数据插值：Lagrange插值方法

数据插值：Lagrange插值方法 数学理论基础

设 y = f ( x ) y = f(x) y=f(x)是区间 [ a , b ] [a,b] [a,b]上的实值函数， x 0 , x 1 , ⋯   , x n x_0,x_1,\cdots,x_n x0​,x1​,⋯,xn​是 [ a , b ] [a,b] [a,b]上的 n + 1 n + 1 n+1个互异节点， l i ( x ) = ∏ j = 0 , j ≠ i n ( x − x j ) ∏ j = 0 , j ≠ i n ( x i − x j ) l_i(x)=\frac{\prod_{j = 0,j\neq i}^{n}(x - x_j)}{\prod_{j = 0,j\neq i}^{n}(x_i - x_j)} li​(x)=∏j=0,j=in​(xi​−xj​)∏j=0,j=in​(x−xj​)​ ( i = 0 , 1 , ⋯   , n ) (i = 0,1,\cdots,n) (i=0,1,⋯,n)是拉格朗日插值基函数。

f ( x ) = ∑ i = 0 n y i l i ( x ) + f ( n + 1 ) ( ξ ) ( n + 1 ) ! ∏ i = 0 n ( x − x i ) f(x)=\sum_{i=0}^{n}y_i l_i(x)+\frac{f^{(n+1)}(\xi)}{(n+1)!} \prod_{i=0}^{n}{(x-x_i)} f(x)=i=0∑n​yi​li​(x)+(n+1)!f(n+1)(ξ)​i=0∏n​(x−xi​)

代码模板 1.插值预测值 import numpy as np def lagrange_interpolation(x_nodes, y_nodes, x): """ 实现拉格朗日插值方法 :param x_nodes: 插值节点的x坐标 :param y_nodes: 插值节点的y坐标 :param x: 要计算插值的点的x坐标 :return: 在点x处的插值结果 """ n = len(x_nodes) result = 0 for i in range(n): l_i = 1 for j in range(n): if j != i: l_i *= (x - x_nodes[j]) / (x_nodes[i] - x_nodes[j]) result += y_nodes[i] * l_i return result 2.插值预测函数表达式 import sympy as sp def lagrange_polynomial(x_nodes, y_nodes): # 定义符号变量 x x = sp.symbols('x') n = len(x_nodes) # 初始化拉格朗日插值多项式为 0 polynomial = 0 for i in range(n): # 初始化拉格朗日插值基函数为 1 l_i = 1 for j in range(n): if j != i: # 计算拉格朗日插值基函数的每一项 l_i *= (x - x_nodes[j]) / (x_nodes[i] - x_nodes[j]) # 累加每一项 y_i * l_i(x) 到多项式中 polynomial += y_nodes[i] * l_i # 展开多项式 polynomial = sp.expand(polynomial) return polynomial 可视化插值函数与数据关系 import sympy as sp import numpy as np import matplotlib.pyplot as plt def lagrange_polynomial(x_nodes, y_nodes): # 定义符号变量 x x = sp.symbols('x') n = len(x_nodes) # 初始化拉格朗日插值多项式为 0 polynomial = 0 for i in range(n): # 初始化拉格朗日插值基函数为 1 l_i = 1 for j in range(n): if j != i: # 计算拉格朗日插值基函数的每一项 l_i *= (x - x_nodes[j]) / (x_nodes[i] - x_nodes[j]) # 累加每一项 y_i * l_i(x) 到多项式中 polynomial += y_nodes[i] * l_i # 展开多项式 polynomial = sp.expand(polynomial) return polynomial # 示例数据 x_nodes = [1, 2, 3, 4] y_nodes = [2, 4, 1, 5] # 计算拉格朗日插值多项式的解析式 poly = lagrange_polynomial(x_nodes, y_nodes) print("拉格朗日插值多项式的解析式为:", poly) # 将符号表达式转换为可调用的函数 x = sp.symbols('x') poly_func = sp.lambdify(x, poly, 'numpy') # 生成用于绘制插值函数的 x 值 x_vals = np.linspace(min(x_nodes) - 1, max(x_nodes) + 1, 400) y_vals = poly_func(x_vals) # 绘制原始数据点 plt.scatter(x_nodes, y_nodes, color='red', label='Original Data') # 绘制插值函数曲线 plt.plot(x_vals, y_vals, color='blue', label='Lagrange Interpolation') # 设置图表标题和坐标轴标签 plt.title('Lagrange Interpolation Visualization') plt.xlabel('x') plt.ylabel('y') # 显示图例 plt.legend() # 显示网格线 plt.grid(True) # 显示图形 plt.show() 不计算基插函数情况下的Neville算法 import numpy as np import matplotlib.pyplot as plt def neville_interpolation_optimized(x_nodes, y_nodes, x): n = len(x_nodes) # 只使用一维数组来存储中间结果 P = y_nodes.copy() for j in range(1, n): for i in range(n - j): numerator = (x - x_nodes[i + j]) * P[i] - (x - x_nodes[i]) * P[i + 1] denominator = x_nodes[i] - x_nodes[i + j] P[i] = numerator / denominator return P[0] # 示例数据 x_nodes = [1, 2, 3, 4] y_nodes = [2, 4, 1, 5] # 生成用于绘制插值函数的 x 值 x_vals = np.linspace(min(x_nodes) - 1, max(x_nodes) + 1, 400) y_vals = [] for x in x_vals: y = neville_interpolation_optimized(x_nodes, y_nodes, x) y_vals.append(y) # 绘制原始数据点 plt.scatter(x_nodes, y_nodes, color='red', label='Original Data') # 绘制插值函数曲线 plt.plot(x_vals, y_vals, color='blue', label='Neville Interpolation') # 设置图表标题和坐标轴标签 plt.title('Neville Interpolation Visualization') plt.xlabel('x') plt.ylabel('y') # 显示图例 plt.legend() # 显示网格线 plt.grid(True) # 显示图形 plt.show()