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Example Code

import numpy as np

import matplotlib

import matplotlib.pyplot as plt

def fun1(x):

return np.abs(np.sin(x*10)/np.exp(x))

upperL=3

lowerL=0

interval=0.01

Xpoints = np.arange(lowerL,upperL,interval)

#print(Xpoints)

#print(fun1(Xpoints))

plt.figure(1)

plt.plot(Xpoints,fun1(Xpoints), 'r.')

plt.show()

Note and Comments

import numpy as np

import math

import matplotlib

import matplotlib.pyplot as plt; 匯入圖形函式庫Matplotlib中的pyplot資料庫

EVN=100000; 宣告要產生多少數據點

posX, posY = (np.random.rand(EVN)-0.5, np.random.rand(EVN)-0.5) 產生兩組-0.5到0.5的亂數, 每組有EVN個亂數點.

poxInCirx, poxInCiry =([],[]) 產生兩組X跟Y的集合,用來存放半徑小於0.5

[(poxInCirx.append(posX[x]), poxInCiry.append(posY[x]) ) for x in range(EVN) if math.hypot(posX[x],posY[x]) <=0.5]

將位於圓內的亂數點用append加到poxInCirx, poxInCiry數組內. math.hypot(x,y)函數將回傳(x*x+y*y)的根號值

plt.plot(posX,posY, 'r.') 以(posX,posY) 爲數據點,2維作圖顏色爲紅色r, 數據點以.表示

plt.plot(poxInCirx,poxInCiry, 'go') 以(poxInCirx,poxInCiry) 爲數據點,2維作圖. 顏色爲綠色g, 數據點以o表示

plt.axes().set_aspect('equal') 將圖的X軸與Y軸設爲相同比例

print("Monte Carlo Pi=", len(poxInCirx)/EVN*4)

print("Error of Monte Carlo Pi=", (len(poxInCirx)/EVN*4-math.pi)/math.pi*100)

plt.text(-0.15, -0.6, r'EVN=%d' %EVN, fontsize=20) 將文字EVN=10000設置在-0.15, -0.6的位置

plt.show() 圖形設定完畢,開始作圖

Example Code

import scipy as sp

import scipy.integrate as spInt

import matplotlib

import matplotlib.pyplot as plt

def fun1(x):

return sp.fabs(sp.sin(x*10)/sp.exp(x))

upperL=3.0

lowerL=0.0

interval=0.01

Xpoints = sp.arange(lowerL,upperL,interval)

Ypoints= fun1(Xpoints)

#print(Xpoints)

#print(Ypoints)

#print(fun1(Xpoints))

#print(Ypoints.max())

#print(np.ceil(Ypoints.max()))

#print(np.ceil(Xpoints.max()))

randLx=sp.ceil(Xpoints.max())

randLy=sp.ceil(Ypoints.max())

EVN=10000;

posAreaX, posAreaY =([],[])

posX, posY = (sp.random.rand(EVN)*randLx, sp.random.rand(EVN)*randLy)

for i in range(EVN):

if(posY[i]<=fun1(posX[i])):

posAreaX.append(posX[i])

posAreaY.append(posY[i])

plt.figure(1)

plt.plot(Xpoints,fun1(Xpoints), 'r-')

plt.plot(posAreaX,posAreaY, 'b.')

plt.show()

print("integrated area =", randLx *randLy *len(posAreaY)/EVN)

result, err = spInt.quad(lambda x: fun1(x), lowerL, upperL)

print("integrated area with sp.integrate.quad =", result)

Note and Comments

import numpy as np

import math

import matplotlib

import matplotlib.pyplot as plt; 匯入圖形函式庫Matplotlib中的pyplot資料庫

EVN=100000; 宣告要產生多少數據點

posX, posY = (np.random.rand(EVN)-0.5, np.random.rand(EVN)-0.5) 產生兩組-0.5到0.5的亂數, 每組有EVN個亂數點.

poxInCirx, poxInCiry =([],[]) 產生兩組X跟Y的集合,用來存放半徑小於0.5

[(poxInCirx.append(posX[x]), poxInCiry.append(posY[x]) ) for x in range(EVN) if math.hypot(posX[x],posY[x]) <=0.5]

將位於圓內的亂數點用append加到poxInCirx, poxInCiry數組內. math.hypot(x,y)函數將回傳(x*x+y*y)的根號值

plt.plot(posX,posY, 'r.') 以(posX,posY) 爲數據點,2維作圖顏色爲紅色r, 數據點以.表示

plt.plot(poxInCirx,poxInCiry, 'go') 以(poxInCirx,poxInCiry) 爲數據點,2維作圖. 顏色爲綠色g, 數據點以o表示

plt.axes().set_aspect('equal') 將圖的X軸與Y軸設爲相同比例

print("Monte Carlo Pi=", len(poxInCirx)/EVN*4)

print("Error of Monte Carlo Pi=", (len(poxInCirx)/EVN*4-math.pi)/math.pi*100)

plt.text(-0.15, -0.6, r'EVN=%d' %EVN, fontsize=20) 將文字EVN=10000設置在-0.15, -0.6的位置

plt.show() 圖形設定完畢,開始作圖

Example Code

from sympy import integrate, erf, exp, sin, log, oo, pi, sinh, symbols, Abs

upperL=3.0

lowerL=0.0

x, y = symbols('x,y', real=True)

print(integrate(sin(x*10), x))

print(integrate(sin(x*10)/exp(x), x))

print(integrate(sin(x*10)/exp(x), x, (x,lowerL, upperL)))

print(integrate(Abs(sin(x*10))/exp(x), x, (x,lowerL, upperL)))

Note and Comments

import numpy as np

import math

import matplotlib

import matplotlib.pyplot as plt; 匯入圖形函式庫Matplotlib中的pyplot資料庫

EVN=100000; 宣告要產生多少數據點

posX, posY = (np.random.rand(EVN)-0.5, np.random.rand(EVN)-0.5) 產生兩組-0.5到0.5的亂數, 每組有EVN個亂數點.

poxInCirx, poxInCiry =([],[]) 產生兩組X跟Y的集合,用來存放半徑小於0.5

[(poxInCirx.append(posX[x]), poxInCiry.append(posY[x]) ) for x in range(EVN) if math.hypot(posX[x],posY[x]) <=0.5]

將位於圓內的亂數點用append加到poxInCirx, poxInCiry數組內. math.hypot(x,y)函數將回傳(x*x+y*y)的根號值

plt.plot(posX,posY, 'r.') 以(posX,posY) 爲數據點,2維作圖顏色爲紅色r, 數據點以.表示

plt.plot(poxInCirx,poxInCiry, 'go') 以(poxInCirx,poxInCiry) 爲數據點,2維作圖. 顏色爲綠色g, 數據點以o表示

plt.axes().set_aspect('equal') 將圖的X軸與Y軸設爲相同比例

print("Monte Carlo Pi=", len(poxInCirx)/EVN*4)

print("Error of Monte Carlo Pi=", (len(poxInCirx)/EVN*4-math.pi)/math.pi*100)

plt.text(-0.15, -0.6, r'EVN=%d' %EVN, fontsize=20) 將文字EVN=10000設置在-0.15, -0.6的位置

plt.show() 圖形設定完畢,開始作圖

Example Code

import scipy as sp

v10= sp.floor(sp.random.rand()*10)

v1= sp.floor(sp.random.rand()*10)

print(v10,v1)

A=0

while(A!=2):

A=0

B=0

a=int(input())

b=int(input())

print(a,v10)

if(a==v10):

A=A+1

if(a==v1):

B=B+1

if(b==v10):

B=B+1

if(b==v1):

A=A+1

print("A=",A,"B=",B)

print("yes the ture valuse is ",a,b)

Note and Comments

import numpy as np

import math

import matplotlib

import matplotlib.pyplot as plt; 匯入圖形函式庫Matplotlib中的pyplot資料庫

EVN=100000; 宣告要產生多少數據點

posX, posY = (np.random.rand(EVN)-0.5, np.random.rand(EVN)-0.5) 產生兩組-0.5到0.5的亂數, 每組有EVN個亂數點.

poxInCirx, poxInCiry =([],[]) 產生兩組X跟Y的集合,用來存放半徑小於0.5

[(poxInCirx.append(posX[x]), poxInCiry.append(posY[x]) ) for x in range(EVN) if math.hypot(posX[x],posY[x]) <=0.5]

將位於圓內的亂數點用append加到poxInCirx, poxInCiry數組內. math.hypot(x,y)函數將回傳(x*x+y*y)的根號值

plt.plot(posX,posY, 'r.') 以(posX,posY) 爲數據點,2維作圖顏色爲紅色r, 數據點以.表示

plt.plot(poxInCirx,poxInCiry, 'go') 以(poxInCirx,poxInCiry) 爲數據點,2維作圖. 顏色爲綠色g, 數據點以o表示

plt.axes().set_aspect('equal') 將圖的X軸與Y軸設爲相同比例

print("Monte Carlo Pi=", len(poxInCirx)/EVN*4)

print("Error of Monte Carlo Pi=", (len(poxInCirx)/EVN*4-math.pi)/math.pi*100)

plt.text(-0.15, -0.6, r'EVN=%d' %EVN, fontsize=20) 將文字EVN=10000設置在-0.15, -0.6的位置

plt.show() 圖形設定完畢,開始作圖

Lesson 6 02: 蒙地卡羅方法求積分值

Lesson 6 01: 函數作圖