利用Python做科學計算

最近有個朋友讓我幫忙做個計算，工具不限。考慮到方便快捷，於是選擇了python。

 其核心要求是做一個函數擬合，但是被擬合函數是個積分表達式。最簡單的方法是利用scipy庫中的函數來做，下面是源代碼。

# -*- coding: utf-8 -*-

from scipy import integrate
from scipy import optimize
from matplotlib import pyplot
import numpy
import pandas
import time

I = complex(0, 1)

def tmp(E, params):
    global I
    Z = params[1]
    delta = params[2]
    Gamma = params[3]
    complex_num = 0.5 + numpy.sqrt(pow(E+I*Gamma, 2)-delta*delta)/(2*(E+I*Gamma))
    alpha = complex_num.real
    eta = complex_num.imag
    beta = 1 - alpha
    gamma = numpy.sqrt( numpy.power(alpha+Z*Z*(alpha-beta), 2)
          + numpy.power(eta*(2*Z*Z+1), 2) )
    return alpha, beta, gamma, eta

def factor(E, params):
    P = params[0]
    Z = params[1]
    alpha, beta, gamma, eta = tmp(E, params)
    numerator1 = numpy.sqrt((alpha*alpha+eta*eta)*(beta*beta+eta*eta))
    denominator1 = gamma*gamma
    AE = numerator1/denominator1
    numerator2 = Z*Z*( numpy.power((alpha-beta)*Z-2*eta, 2) 
              + numpy.power(2*eta*Z+(alpha-beta), 2) )
    denominator2 = gamma*gamma
    BE = numerator2/denominator2
    return 1+(1-P)*AE-BE

def dfdV_mod(E, V):
    variable = E - V
    if (variable >= 0):
        exp = numpy.exp(-variable)
    else:
        exp = numpy.exp(variable)
    numerator = -exp
    denominator = numpy.power(exp+1, 2)
    return numerator/denominator

# 被積函數
def integrand(E, V, params):
    return dfdV_mod(E, V)*factor(E, params)

# 積分
def integral(V, params):
    result = integrate.quad(integrand, -numpy.inf, numpy.inf, args=(V, params))
    return result[0]

# 畫圖時計算積分
def integral_all(V, params):
    result = numpy.zeros(V.size)
    for i in range(0, V.size):
        res = integrate.quad(integrand, -numpy.inf, numpy.inf, args=(V[i], params))
        result[i] = res[0]
    return result

# 實驗測量值與理論值的偏差
def residual(params, g, V):
    res = numpy.zeros(g.size)
    for i in range(0, g.size):
        res[i] = g[i] - integral(V[i], params)
    return res

# 將實驗數據讀入，需在實驗數據中添加表頭
def ReadData(path):
    dataframe = pandas.read_excel(path, sheet_name=0)
    x = numpy.array(dataframe.iloc[:, 0])
    y = numpy.array(dataframe.iloc[:, 1])
    return y, x

if __name__ == '__main__':
    start = time.time()
    # 賦初值
    P = 0.5
    Z = 1
    delta = 0
    Gamma = 0

    # 讀入數據
    g, V = ReadData("data.xlsx")

    # 最小二乘法擬合
    params0 = numpy.array([P, Z, delta, Gamma])
    result = optimize.least_squares(residual, params0, bounds=([-1, -5, -1, -1], [1, 5, 1, 1]), args=(g, V))
    print("result: " + str(result))
    
    # 畫出結果
    params = result.x
    V_test = numpy.linspace(V.min(), V.max(), 100)
    g_test = integral_all(V_test, params)
    pyplot.plot(V, g, 'o', markersize=1, label='data')
    pyplot.plot(V_test, g_test, label='fitted curve')
    pyplot.xlabel('V')
    pyplot.ylabel('g')
    pyplot.show()
    
    end = time.time()
    print("time elapsed: " + str(end-start) + "s")

從代碼來看還是很清晰的，主要用到兩個函數，一個是integrate.quad，用來算積分，另一個是optimize.least_squares，利用最小二乘法給出函數參數值。