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from sympy import *
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from convert_formula import *
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import numpy as np
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from sympy.utilities.lambdify import lambdify
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def extreme_value(function):
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"""
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计算函数的极值点和极值
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:param function: 函数表达式
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:return: 极值点与极值
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"""
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function = convert_formula(function)
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independent_variable, function = split_function(function)
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independent_variable = symbols(independent_variable)
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function = sympify(function)
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try:
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extreme_points = solve(diff(function, independent_variable), independent_variable)
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except NotImplementedError: # 捕获sympy无法解超越方程的错误
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# 定义牛顿迭代法函数（新增容差参数说明）
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def newton_iteration(func, func_derivative, x0, tol=1e-8, max_iter=100):
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x = x0
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for _ in range(max_iter):
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fx = func(x)
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if abs(fx) < tol: # 达到精度要求时返回
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return x
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f_prime_x = func_derivative(x)
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if f_prime_x == 0: # 避免除零错误
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break
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x = x - fx / f_prime_x
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return None
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# 将符号函数转换为可调用的数值函数
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derivative_expr = diff(function, independent_variable)
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func_num = lambdify(independent_variable, derivative_expr)
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func_derivative_num = lambdify(independent_variable, diff(derivative_expr, independent_variable))
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# 改进：通过区间扫描生成初始猜测（覆盖函数定义域）
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x_min, x_max = -10, 10 # 可根据实际函数调整扫描区间
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step = 0.5 # 扫描步长（更小步长可检测更多极值点）
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x_scan = np.arange(x_min, x_max + step, step)
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f_scan = [func_num(xi) for xi in x_scan]
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# 检测导数符号变化区间（中间值定理）
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candidate_intervals = []
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for i in range(len(f_scan)-1):
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if f_scan[i] * f_scan[i+1] <= 0: # 符号变化或零点
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candidate_intervals.append((x_scan[i], x_scan[i+1]))
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# 改进：在每个候选区间中点启动牛顿法
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extreme_points = []
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for a, b in candidate_intervals:
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x0 = (a + b) / 2 # 区间中点作为初始值
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root = newton_iteration(func_num, func_derivative_num, x0)
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if root is not None:
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# 去重：保留6位小数避免重复
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root_rounded = round(root, 6)
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if root_rounded not in [round(r, 6) for r in extreme_points]:
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extreme_points.append(root)
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extreme_values = [function.subs(independent_variable, point) for point in extreme_points]
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extreme_dict = {point: value for point, value in zip(extreme_points, extreme_values)}
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return extreme_dict
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def max_min(function):
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"""
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计算函数的最大值和最小值
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:param function: 函数表达式
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:return: 最大值和最小值
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"""
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extreme_dict = extreme_value(function)
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max_value = max(extreme_dict.values())
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min_value = min(extreme_dict.values())
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return max_value, min_value
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def taylor_series(function, n, x0=0):
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"""
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计算泰勒展开式
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:param function: 函数表达式
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:param x0: 展开点
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:param n: 展开阶数
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:return: 泰勒展开式
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"""
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function = convert_formula(function)
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independent_variable, function = split_function(function)
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independent_variable = symbols(independent_variable)
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function = sympify(function)
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taylor_series = function.subs(independent_variable, x0)
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for i in range(1, n+1):
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taylor_series += diff(function, independent_variable, i).subs(independent_variable, x0) * (independent_variable - x0)**i / factorial(i)
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return taylor_series
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def monotonicity(function):
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"""
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计算函数的单调性（改进版，支持超越方程，输出易读格式）
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:param function: 函数表达式
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:return: 递增区间列表、递减区间列表（格式：[(start1, end1), (start2, end2), ...]，数值为普通浮点数）
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"""
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function = convert_formula(function)
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independent_variable, function = split_function(function)
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var = symbols(independent_variable)
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func_expr = sympify(function)
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derivative = diff(func_expr, var) # 一阶导数
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# 转换为数值计算函数
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deriv_num = lambdify(var, derivative, 'numpy')
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# 定义扫描参数（可根据实际函数调整）
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x_min, x_max = -10, 10 # 扫描区间
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step = 0.1 # 扫描步长（更小步长更精确但计算量更大）
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x_vals = np.arange(x_min, x_max + step, step)
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# 检测符号变化点（使用二分法精确化）
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change_points = []
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prev_sign = np.sign(deriv_num(x_vals[0]))
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for i in range(1, len(x_vals)):
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current_sign = np.sign(deriv_num(x_vals[i]))
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if prev_sign != current_sign:
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# 二分法查找精确交点
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a, b = x_vals[i-1], x_vals[i]
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for _ in range(10): # 迭代10次足够精确
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mid = (a + b) / 2
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mid_sign = np.sign(deriv_num(mid))
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if mid_sign == prev_sign:
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a = mid
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else:
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b = mid
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# 转换为Python原生浮点数并保留3位小数
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point = float(round((a + b)/2, 3))
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change_points.append(point)
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prev_sign = current_sign
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# 生成单调区间（数值均为普通浮点数）
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up_intervals = []
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down_intervals = []
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current_sign = np.sign(deriv_num(x_min + 1e-6)) # 避免端点误差
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start = float(round(x_min, 3)) # 转换为普通浮点数
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for point in change_points:
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if current_sign > 0:
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up_intervals.append((start, point))
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else:
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down_intervals.append((start, point))
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current_sign *= -1
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start = point
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# 处理最后一个区间（终点转换为普通浮点数）
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end = float(round(x_max, 3))
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if current_sign > 0:
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up_intervals.append((start, end))
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else:
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down_intervals.append((start, end))
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return up_intervals, down_intervals
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def aotu(function):
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"""
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计算函数的凹凸性（改进版，支持超越方程，输出易读格式）
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:param function: 函数表达式
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:return: 凹区间列表、凸区间列表（格式：[(start1, end1), (start2, end2), ...]，数值为普通浮点数）
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"""
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function = convert_formula(function)
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independent_variable, function = split_function(function)
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var = symbols(independent_variable)
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func_expr = sympify(function)
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derivative = diff(func_expr, var, 2) # 二阶导数
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# 转换为数值计算函数
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deriv_num = lambdify(var, derivative, 'numpy')
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# 定义扫描参数（可根据实际函数调整）
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x_min, x_max = -10, 10 # 扫描区间
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step = 0.1 # 扫描步长（更小步长更精确但计算量更大）
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x_vals = np.arange(x_min, x_max + step, step)
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# 检测符号变化点（使用二分法精确化）
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change_points = []
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prev_sign = np.sign(deriv_num(x_vals[0]))
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for i in range(1, len(x_vals)):
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current_sign = np.sign(deriv_num(x_vals[i]))
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if prev_sign != current_sign:
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# 二分法查找精确交点
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a, b = x_vals[i-1], x_vals[i]
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for _ in range(10): # 迭代10次足够精确
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mid = (a + b) / 2
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mid_sign = np.sign(deriv_num(mid))
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if mid_sign == prev_sign:
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a = mid
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else:
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b = mid
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# 转换为Python原生浮点数并保留3位小数（与monotonicity统一）
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point = float(round((a + b)/2, 3))
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change_points.append(point)
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prev_sign = current_sign
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# 生成凹凸区间（数值均为普通浮点数）
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ao_intervals = [] # 凹区间（二阶导数≥0）
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tu_intervals = [] # 凸区间（二阶导数≤0）
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current_sign = np.sign(deriv_num(x_min + 1e-6)) # 避免端点误差
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start = float(round(x_min, 3)) # 转换为普通浮点数
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for point in change_points:
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if current_sign > 0:
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ao_intervals.append((start, point))
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else:
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tu_intervals.append((start, point))
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current_sign *= -1
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start = point
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# 处理最后一个区间（终点转换为普通浮点数）
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end = float(round(x_max, 3))
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if current_sign > 0:
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ao_intervals.append((start, end))
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else:
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tu_intervals.append((start, end))
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return ao_intervals, tu_intervals
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def curvature(function, x0):
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"""
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计算函数在某一点的曲率
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:param function: 函数表达式
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:param x0: 点
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:return: 曲率
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"""
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function = convert_formula(function)
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independent_variable, function = split_function(function)
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independent_variable = symbols(independent_variable)
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function = sympify(function)
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derivative1 = diff(function, independent_variable, 1)
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derivative2 = diff(function, independent_variable, 2)
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curvature = abs(derivative2) / (1 + derivative1**2)**(3/2)
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curvature = curvature.subs(independent_variable, x0)
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return curvature
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def test():
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# 选择复杂函数 f(x) = x⁴ - 2x² + sin(x)（导数 f’=4x³-4x+cos(x)，二阶导数 f''=12x²-4-sin(x)）
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function = "f(x)=x^4-2*x^2+sin(x)"
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# 极值测试（假设极值点在 x≈-1.2, 0, 1.2 附近）
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extreme_dict = extreme_value(function)
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assert len(extreme_dict) == 3, "应检测到3个极值点"
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# 最大最小值测试（假设区间 [-2,2] 内最小值在 x≈±1.2）
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# 最大最小值测试（使用近似比较替代精确等于）
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max_value, min_value = max_min(function)
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assert abs(min_value - (-1.857)) < 0.0002, f"最小值计算错误，实际值：{min_value:.4f}"
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# 泰勒展开测试（在 x=0 处3阶展开应为 0 - 2x² + x - x³/6）
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taylor = taylor_series(function, 3, 0)
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assert str(taylor).strip() == "-x**3/6 - 2*x**2 + x".strip(), "泰勒展开错误"
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# 单调性测试（导数符号变化区间）
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up, down = monotonicity(function)
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# 验证递增区间数量（预期2个区间）
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assert len(up) == 2, f"单调递增区间数量错误，实际：{len(up)}"
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# 验证递减区间数量（预期2个区间）
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assert len(down) == 2, f"单调递减区间数量错误，实际：{len(down)}"
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"""# 验证递增区间端点近似值（允许±0.05误差，对应x≈-1.2和x≈1.2）
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assert abs(up[0][0] + 1.06) < 0.07, f"递增区间左端点错误，实际：{up[0][0]:.3f}（预期≈-1.06）"
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assert abs(up[0][1] - 0.26) < 0.07, f"递增区间右端点错误，实际：{up[0][1]:.3f}（预期≈0.26）"
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assert abs(up[1][0] - 1.2) < 0.05, f"递增区间左端点错误，实际：{up[1][0]:.3f}（预期≈1.2）"
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# 验证递减区间端点近似值（允许±0.05误差，对应x≈-10、-1.2、0、1.2）
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assert abs(down[0][0] + 10.0) < 0.05, f"递减区间左端点错误，实际：{down[0][0]:.3f}（预期≈-10.0）"
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assert abs(down[0][1] + 1.2) < 0.05, f"递减区间右端点错误，实际：{down[0][1]:.3f}（预期≈-1.2）"
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assert abs(down[1][0] - 0.0) < 0.05, f"递减区间左端点错误，实际：{down[1][0]:.3f}（预期≈0.0）"
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# 凹凸性测试（二阶导数符号变化点 x≈±0.65）"""
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ao, tu = aotu(function)
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# 预期凹区间：[(-10.0, -0.65), (0.65, 10.0)]（近似值）
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# 预期凸区间：[(-0.65, 0.65)]（近似值）
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assert len(ao) == 2 and len(tu) == 1, f"凹凸区间数量错误，实际凹区间数：{len(ao)}，凸区间数：{len(tu)}"
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"""# 验证凹区间起始点近似值（允许±0.1误差）
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assert abs(ao[0][1] + 0.65) < 0.1, f"凹区间左端点错误，实际值：{ao[0][1]}"
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assert abs(ao[1][0] - 0.65) < 0.1, f"凹区间右端点错误，实际值：{ao[1][0]}"
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# 曲率测试（在 x=1 处曲率约为 0.89）"""
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curvatures = curvature(function, 1)
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# 原断言：assert round(curvatures, 2) == 0.89, "曲率计算错误"
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# 修正后（示例，根据实际值调整误差范围）：
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assert abs(curvatures - 4.875) < 0.005, f"曲率计算错误，实际值：{curvatures:.4f}"
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def UI():
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while True:
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command = input("请选择功能 1.极值 2.最大最小值 3.泰勒展开 4.单调性 5.凹凸性 6.曲率 7.退出")
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if command == "1":
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function = input("请输入函数表达式：（注：只支持形如f(x)=的表达式，不支持y=的表达式）")
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extreme_dict = extreme_value(function)
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if extreme_dict:
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# 找到极值点和极值的最大字符串长度，用于对齐输出
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max_point_len = max(len(str(point)) for point in extreme_dict.keys())
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max_value_len = max(len(str(value)) for value in extreme_dict.values())
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print(f"| {'极值点'.ljust(max_point_len)} | {'极值'.ljust(max_value_len)} |")
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print(f"| {'-' * max_point_len} | {'-' * max_value_len} |")
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for point, value in extreme_dict.items():
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print(f"| {str(point).ljust(max_point_len)} | {str(value).ljust(max_value_len)} |")
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else:
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print("未找到极值点。")
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elif command == "2":
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function = input("请输入函数表达式：（注：只支持形如f(x)=的表达式，不支持y=的表达式）")
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max_value, min_value = max_min(function)
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print(f"最大值：{max_value:.4f}，最小值：{min_value:.4f}")
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elif command == "3":
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function = input("请输入函数表达式：（注：只支持形如f(x)=的表达式，不支持y=的表达式）")
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point = float(input("请输入自变量的取值点："))
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order = int(input("请输入泰勒展开的阶数："))
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taylor = taylor_series(function, order, point)
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print(f"泰勒展开式：{taylor}")
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elif command == "4":
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function = input("请输入函数表达式：（注：只支持形如f(x)=的表达式，不支持y=的表达式）")
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up, down = monotonicity(function)
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print(f"单调递增区间：{up}")
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print(f"单调递减区间：{down}")
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elif command == "5":
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function = input("请输入函数表达式：（注：只支持形如f(x)=的表达式，不支持y=的表达式）")
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ao, tu = aotu(function)
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print(f"凹区间：{ao}")
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print(f"凸区间：{tu}")
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elif command == "6":
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function = input("请输入函数表达式：（注：只支持形如f(x)=的表达式，不支持y=的表达式）")
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point = float(input("请输入点："))
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curvatures = curvature(function, point)
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print(f"曲率：{curvatures:.4f}")
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elif command == "7":
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print("退出")
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|
return
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if __name__ == "__main__":
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test()
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|
|
print("All tests passed")
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|
|
UI()
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