插值 ：在离散数据的基础上补插连续函数，使得这条连续曲线通过全部给定的离散数据点。
二、多项式插值
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f(x) simeq a_{0}+a_{1} x+a_{2} x^{2}+cdots+a_{n-1} x^{n-1}, quadleft(x simeq x_{i}right)
f(x)≃a0+a1x+a2x2+⋯+an−1xn−1,(x≃xi).
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left(x_{0}, y_{0}right),left(x_{1}, y_{1}right)
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y=frac{x-x_{1}}{x_{0}-x_{1}} y_{0}+frac{x-x_{0}}{x_{1}-x_{0}} y_{1}
y=x0−x1x−x1y0+x1−x0x−x0y1
将上式改写成
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quad L_{1}(x)=l_{0}(x) y_{0}+l_{1}(x) y_{1}
L1(x)=l0(x)y0+l1(x)y1
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begin{array}{lll}l_{0}(x), l_{1}(x)text{满足：}& l_{0}left(x_{0}right)=1, & l_{0}left(x_{1}right)=0 \ & l_{1}left(x_{0}right)=0, & l_{1}left(x_{1}right)=1end{array}
l0(x),l1(x)满足：l0(x0)=1,l1(x0)=0,l0(x1)=0l1(x1)=1
是线性揷值函数的基函数，这种插值称为拉格朗日插值.
利用3点构造二次拉格朗日揷值多项式:
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L_{2}(x)=l_{0}(x) y_{0}+l_{1}(x) y_{1}+l_{2}(x) y_{2}
L2(x)=l0(x)y0+l1(x)y1+l2(x)y2 是通过
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left(x_{0}, y_{0}right),left(x_{1}, y_{1}right),left(x_{2}, y_{2}right)
(x0,y0),(x1,y1),(x2,y2)三点的抛物线.则二次拉格朗日插值基函数：
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left{begin{array}{l} l_{0}(x)=frac{left(x-x_{1}right)left(x-x_{2}right)}{left(x_{0}-x_{1}right)left(x_{0}-x_{2}right)} \ \ l_{1}(x)=frac{left(x-x_{0}right)left(x-x_{2}right)}{left(x_{1}-x_{0}right)left(x_{1}-x_{2}right)} \ \ l_{2}(x)=frac{left(x-x_{0}right)left(x-x_{1}right)}{left(x_{2}-x_{0}right)left(x_{2}-x_{1}right)} end{array}right.
⎩⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎧l0(x)=(x0−x1)(x0−x2)(x−x1)(x−x2)l1(x)=(x1−x0)(x1−x2)(x−x0)(x−x2)l2(x)=(x2−x0)(x2−x1)(x−x0)(x−x1)
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begin{array}{lll} l_{0}left(x_{0}right)=1 & l_{0}left(x_{1}right)=0 & l_{0}left(x_{2}right)=0 \ l_{1}left(x_{0}right)=0 & l_{1}left(x_{1}right)=1 & l_{1}left(x_{2}right)=0 \ l_{2}left(x_{0}right)=0 & l_{2}left(x_{1}right)=0 & l_{2}left(x_{2}right)=1 end{array}
l0(x0)=1l1(x0)=0l2(x0)=0l0(x1)=0l1(x1)=1l2(x1)=0l0(x2)=0l1(x2)=0l2(x2)=1
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n次的拉格朗日插值多项式：
拉格朗日插值多项式：
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L_{n}(x)=sum_{k=0}^{n} l_{k}(x) y_{k}
Ln(x)=∑k=0nlk(x)yk
拉格朗日插值基函数：
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l_{k}(x)=frac{left(x-x_{0}right) cdotsleft(x-x_{k-1}right)left(x-x_{k+1}right) cdotsleft(x-x_{n}right)}{left(x_{k}-x_{0}right) cdotsleft(x_{k}-x_{k-1}right)left(x_{k}-x_{k+1}right) cdotsleft(x_{k}-x_{n}right)}
lk(x)=(xk−x0)⋯(xk−xk−1)(xk−xk+1)⋯(xk−xn)(x−x0)⋯(x−xk−1)(x−xk+1)⋯(x−xn)
或写成： l k ( x ) = ∏ j = 0 , j ≠ k n ( x − x j ) ( x k − x j ) l_{k}(x)=prod_{j=0, j neq k}^{n} frac{left(x-x_{j}right)}{left(x_{k}-x_{j}right)} lk(x)=∏j=0,j=kn(xk−xj)(x−xj)
3. Python代码实现Lagrange插值多项式import numpy as np
import matplotlib.pyplot as plt
def Lagrange_interp(xdata,ydata,x):
'''
Lagrange插值代码,
参数设置：
xdata，ydata：已知数据
x:要插入数据的x坐标
'''
v = np.zeros(x.size) #插值结果数值
n = xdata.size
for k in range(n): #n个Lagrange多项式
w = np.ones(x.size) #记录连乘
for j in range(n):
if j != k:
w *= (x-xdata[j])/(xdata[k]-xdata[j])
v += w*ydata[k] # L_k(x)乘以对应函数值f(x_k)
return v
4. 实例：
import numpy as np
import matplotlib.pyplot as plt
'''
Lagrange插值代码,参数设置：xdata，ydata：已知数据x:要插入数据的x坐标
'''
def Lagrange_interp(xdata,ydata,x):
v = np.zeros(x.size) #插值结果数值
n = xdata.size
for k in range(n): #n个Lagrange多项式
w = np.ones(x.size) #记录连乘
for j in range(n):
if j != k:
w *= (x-xdata[j])/(xdata[k]-xdata[j])
v += w*ydata[k] # L_k(x)乘以对应函数值f(x_k)
return v
x = np.arange(1,7)
y = np.array([16,18,21,17,15,12])
u = np.arange(0.75,6.3,0.05)
v = Lagrange_interp(x,y,u)
plt.plot(x,y,'o',u,v,'-')
结果：
