768263
123 3.883577 0.324248 -0.747674
124 3.915151 0.301315 -0.726340
125 3.946724 0.279078 -0.704281
126 3.978298 0.257560 -0.681521
127 4.009872 0.236782 -0.658081
128 4.041446 0.216764 -0.633986
129 4.073020 0.197528 -0.609258
130 4.104593 0.179091 -0.583923
131 4.136167 0.161473 -0.558006
132 4.167741 0.144690 -0.531533
133 4.199315 0.128760 -0.504529
134 4.230889 0.113699 -0.477023
135 4.262462 0.099521 -0.449042
136 4.294036 0.086241 -0.420613
137 4.325610 0.073871 -0.391764
138 4.357184 0.062425 -0.362525
139 4.388758 0.051913 -0.332925
140 4.420331 0.042347 -0.302993
141 4.451905 0.033735 -0.272758
142 4.483479 0.026086 -0.242252
143 4.515053 0.019408 -0.211505
144 4.546627 0.013707 -0.180546
145 4.578200 0.008990 -0.149408
146 4.609774 0.005260 -0.118120
147 4.641348 0.002522 -0.086715
148 4.672922 0.000779 -0.055224
149 4.704496 0.000031 -0.023677
150 4.736069 0.000280 0.007893
151 4.767643 0.001526 0.039455
152 4.799217 0.003767 0.070978
153 4.830791 0.007001 0.102431
154 4.862365 0.011225 0.133781
155 4.893938 0.016435 0.164998
156 4.925512 0.022625 0.196050
157 4.957086 0.029789 0.226907
158 4.988660 0.037921 0.257537
159 5.020233 0.047011 0.287911
160 5.051807 0.057052 0.317998
161 5.083381 0.068032 0.347768
162 5.114955 0.079941 0.377192
163 5.146529 0.092768 0.406239
164 5.178102 0.106499 0.434881
165 5.209676 0.121120 0.463090
166 5.241250 0.136618 0.490838
167 5.272824 0.152976 0.518096
168 5.304398 0.170179 0.544837
169 5.335971 0.188208 0.571035
170 5.367545 0.207047 0.596665
171 5.399119 0.226677 0.621699
172 5.430693 0.247077 0.646114
173 5.462267 0.268228 0.669885
174 5.493840 0.290108 0.692987
175 5.525414 0.312696 0.715400
176 5.556988 0.335969 0.737099
177 5.588562 0.359904 0.758063
178 5.620136 0.384477 0.778271
179 5.651709 0.409663 0.797704
180 5.683283 0.435438 0.816342
181 5.714857 0.461776 0.834166
182 5.746431 0.488650 0.851158
183 5.778005 0.516034 0.867302
184 5.809578 0.543901 0.882581
185 5.841152 0.572222 0.896981
186 5.872726 0.600969 0.910486
187 5.904300 0.630115 0.923084
188 5.935874 0.659629 0.934762
189 5.967447 0.689482 0.945508
190 5.999021 0.719645 0.955311
191 6.030595 0.750087 0.964162
192 6.062169 0.780778 0.972052
193 6.093743 0.811688 0.978974
194 6.125316 0.842786 0.984919
195 6.156890 0.874040 0.989882
196 6.188464 0.905420 0.993859
197 6.220038 0.936894 0.996845
198 6.251612 0.968431 0.998837
199 6.283185 1.000000 0.999834