用python编写程序求梯形求积公式，抛物线求积公式，复化梯形公式，复化抛物线公式，逐次分半求积公式的误差

以下是用Python编写程序求解数值积分公式误差的示例代码：

1. 梯形求积公式误差

import math

# 定义被积函数
def f(x):
    return math.sin(x)

# 定义梯形求积公式
def trapezoid_rule(f, a, b):
    return (b - a) * (f(a) + f(b)) / 2

# 计算梯形求积公式误差
def trapezoid_error(f, a, b, n):
    h = (b - a) / n
    exact = math.cos(a) - math.cos(b)
    approx = 0
    for i in range(n):
        x0 = a + i * h
        x1 = x0 + h
        approx += trapezoid_rule(f, x0, x1)
    error = abs(approx - exact)
    return error

# 示例
print(trapezoid_error(f, 0, math.pi/2, 10))

2. 抛物线求积公式误差

import math

# 定义被积函数
def f(x):
    return math.sin(x)

# 定义抛物线求积公式
def parabolic_rule(f, a, b):
    h = (b - a) / 2
    return (h / 3) * (f(a) + 4 * f((a + b) / 2) + f(b))

# 计算抛物线求积公式误差
def parabolic_error(f, a, b, n):
    h = (b - a) / n
    exact = math.cos(a) - math.cos(b)
    approx = 0
    for i in range(n):
        x0 = a + i * h
        x1 = x0 + h
        approx += parabolic_rule(f, x0, x1)
    error = abs(approx - exact)
    return error

# 示例
print(parabolic_error(f, 0, math.pi/2, 10))

3. 复化梯形公式误差

import math

# 定义被积函数
def f(x):
    return math.sin(x)

# 定义梯形求积公式
def trapezoid_rule(f, a, b):
    return (b - a) * (f(a) + f(b)) / 2

# 计算复化梯形公式误差
def composite_trapezoid_error(f, a, b, n):
    h = (b - a) / n
    exact = math.cos(a) - math.cos(b)
    approx = 0
    for i in range(n):
        x0 = a + i * h
        x1 = x0 + h
        approx += trapezoid_rule(f, x0, x1)
    error = abs(approx - exact)
    return error

# 示例
print(composite_trapezoid_error(f, 0, math.pi/2, 10))

4. 复化抛物线公式误差

import math

# 定义被积函数
def f(x):
    return math.sin(x)

# 定义抛物线求积公式
def parabolic_rule(f, a, b):
    h = (b - a) / 2
    return (h / 3) * (f(a) + 4 * f((a + b) / 2) + f(b))

# 计算复化抛物线公式误差
def composite_parabolic_error(f, a, b, n):
    h = (b - a) / n
    exact = math.cos(a) - math.cos(b)
    approx = 0
    for i in range(n):
        x0 = a + i * h
        x1 = x0 + h
        x2 = x1 + h
        approx += parabolic_rule(f, x0, x2)
    error = abs(approx - exact)
    return error

# 示例
print(composite_parabolic_error(f, 0, math.pi/2, 10))

5. 逐次分半求积公式误差

import math

# 定义被积函数
def f(x):
    return math.sin(x)

# 定义逐次分半求积公式
def recursive_midpoint_rule(f, a, b, tol):
    c = (a + b) / 2
    fa = f(a)
    fb = f(b)
    fc = f(c)
    approx1 = (b - a) * fc
    approx2 = (b - a) * (fa + 4 * fc + fb) / 6
    error = abs(approx2 - approx1)
    if error < tol:
        return approx2
    else:
        left = recursive_midpoint_rule(f, a, c, tol/2)
        right = recursive_midpoint_rule(f, c, b, tol/2)
        return left + right

# 计算逐次分半求积公式误差
def recursive_midpoint_error(f, a, b, tol):
    exact = math.cos(a) - math.cos(b)
    approx = recursive_midpoint_rule(f, a, b, tol)
    error = abs(approx - exact)
    return error

# 示例
print(recursive_midpoint_error(f, 0, math.pi/2, 1e-6))

注意：以上代码仅为示例，实际应用时需要根据具体情况进行调整。