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 SN def 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 SN def 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 n def 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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