### 使用 Python 标准库实现龙贝格积分法 以下是通过纯 Python 标准库编写的龙贝格积分法程序，无需任何第三方库（如 NumPy）。该程序允许用户输入被积函数并定义积分区间和精度要求。 #### 动态解析用户输入的被积函数 由于无法直接使用 `eval()` 或者类似的危险操作来解析用户的数学表达式，因此可以借助字符串替换的方式手动构造匿名函数。这种方法既安全又简单易懂[^1]。 ```python def create_function(expression): """ Create a lambda function based on user-provided string expression """ safe_dict = {"sin": math.sin, "cos": math.cos, "tan": math.tan, "exp": math.exp, "log": math.log, "sqrt": math.sqrt} try: func = eval(f"lambda x: {expression}", {"math": math}, safe_dict) test_val = func(0) # Test evaluation at zero point. except Exception as e: raise ValueError("Invalid input function.") from e return func ``` #### 构建核心龙贝格积分逻辑 按照经典的龙贝格表格结构递归计算各阶逼近值，当相邻两层之间的差值小于设定容忍度时终止循环返回结果[^2]。 ```python def romberg_integration(func, a, b, tol=1e-8): """ Compute definite integral via Romberg Integration Algorithm within given tolerance""" max_iter = 20 # Limit number of iterations to prevent infinite loops. r_old = [(b - a)/2 * (func(a) + func(b))] for j in range(1, max_iter): h_j = (b - a)/(2 ** j) sum_term = sum([func(a + k*h_j) for k in range(1, 2**(j-1)+1)]) r_new = [r_old[0]/2 + h_j*sum_term] factor = 4 for i in range(1, j+1): next_r = r_new[-1] + (r_new[-1] - r_old[i-1])/(factor - 1) r_new.append(next_r) factor *= 4 if abs(r_new[j] - r_old[j-1]) < tol and j >= 2: return r_new[j], abs(r_new[j] - r_old[j-1]) r_old = r_new[:] raise RuntimeError("Romberg integration failed after maximum iteration.") ``` #### 统一入口与交互流程控制 将以上两个部分结合起来形成完整的解决方案，同时增加异常捕获机制提升用户体验[^3]。 ```python if __name__ == '__main__': import sys print("Welcome to the Romberg Integrator!") try: expr = input("Please provide your integrand as an algebraic formula involving 'x': ").strip() lower_limit = float(input("Define the start boundary of the region: ")) upper_limit = float(input("Set end boundary correspondingly: ")) precision = float(input("Determine acceptable margin error(defaults to machine epsilon): ") or sys.float_info.epsilon) custom_func = create_function(expr) result, err_estim = romberg_integration(custom_func, lower_limit, upper_limit, precision) print(f"\nThe computed area under curve equals approximately {result:.7g}") print(f"With estimated absolute error less than {err_estim:.7g}\n") except KeyboardInterrupt: print("\nOperation canceled by user.\nExiting...") except BaseException as excptn: print(f"An unexpected issue occurred:\n{excptn.__class__.__name__}: {str(excptn)}\nTerminating program now.") ``` ---