数值分析之牛顿插值方法详解

数据点通用格式

$x_{0}$	$x_{1}$	$x_{2}$	$\dots$	$x_{n}$
$y_{0}$	$y_{1}$	$y_{2}$	$\dots$	$y_{n}$

算法描述

数据点有 $n + 1$ 个
设置常系数项 $n + 1$ 个，即:

[a_{0}, a_{1}, a_{2}, \dots, a_{n - 2}, a_{n_{1}}, a_{n}]

设置系数函数

P_{n} (x)

，如下：

P_{n} (x) = a_{0} + (x - x_{0}) a_{1} + (x - x_{0}) (x - x_{2}) a_{2} + \dots + (x - x_{0}) (x - x_{1}) \dots (x - x_{n - 1}) a_{n}

利用合并同类项找出规律：
假设

n = 3

，则

P_{3} (x)

的表达式如下：

\begin{array}{rcl} P_{3} (x) & = & a_{0} + (x - x_{0}) a_{1} + (x - x_{0}) (x - x_{1}) a_{2} + (x - x_{0}) (x - x_{1}) (x - x_{2}) a_{3} \\ = & a_{0} + (x - x_{0}) {a_{1} + (x - x_{1}) [a_{2} + (x - x_{2}) a_{3}]} . \end{array}

于是：假设

P_{0} (x) = a_{3}

P_{1} (x) = a_{2} + (x - x_{2}) P_{0} (x)

P_{2} (x) = a_{1} + (x - x_{1}) P_{1} (x)

P_{3} (x) = a_{0} + (x - x_{0}) P_{2} (x)

将其推广到一般情况：
$P_{0} (x) = a_{n}$
$P_{1} (x) = a_{n - 1} + (x - x_{n - 1}) P_{0} (x)$
$P_{2} (x) = a_{n - 2} + (x - x_{n - 2}) P_{1} (x)$
$P_{3} (x) = a_{n - 3} + (x - x_{n - 3}) P_{2} (x)$
$P_{4} (x) = a_{n - 4} + (x - x_{n - 4}) P_{3} (x)$
$\dots$
$P_{k} (x) = a_{n - k} + (x - x_{n - k}) P_{k - 1} (x)$
$\dots$
$P_{n - 2} (x) = a_{2} + (x - x_{2}) P_{n - 3} (x)$
$P_{n - 1} (x) = a_{1} + (x - x_{1}) P_{n - 2} (x)$
$P_{n} (x) = a_{0} + (x - x_{0}) P_{n - 1} (x)$
充分利用公式： $y_{i} = P_{n} (x_{i})$ ，使得多项式穿过每一个点，利用数据，最终求得每一个常系数 $a_{i}$ 。
$y_{0} = P_{n} (x_{0}) = a_{0}$
$y_{1} = P_{n} (x_{1}) = a_{0} + (x_{1} - x_{0}) P_{n - 1} (x_{1}) = a_{0} + (x_{1} - x_{0}) (a_{1} + (x_{1} - x_{1}) P_{n - 2} (x_{1}))$
$y_{2} = P_{n} (x_{2}) = a_{0} + (x_{2} - x_{0}) a_{1} + (x_{2} - x_{0}) (x_{2} - x_{1}) a_{2}$
$\dots$
$y_{n - 2} = P_{n} (x_{n - 2}) = a_{0} + (x_{n - 2} - x_{0}) a_{1} + \dots + (x_{n - 2} - x_{0}) (x_{n - 2} - x_{1}) \dots (x_{n - 2} - x_{n - 3}) a_{n - 2}$
$y_{n - 1} = P_{n} (x_{n - 1}) = a_{0} + (x_{n - 1} - x_{0}) a_{1} + \dots + (x_{n - 1} - x_{0}) (x_{n - 1} - x_{1}) \dots (x_{n - 1} - x_{n - 2}) a_{n - 1}$
$y_{n} = P_{n} (x_{n}) = a_{0} + (x_{n} - x_{0}) a_{1} + \dots + (x_{n} - x_{0}) (x_{n} - x_{1}) \dots (x_{n} - x_{n - 1}) a_{n}$
知识点补充Introducing the divided differences
$▽ y_{i} = \frac{y_{i} - y_{0}}{x_{i} - x_{0}}$ ， $i = 1, 2, \dots, n$
$▽^{2} y_{i} = \frac{▽ y_{i} - ▽ y_{1}}{x_{i} - x_{1}}$ ， $i = 2, 3, \dots, n$
$▽^{3} y_{i} = \frac{▽^{2} y_{i} - ▽^{2} y_{2}}{x_{i} - x_{2}}$ ， $i = 3, 4, \dots, n$
$\dots$
$▽^{n} y_{n} = \frac{▽^{n - 1} y_{n} - ▽^{n - 1} y_{n - 1}}{x_{n} - x_{n - 1}}$ ， $n$
求常系数 $a_{i}$ ，
$a_{0} = y_{0}$ ， $a_{1} = ▽ y_{1}$ ， $a_{2} = ▽^{2} y_{2}$ ， $\dots$ ， $a_{n} = ▽^{n} y_{n}$
列表分析：当 $n = 4$ 时

$x_{0}$	$y_{0}$
$x_{1}$	$y_{1}$	$▽ y_{1}$
$x_{2}$	$y_{2}$	$▽ y_{2}$	$▽^{2} y_{2}$
$x_{3}$	$y_{3}$	$▽ y_{3}$	$▽^{2} y_{3}$	$▽^{3} y_{3}$
$x_{4}$	$y_{4}$	$▽ y_{4}$	$▽^{2} y_{4}$	$▽^{3} y_{4}$	$▽^{4} y_{4}$

牛顿插值算法Python代码

import numpy as np
def coeffts(xData,yData):
    m = len(xData)
    Table = np.zeros((m,m))
    Table[:,0] = np.array(yData)
    for k in range(1,m):
        for i in range(k,m):
            Table[i,k] = (Table[i,k-1]-Table[k-1,k-1])/(xData[i]-xData[k-1])
    return np.diag(Table)


def evalPoly(a,xData,x):
    n = len(xData) - 1 # Degree of polynomial
    p = a[n]
    for k in range(1,n+1):
        p = a[n-k] + (x -xData[n-k])*p
    return p

案例分析

import numpy as np
xData = np.array([0.15,2.3,3.15,4.85,6.25,7.95])
yData = np.array([4.79867,4.49013,4.2243,3.47313,2.66674,1.51909])
a = coeffts(xData,yData)
print(" x    yInterp")
print('-'*20)
for x in np.arange(0.0,8.1,0.5):
    y = evalPoly(a,xData,x)
    print('{:3.1f} {:9.5f}'.format(x,y))

插值结果：