梯度下降法
基本概念
1.微分
在梯度下降法中，我們所要優化的函數必須是一個連續可微的函數，可微，既可微分，意思是在函數的任意定義域上導數存在。如果導數存在且是連續函數，則原函數是連續可微的（可微必可導，可導必連續）。在高等數學中，我們知道函數的導數（近似於函數的微分）可以有以下兩種理解：
1.函數上某一點的切線的斜率就是該點的導數值。
2.函數在某點的導數值能反映出函數在該點的變化率，即導數值越大，函數值變化越快。
下面舉幾個連續可微函數求微分的例子：
下面是多元函數的微分：
2.梯度
研究梯度下降法怎麼能不知道梯度這個概念呢？梯度的本意是一個向量（矢量），表示某一函數在該點處的方向導數沿着該方向取得最大值，即函數在該點處沿着該方向（此梯度的方向）變化最快，變化率最大（爲該梯度的模），這是一個微積分學中的概念。
設二元函數
在平面區域D上具有一階連續偏導數，則對於每一個點P（x，y）都可得出兩個一階偏導數
而兩個偏導數構成的向量則爲該二元函數的梯度向量，記作gradf（x，y）或
例如：求函數u=x²+2y²+3z²+3x-2y在（1，1，2）點處的梯度？
3.學習率
學習率也被稱爲迭代的步長，優化函數的梯度一般是不斷變化的（梯度的方向隨梯度的變化而變化），因此需要一個適當的學習率約束着每次下降的距離不會太多也不會太少。
梯度下降法的一般求解步驟：
1.給定待優化連續可微函數J(Θ)，學習率α以及一組初始值
2.計算待優化函數梯度：gradJ(Θ)
3.更新迭代公式：
4.計算Θ^0+1處函數梯度gradJ(Θ0+1)
5.計算梯度向量的模來判斷算法是否收斂：||gradJ(Θ)||≤ε
6.若收斂，算法停止，否則根據迭代公式繼續迭代
import numpy as np
import matplotlib.pyplot as plt
import matplotlib as mpl
import math
from mpl_toolkits.mplot3d import Axes3D
import warnings
# 解決中文顯示問題
mpl.rcParams['font.sans-serif'] = [u'SimHei']
mpl.rcParams['axes.unicode_minus'] = False
"""
二維原始圖像
1、構建一個函數。
2、隨機生成X1,X2點，根據X1,X2點生成Y點。
3、畫出圖像。
"""
def f2(x1,x2):
return x1**2 + 2*x2**2 - 4*x1 - 2*x1*x2
X1 = np.arange(-4,4,0.2)
X2 = np.arange(-4,4,0.2)
X1, X2 = np.meshgrid(X1, X2) # 生成xv、yv，將X1、X2變成n*m的矩陣，方便後面繪圖
Y = np.array(list(map(lambda t : f2(t[0],t[1]),zip(X1.flatten(),X2.flatten()))))
Y.shape = X1.shape # 1600的Y圖還原成原來的（40,40）
%matplotlib inline
#作圖
fig = plt.figure(facecolor='w')
ax = Axes3D(fig)
ax.plot_surface(X1,X2,Y,rstride=1,cstride=1,cmap=plt.cm.jet)
ax.set_title(u'$ y = x1^2+2x2^2-4x1-2x1x2 $')
plt.show()
"""
對當前二維圖像求最小點¶
1、隨機取一個點（x1,x2）,設定α參數值
2、對這個點分別求關於x1、x2的偏導數
3、重複第二補，設置 y的變化量 小於多少時 不再重複。
"""
# 二維原始圖像
def f2(x, y):
return x**2 + 2*y**2 - 4*x - 2*x*y
## 偏函數
def hx1(x, y):
return 2*x - 4 -2*y
def hx2(x, y):
return 4*y-2*x
x1 = -1
x2 = -1
alpha = 0.05
#保存梯度下降經過的點
GD_X1 = [x1]
GD_X2 = [x2]
GD_Y = [f2(x1,x2)]
# 定義y的變化量和迭代次數
y_change = f2(x1,x2)
iter_num = 0
while(y_change > 1e-10 and iter_num < 500) :
tmp_x1 = x1 - alpha * hx1(x1,x2)
tmp_x2 = x2 - alpha * hx2(x1,x2)
tmp_y = f2(tmp_x1,tmp_x2)
f_change = np.absolute(tmp_y - f2(x1,x2))
x1 = tmp_x1
x2 = tmp_x2
GD_X1.append(x1)
GD_X2.append(x2)
GD_Y.append(tmp_y)
iter_num += 1
print(u"最終結果爲:(%.5f, %.5f, %.5f)" % (x1, x2, f2(x1,x2)))
print(u"迭代過程中X的取值，迭代次數:%d" % iter_num)
print(GD_X1)
最終結果爲:(4.00000, 2.00000, -8.00000)
迭代過程中X的取值，迭代次數:500
[-1, -0.8, -0.6100000000000001, -0.42900000000000005, -0.2562, -0.09094999999999998, 0.06728700000000001, 0.21896240000000003, 0.36446231, 0.504122623, 0.638240219, 0.76708130997, 0.890887671459, 1.0098813114016, 1.12426798264683, 1.2342398394044751, 1.3399774593083582, 1.4416513948470318, 1.5394233751310196, 1.6334472473813408, 1.7238697242052554, 1.8108309855081823, 1.8944651711781786, 1.9749007912920942, 2.0522610736600533, 2.1266642634047037, 2.1982238854893583, 2.2670489783145698, 2.333244304437324, 2.3969105429401063, 2.458144466847681, 2.5170391081535817, 2.5736839123992357, 2.6281648842896574, 2.6805647254889604, 2.730962965485576, 2.779436086228317, 2.82605764109338, 2.870898368636641, 2.91402630150599, 2.955506870828168, 2.9954030063386328, 3.033775232487676, 3.07068176072862, 3.106178578172404, 3.140319532775767, 3.173156415216397, 3.20473903759708, 3.235115309111394, 3.2643313087954433, 3.292431355483164, 3.319458075076614, 3.345452465237197, 3.3704539575988393, 3.3945004775996166, 3.4176285020241726, 3.4398731143453656, 3.461268057949959, 3.481845787329721, 3.5016375173160545, 3.520673270433191, 3.5389819224420256, 3.5565912461438782, 3.5735279535107547, 3.5898177362061294, 3.605485304557784, 3.620554425041881, 3.635047956335171, 3.6489878839900554, 3.6623953537851226, 3.6752907038017693, 3.6876934952755707, 3.6996225422692137, 3.7110959402120085, 3.7221310933492724, 3.7327447411432373, 3.74295298366552, 3.7527713060196852, 3.7622146018309457, 3.771297195838631, 3.7800328656257016, 3.7884348625182644, 3.796515931686802, 3.804288331479595, 3.811763852017682, 3.818953833079547, 3.825869181302676, 3.8325203867280706, 3.83891753871282, 3.845070341234864, 3.850988127613167, 3.85667987466563, 3.8621542163262226, 3.867419456741981, 3.8724835828697497, 3.877354276591768, 3.8820389263684834, 3.8865446384462663, 3.8908782476370294, 3.895046327686101, 3.899055201244081, 3.902910949457806, 3.9066194211949727, 3.9101862419164113, 3.9136168222094687, 3.9169163659954447, 3.920089878423533, 3.9231421734632406, 3.9260778812068007, 3.92890145489266, 3.931617177660694, 3.9342291690493907, 3.9367413912448717, 3.9391576550912144, 3.9414816258712055, 3.9437168288662874, 3.9458666547041323, 3.947934364501961, 3.9499230948134, 3.951835862386387, 3.9536755687393446, 3.955445004562551, 3.957146853951402, 3.958783698477972, 3.960358021107057, 3.9618722099626367, 3.963328561950472, 3.9647292862423305, 3.9660765076271267, 3.9673722697340605, 3.968618538132643, 3.96981720331431, 3.9709700835601507, 3.972078927699096, 3.9731454177607564, 3.9741711715269274, 3.9751577449856397, 3.976106634691469, 3.977019280035693, 3.9778970654297354, 3.978741322405208, 3.9795533316337415, 3.9803343248696628, 3.98108548681847, 3.981807956933939, 3.9825028311465824, 3.9831711635260936, 3.9838139678802853, 3.984432219292959, 3.9850268556030275, 3.985598778827146, 3.9861488565279983, 3.9866779231303235, 3.9871867811866712, 3.9876762025948116, 3.988146929768643, 3.9885996767643768, 3.989035130363704, 3.9894539511155895, 3.9898567743382722, 3.9902442110829943, 3.990616849060917, 3.9909752535346326, 3.9913199681756244, 3.9916515158889725, 3.99197039960656, 3.9922771030499815, 3.992572091464311, 3.9928558123238416, 3.99312869601087, 3.9933911564685514, 3.99364359182882, 3.993886385016322, 3.9941199043292857, 3.994344503998197, 3.9945605247231417, 3.9947682941906213, 3.9949681275706257, 3.9951603279947223, 3.9953451870158836, 3.9955229850507497, 3.995693991804997, 3.9958584666824626, 3.9960166591786384, 3.996168809259137, 3.9963151477236996, 3.996455896556302, 3.996591269261887, 3.9967214711902335, 3.996846699847457, 3.9969671451956112, 3.9970829899408447, 3.9971944098105516, 3.9973015738199384, 3.9974046445284035, 3.99750377828613, 3.997599125471254, 3.9976908307179797, 3.9977790331359753, 3.997863866521392, 3.9979454595598245, 3.998023936021513, 3.9980994149490967, 3.99817201083819, 3.998241833811065, 3.998308989783695, 3.9983735806264256, 3.9984357043185, 3.998495455096688, 3.9985529235982344, 3.99860819699835, 3.9986613591424485, 3.9987124906733342, 3.9987616691535295, 3.998808969182933, 3.99885446251198, 3.9988982181504835, 3.9989403024723162, 3.9989807793160943, 3.999019710082016, 3.999057153825, 3.9990931673442685, 3.9991278052695067, 3.999161120143731, 3.999193162502993, 3.9992239809530385, 3.9992536222430406, 3.999282131336512, 3.999309551479511, 3.9993359242662456, 3.999361289702165, 3.9993856862646457, 3.9994091509613607, 3.999431719386415, 3.999453425774339, 3.9994743030520215, 3.999494382888656, 3.9995136957437802, 3.9995322709134804, 3.999550136574833, 3.9995673198286448, 3.9995838467405647, 3.999599742380622, 3.999615030861257, 3.999629735373895, 3.9996438782241293, 3.999657480865554, 3.9996705639323102, 3.999683147270384, 3.9996952499677123, 3.9997068903831385, 3.99971808617426, 3.9997288543242133, 3.999739211167438, 3.9997491724144534, 3.9997587531756897, 3.9997679679844107, 3.9997768308187585, 3.999785355122958, 3.99979355382771, 3.9998014393698065, 3.999809023710997, 3.9998163183561326, 3.9998233343706175, 3.9998300823971955, 3.999836572672094, 3.9998428150405507, 3.9998488189717496, 3.9998545935731835, 3.9998601476044695, 3.999865489490638, 3.999870627334911, 3.999875568930996, 3.9998803217749064, 3.9998848930763335, 3.9998892897695835, 3.999893518524095, 3.999897585754557, 3.9999014976306397, 3.999905260086352, 3.9999088788290442, 3.999912359348065, 3.9999157069230895, 3.999918926632126, 3.9999220233592205, 3.9999250018018655, 3.999927866478125, 3.999930621733488, 3.9999332717474605, 3.9999358205399065, 3.999938271977144, 3.9999406297778113, 3.999942897518507, 3.999945078639216, 3.999947176448527, 3.9999491941286527, 3.999951134740255, 3.9999530012270905, 3.999954796420473, 3.9999565230435694, 3.9999581837155325, 3.9999597809554706, 3.9999613171862722, 3.9999627947382788, 3.9999642158528204, 3.9999655826856166, 3.999966897310046, 3.9999681617202905, 3.9999693778343612, 3.9999705474970075, 3.9999716724825163, 3.9999727544974024, 3.9999737951829974, 3.99997479611794, 3.99997575882057, 3.9999766847512315, 3.999977575314489, 3.9999784318612566, 3.999979255690849, 3.9999800480529513, 3.9999808101495145, 3.9999815431365793, 3.999982248126029, 3.9999829261872786, 3.9999835783488926, 3.99998420560015, 3.999984808892541, 3.999985389141213, 3.9999859472263584, 3.999986483994548, 3.999987000260017, 3.9999874968059, 3.9999879743854176, 3.999988433723021, 3.9999888755154895, 3.999989300432987, 3.9999897091200807, 3.9999901021967164, 3.9999904802591604, 3.999990843880904, 3.9999911936135333, 3.9999915299875646, 3.9999918535132513, 3.9999921646813563, 3.9999924639638973, 3.999992751814862, 3.999993028670899, 3.9999932949519756, 3.9999935510620204, 3.999993797389532, 3.99999403430817, 3.9999942621773212, 3.9999944813426453, 3.999994692136599, 3.9999948948789403, 3.999995089877213, 3.9999952774272147, 3.999995457813444, 3.999995631309532, 3.9999957981786594, 3.9999959586739533, 3.9999961130388724, 3.999996261507576, 3.99999640430528, 3.999996541648597, 3.999996673745866, 3.9999968007974687, 3.999996922996132, 3.999997040527221, 3.9999971535690224, 3.999997262293011, 3.999997366864113, 3.999997467440954, 3.999997564176102, 3.9999976572162956, 3.9999977467026704, 3.9999978327709695, 3.999997915551752, 3.9999979951705904, 3.9999980717482595, 3.999998145400922, 3.999998216240303, 3.9999982843738606, 3.999998349904948, 3.99999841293297, 3.9999984735535365, 3.999998531858603, 3.9999985879366147, 3.9999986418726365, 3.999998693748486, 3.999998743642854, 3.999998791631427, 3.999998837786999, 3.9999988821795855, 3.999998924876526, 3.9999989659425887, 3.9999990054400674, 3.9999990434288764, 3.999999079966642, 3.999999115108789, 3.9999991489086257, 3.9999991814174236, 3.9999992126844957, 3.999999242757272, 3.9999992716813706, 3.999999299500667, 3.9999993262573605, 3.9999993519920394, 3.999999376743741, 3.9999994005500117, 3.9999994234469636, 3.9999994454693297, 3.9999994666505168, 3.999999487022654, 3.9999995066166454, 3.9999995254622127, 3.9999995435879434, 3.999999561021333, 3.999999577788826, 3.9999995939158577, 3.9999996094268915, 3.9999996243454565, 3.999999638694183, 3.9999996524948376, 3.9999996657683536, 3.9999996785348664, 3.999999690813742, 3.9999997026236063, 3.9999997139823735, 3.9999997249072745, 3.9999997354148817, 3.999999745521134, 3.9999997552413618, 3.99999976459031, 3.99999977358216, 3.9999997822305517, 3.9999997905486047, 3.999999798548936, 3.999999806243682, 3.9999998136445147, 3.999999820762661, 3.999999827608918, 3.9999998341936713, 3.9999998405269097, 3.99999984661824, 3.999999852476902, 3.999999858111783, 3.9999998635314307, 3.9999998687440663, 3.999999873757597, 3.9999998785796276, 3.9999998832174732, 3.999999887678169, 3.999999891968481, 3.999999896094918, 3.999999900063739, 3.9999999038809646, 3.999999907552385, 3.99999991108357, 3.9999999144798752, 3.9999999177464534, 3.9999999208882593, 3.999999923910059, 3.999999926816436, 3.9999999296117994, 3.9999999323003896, 3.9999999348862847, 3.999999937373407, 3.99999993976553, 3.999999942066282, 3.9999999442791534, 3.9999999464075002, 3.9999999484545516, 3.9999999504234127, 3.99999995231707, 3.9999999541383957, 3.999999955890153, 3.999999957574999, 3.99999995919549, 3.9999999607540837, 3.999999962253144, 3.999999963694946, 3.9999999650816758, 3.999999966415437, 3.999999967698253, 3.99999996893207, 3.9999999701187594, 3.999999971260121, 3.999999972357887, 3.999999973413722, 3.9999999744292274, 3.999999975405944, 3.9999999763453533, 3.9999999772488803, 3.999999978117896, 3.999999978953718, 3.9999999797576145, 3.999999980530805, 3.999999981274462, 3.999999981989714, 3.999999982677646]
# 作圖
fig = plt.figure(facecolor='w',figsize=(20,18))
ax = Axes3D(fig)
ax.plot_surface(X1,X2,Y,rstride=1,cstride=1,cmap=plt.cm.jet)
ax.plot(GD_X1,GD_X2,GD_Y,'ko-')
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('z')
ax.set_title(u'函數;\n學習率:%.3f; 最終解:(%.3f, %.3f, %.3f);迭代次數:%d' % (alpha, x1, x2, f2(x1,x2), iter_num))
plt.show()
import numpy as np
from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
data=np.genfromtxt('C:/Users/asus/Desktop/data.csv',delimiter=',')
x_data=data[:,:-1]
y_data=data[:,2]
#定義學習率、斜率、截據
#設方程爲y=theta1x1+theta2x2+theta0
lr=0.00001
theta0=0
theta1=0
theta2=0
#定義最大迭代次數，因爲梯度下降法是在不斷迭代更新k與b
epochs=10000
#定義最小二乘法函數-損失函數（代價函數）
def compute_error(theta0,theta1,theta2,x_data,y_data):
totalerror=0
for i in range(0,len(x_data)):#定義一共有多少樣本點
totalerror=totalerror+(y_data[i]-(theta1*x_data[i,0]+theta2*x_data[i,1]+theta0))**2
return totalerror/float(len(x_data))/2
#梯度下降算法求解參數
def gradient_descent_runner(x_data,y_data,theta0,theta1,theta2,lr,epochs):
m=len(x_data)
for i in range(epochs):
theta0_grad=0
theta1_grad=0
theta2_grad=0
for j in range(0,m):
theta0_grad-=(1/m)*(-(theta1*x_data[j,0]+theta2*x_data[j,1]+theta2)+y_data[j])
theta1_grad-=(1/m)*x_data[j,0]*(-(theta1*x_data[j,0]+theta2*x_data[j,1]+theta0)+y_data[j])
theta2_grad-=(1/m)*x_data[j,1]*(-(theta1*x_data[j,0]+theta2*x_data[j,1]+theta0)+y_data[j])
theta0=theta0-lr*theta0_grad
theta1=theta1-lr*theta1_grad
theta2=theta2-lr*theta2_grad
return theta0,theta1,theta2
#進行迭代求解
theta0,theta1,theta2=gradient_descent_runner(x_data,y_data,theta0,theta1,theta2,lr,epochs)
print('結果：迭代次數：{0} 學習率：{1}之後 a0={2},a1={3},a2={4},代價函數爲{5}'.format(epochs,lr,theta0,theta1,theta2,compute_error(theta0,theta1,theta2,x_data,y_data)))
print("多元線性迴歸方程爲:y=",theta1,"X1+",theta2,"X2+",theta0)
#畫圖
ax=plt.figure().add_subplot(111,projection='3d')
ax.scatter(x_data[:,0],x_data[:,1],y_data,c='r',marker='o')
x0=x_data[:,0]
x1=x_data[:,1]
#生成網格矩陣
x0,x1=np.meshgrid(x0,x1)
z=theta0+theta1*x0+theta2*x1
#畫3d圖
ax.plot_surface(x0,x1,z)
ax.set_xlabel('area')
ax.set_ylabel('distance')
ax.set_zlabel("Monthly turnover")
plt.show()
結果：迭代次數：10000 學習率：1e-05之後 a0=5.3774162274868,a1=45.0533119768975,a2=-0.19626929358281256,代價函數爲366.7314528822914
多元線性迴歸方程爲:y= 45.0533119768975 X1+ -0.19626929358281256 X2+ 5.3774162274868
牛頓法
設r是f(x)=0的根，選取x0作爲r的初始近似值，過點(x0,f(x0))做曲線y=f(x)的切線L
L:y=f(x0)+f’(x0)(x-x0),則L與x軸交點的橫座標
x1=x0-f(x0)/f’(x0),稱x1爲r的一次近似值。以此類推，
則爲牛頓迭代公式。
運用牛頓法的一般步驟
一、確定迭代變量
在可以用迭代算法解決的問題中，至少存在一個可直接或間接地不斷由舊值遞推出新值的變量，這個變量就是迭代變量。
二、建立迭代關係式
所謂迭代關係式，指如何從變量的前一個值推出其下一個值的公式（或關係）。迭代關係式的建立是解決迭代問題的關鍵，通常可以使用遞推或倒推的方法來完成。
三、對迭代過程進行控制
在什麼時候結束迭代過程？這是編寫迭代程序必須考慮的問題。不能讓迭代過程無休止地執行下去。迭代過程的控制通常可分爲兩種情況：一種是所需的迭代次數是個確定的值，可以計算出來；另一種是所需的迭代次數無法確定。對於前一種情況，可以構建一個固定次數的循環來實現對迭代過程的控制；對於後一種情況，需要進一步分析得出可用來結束迭代過程的條件。
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
%matplotlib inline
dataset = pd.read_excel('C:/Users/asus/Desktop/data.xlsx')
data = np.genfromtxt("C:/Users/asus/Desktop/data.csv",delimiter=",")
X1=data[0:10,0]#自變量溫度
X2=data[0:10,1]#因變量銷售量
Y=data[0:10,2]#自變量溫度
#將因變量賦值給矩陣Y1
Y1=np.array([Y]).T
#爲自變量係數矩陣X賦值
X11=np.array([X1]).T
X22=np.array([X2]).T
A=np.array([[1],[1],[1],[1],[1],[1],[1],[1],[1],[1]])#創建係數矩陣
B=np.hstack((A,X11))#將矩陣a與矩陣X11合併爲矩陣b
X=np.hstack((B,X22))#將矩陣b與矩陣X22合併爲矩陣X
#求矩陣X的轉置矩陣
X_=X.T
X_X=np.dot(X_,X)
X_X_=np.linalg.inv(X_X)
#求解係數矩陣W，分別對應截距b、a1、和a2
W=np.dot(np.dot((X_X_),(X_)),Y1)
b=W[0][0]
a1=W[1][0]
a2=W[2][0]
print("係數a1=",a1)
print("係數a2=",a2)
print("截距爲=",b)
print("多元線性迴歸方程爲:y=",a1,"X1+",a2,"X2+",b)
#求月銷售量Y的和以及平均值y1
sumy=0#因變量的和
y1=0#因變量的平均值
for i in range(0,len(Y)):
sumy=sumy+Y[i]
y1=sumy/len(Y)
#求月銷售額y-他的平均值的和
y_y1=0#y-y1的值的和
for i in range(0,len(Y)):
y_y1=y_y1+(Y[i]-y1)
print("銷售量-銷售量平均值的和爲:",y_y1)
#求預測值sales1
sales1=[]
for i in range(0,len(Y)):
sales1.append(a1*X1[i]+a2*X2[i]+b)
#求預測值的平均值y2
y2=0
sumy2=0
for i in range(len(sales1)):
sumy2=sumy2+sales1[i]
y2=sumy2/len(sales1)
#求預測值-平均值的和y11_y2
y11_y2=0
for i in range(0,len(sales1)):
y11_y2=y11_y2+(sales1[i]-y2)
print("預測銷售值-預測銷售平均值的和爲:",y11_y2)
#求月銷售額y-他的平均值的平方和
Syy=0#y-y1的值的平方和
for i in range(0,len(Y)):
Syy=Syy+((Y[i]-y1)*(Y[i]-y1))
print("Syy=",Syy)
#求y1-y1平均的平方和
Sy1y1=0
for i in range(0,len(sales1)):
Sy1y1=Sy1y1+((sales1[i]-y2)*(sales1[i]-y2))
print("Sy1y1=",Sy1y1)
#（y1-y1平均）*（y-y平均）
Syy1=0
for i in range(0,len(sales1)):
Syy1=Syy1+((Y[i]-y1)*(sales1[i]-y2))
print("Syy1=",Syy1)
#（y1-y1平均）*（y-y平均）
Syy1=0
for i in range(0,len(sales1)):
Syy1=Syy1+((Y[i]-y1)*(sales1[i]-y2))
print("Syy1=",Syy1)
係數a1= 41.51347825643848
係數a2= -0.34088268566362023
截距爲= 65.32391638894899
多元線性迴歸方程爲:y= 41.51347825643848 X1+ -0.34088268566362023 X2+ 65.32391638894899
銷售量-銷售量平均值的和爲: -1.1368683772161603e-13
預測銷售值-預測銷售平均值的和爲: -3.410605131648481e-13
Syy= 76199.6
Sy1y1= 72026.59388000543
Syy1= 72026.59388000538
Syy1= 72026.59388000538
參考文獻
https://blog.csdn.net/weixin_44606638/article/details/105326348
https://www.jianshu.com/p/424b7b70df7b
https://blog.csdn.net/qq_35946969/article/details/84446000