内插法计算公式是什么：从基础原理到高级应用的全面指南

def linear_interpolation(x1, y1, x2, y2, x):
    """
    计算线性内插值
    :param x1, y1: 第一个已知点
    :param x2, y2: 第二个已知点
    :param x: 目标横坐标
    :return: 估算的纵坐标 y
    """
    if x1 == x2:
        raise ValueError("x1 and x2 cannot be the same")
    y = y1 + (x - x1)  (y2 - y1) / (x2 - x1)
    return y
示例调用
result = linear_interpolation(10, 100, 11, 102, 10.5)
print(f"线性内插结果: {result}")
            

def lagrange_interpolation(x_points, y_points, x):
    """
    计算拉格朗日内插值
    :param x_points: 已知点的横坐标列表
    :param y_points: 已知点的纵坐标列表
    :param x: 目标横坐标
    :return: 估算的纵坐标 y
    """
    n = len(x_points)
    result = 0.0
    for i in range(n):
        basis = 1.0
        for j in range(n):
            if i != j:
                basis = (x - x_points[j]) / (x_points[i] - x_points[j])
        result += y_points[i]  basis
    return result
示例调用
x_pts = [1, 2, 3]
y_pts = [1, 4, 9] # f(x) = x^2
print(f"拉格朗日内插结果 (x=2.5): {lagrange_interpolation(x_pts, y_pts, 2.5)}")