Python实现龙贝格积分法：从复合梯形到高精度求解

# 龙贝格法求积分
import math
a=0    # 积分下限
b=1    # 积分上限
eps=10**-5  # 精度
T=[]   # 复化梯形序列
S=[]   # Simpson序列
C=[]   # Cotes序列
R=[]   # Romberg序列
def func(x): # 被积函数
 y=math.exp(-x)
 return y
def Romberg(a,b,eps,func):
 h = b - a
 T.append(h * (func(a) + func(b)) / 2)
 ep=eps+1
 m=0
 while(ep>=eps):
  m=m+1
  t=0
  for i in range(2**(m-1)-1):
   t=t+func(a+(2*(i+1)-1)*h/2**m)*h/2**m
  t=t+T[-1]/2
  T.append(t)
  if m>=1:
   S.append((4**m*T[-1]-T[-2])/(4**m-1))
  if m>=2:
   C.append((4**m*S[-1]-S[-2])/(4**m-1))
  if m>=3:
   R.append((4**m*C[-1]-C[-2])/(4**m-1))
  if m>4:
   ep=abs(10*(R[-1]-R[-2]))
Romberg(a,b,eps,func)
# print(T)
# print(S)
# print(C)
# print(R)
# 计算机参考值0.6321205588
print("积分结果为：{:.5f}".format(R[-1]))

# coding=UTF-8
# Author:winyn
'''
给定一个函数，如：f(x)= x^(3/2），和积分上下限a,b，用机械求积Romberg公式求积分。

'''
import numpy as np


def func(x):
 return x**(3/2)

class Romberg:
 def __init__(self, integ_dowlimit, integ_uplimit):
  '''
  初始化积分上限integ_uplimit和积分下限integ_dowlimit
  输入一个函数，输出函数在积分上下限的积分

  '''
  self.integ_uplimit = integ_uplimit
  self.integ_dowlimit = integ_dowlimit



 def calc(self):
  '''
  计算Richardson外推算法的四个序列

  '''
  t_seq1 = np.zeros(5, 'f')
  s_seq2 = np.zeros(4, 'f')
  c_seq3 = np.zeros(3, 'f')
  r_seq4 = np.zeros(2, 'f')
  # 循环生成hm间距序列
  hm = [(self.integ_uplimit - self.integ_dowlimit) / (2 ** i) for i in range(0,5)]
  print(hm)
  # 循环生成t_seq1
  fa = func(self.integ_dowlimit)
  fb = func(self.integ_uplimit)

  t0 = (1 / 2) * (self.integ_uplimit - self.integ_dowlimit) * (fa+fb)
  t_seq1[0] = t0

  for i in range(1, 5):
   sum = 0
   # 多出来的点的累加和
   for each in range(1, 2**i,2):
    sum =sum + hm[i]*func( self.integ_dowlimit+each * hm[i])#计算两项值
   temp1 = 1 / 2 * t_seq1[i - 1]
   temp2 =sum
   temp = temp1 + temp2
   # 求t_seql的1-4位
   t_seq1[i] = temp
  print('T序列：'+ str(list(t_seq1)))
  # 循环生成s_seq2
  s_seq2 = [round((4 * t_seq1[i + 1] - t_seq1[i]) / 3,6) for i in range(0, 4)]
  print('S序列：' + str(list(s_seq2)))
  # 循环生成c_seq3
  c_seq3 = [round((4 ** 2 * s_seq2[i + 1] - s_seq2[i]) / (4 ** 2 - 1),6) for i in range(0, 3)]
  print('C序列：' + str(list(c_seq3)))
  # 循环生成r_seq4
  r_seq4 = [round((4 ** 3 * c_seq3[i + 1] - c_seq3[i]) / (4 ** 3 - 1),6) for i in range(0, 2)]
  print('R序列：' + str(list(r_seq4)))
  return 'end'


rom = Romberg(0, 1)
print(rom.calc())