原理：

利用复化梯形公式，复化Simpson公式，计算积分。

步骤：

import math"""测试函数"""def f(x,i): if i == 1: return (4 - (math.sin(x)) ** 2) ** 0.5 if i == 2: if x == 0: return 1 else: return math.sin(x) / x if i == 3: return (math.exp(x)) / (4 + x ** 2) if i == 4: return math.log(1+x,math.e) / (1 + x ** 2)"""打印显示函数""" def p(i,n): return "第" + str(i) + "题,n=" + str(n) + "时的积分值为：""""复化Simpson函数"""def Simpson(a, b, n, i): h = (b - a) / (2 * n) F0 = f(a,i) + f(b,i) F1 = 0 F2 = 0 for j in range(1,2 * n): x = a + (j * h) if j % 2 == 0: F2 = F2 + f(x,i) else: F1 = F1 + f(x,i) SN = (h * (F0 + 2 * F2 + 4 * F1)) / 3 print("复化Simpson函数" + p(i,n) + str("%-10.7f"%(SN))) return SNdef T(a, b, n, i): h = (b - a) / n F0 = f(a,i) + f(b,i) F = 0 for j in range(1,n): x = a + (j * h) F = F + f(x,i) SN = (h * (F0 + 2 * F)) / 2 print("复化梯形函数" + p(i,n) + str("%-10.7f"%(SN))) return SNdef SimpsonTimes(x): n = 1 y = Simpson(0, math.pi/4, n, 1) while(abs(y - 1.5343916) > x): n = n + 1 y = Simpson(0, math.pi/4, n, 1) else: return ndef Times(x): n = 1 y = T(0, math.pi/4, n, 1) while(abs(y - 1.5343916) > x): n = n + 1 y = T(0, math.pi/4, n, 1) else: return n""" 测试部分 """Simpson(0, math.pi/4, 10, 1)Simpson(0, 1, 10, 2)Simpson(0, 1, 10, 3)Simpson(0, 1, 10, 4)Simpson(0, math.pi/4, 20, 1)Simpson(0, 1, 20, 2)Simpson(0, 1, 20, 3)Simpson(0, 1, 20, 4)T(0, math.pi/4, 10, 1)T(0, 1, 10, 2)T(0, 1, 10, 3)T(0, 1, 10, 4)T(0, math.pi/4, 20, 1)T(0, 1, 20, 2)T(0, 1, 20, 3)T(0, 1, 20, 4)print("复化梯形函数求解第一问，精度为0.00001时需要" + str(Times(0.00001)) + "个步数")print("复化Simpson函数求解第一问，精度为0.00001时需要" + str(SimpsonTimes(0.00001)) + "个步数")print("复化梯形函数求解第一问，精度为0.000001时需要" + str(Times(0.000001)) + "个步数")print("复化Simpson函数求解第一问，精度为0.000001时需要" + str(SimpsonTimes(0.000001)) + "个步数")

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