python用牛顿插值法实现缺失值的插值

牛顿插值法是一种多项式插值方法，可以用于缺失值的插值。具体步骤如下：

1. 确定已知数据点的横坐标和纵坐标，将其分别存放在数组x和数组y中。

2. 计算差商表：

def diff_quotient(x, y):
    n = len(x)
    table = [[0] * n for i in range(n)]
    for i in range(n):
        table[i][0] = y[i]
    for j in range(1, n):
        for i in range(n - j):
            table[i][j] = (table[i+1][j-1] - table[i][j-1]) / (x[i+j] - x[i])
    return table

3. 计算插值多项式的系数：

def newton_coefficient(x, y):
    n = len(x)
    table = diff_quotient(x, y)
    coeff = [0] * n
    for i in range(n):
        coeff[i] = table[0][i]
    return coeff

4. 用插值多项式计算缺失值：

def newton_interpolation(x, y, x_missing):
    n = len(x)
    coeff = newton_coefficient(x, y)
    res = coeff[-1]
    for i in range(n-2, -1, -1):
        res = res * (x_missing - x[i]) + coeff[i]
    return res

完整代码如下：

def diff_quotient(x, y):
    n = len(x)
    table = [[0] * n for i in range(n)]
    for i in range(n):
        table[i][0] = y[i]
    for j in range(1, n):
        for i in range(n - j):
            table[i][j] = (table[i+1][j-1] - table[i][j-1]) / (x[i+j] - x[i])
    return table

def newton_coefficient(x, y):
    n = len(x)
    table = diff_quotient(x, y)
    coeff = [0] * n
    for i in range(n):
        coeff[i] = table[0][i]
    return coeff

def newton_interpolation(x, y, x_missing):
    n = len(x)
    coeff = newton_coefficient(x, y)
    res = coeff[-1]
    for i in range(n-2, -1, -1):
        res = res * (x_missing - x[i]) + coeff[i]
    return res

x = [1, 2, 4, 5, 7, 9]
y = [6, 2, 7, 8, 3, 5]
x_missing = 3
y_missing = newton_interpolation(x, y, x_missing)
print('Missing value at x =', x_missing, 'is', y_missing)