编程

Wang Haihua

🍈 🍉🍊 🍋 🍌

---

Write raw LaTeX or other formats here, for use with nbconvert. It will not be rendered in the notebook. When passing through nbconvert, a Raw Cell's content is added to the output unmodified.

插值是离散函数逼近的重要方法，利用它可通过函数在有限个点处的取值状况，估算出函数在其他点处的近似值。

早在6世纪，中国的刘焯已将等距二次插值用于天文计算。17世纪之后，I.牛顿，J.-L.拉格朗日分别讨论了等距和非等距的一般插值公式。在近代，插值法仍然是数据处理和编制函数表的常用工具，又是数值积分、数值微分、非线性方程求根和微分方程数值解法的重要基础，许多求解计算公式都是以插值为基础导出的。

现在已知数据

x = [1,2,3,4,5,6]
y = [300,500,800,1300,3000,5000]

绘制函数图像：

import matplotlib.pyplot as plt # 导入matplotlib库中的pyplot模块，并命名为plt
%matplotlib inline

x = [1,2,3,4,5,6]
y = [300,500,800,1300,3000,5000]

plt.scatter(x,y) # 绘制散点图

我们希望对x，y关系进行插值，scipy中一维插值函数的调用方式为： interp1d(x, y, kind='linear', axis=-1, copy=True, bounds_error=None, fill_value=nan, assume_sorted=False)

其中:

• x: 自变量数据
• y: 因变量数据
• kind：插值类型，可选的插值类型包括：
  • 'linear'：线性插值
  • 'nearest'：最近点插值
  • 'nearest-up'： 最近点插值（取上）
  • 'zero'：零插值
  • 'quadratic'：二阶样条插值
  • 'cubic'：三阶样条插值

等。

接下来我们来看一下这几种插值形式的样子：

• 拟合函数

  from scipy.interpolate import interp1d
  func_linear = interp1d(x,y,'linear')
  func_nearest = interp1d(x,y,'nearest')
  func_nearest_up = interp1d(x,y,'nearest-up')
  func_zero = interp1d(x,y,'zero')
  func_quadratic = interp1d(x,y,'quadratic')
  func_cubic = interp1d(x,y,'cubic')
  

• 求插值

import numpy as np
x1 = np.linspace(1,6,100)
y_linear = func_linear(x1)
y_nearest = func_nearest(x1)
y_nearest_up = func_nearest_up(x1)
y_zero = func_zero(x1)
y_quadratic = func_quadratic(x1)
y_cubic = func_cubic(x1)

• 比较图像

plt.figure(figsize=(20,10))
plt.subplot(2,3,1)
plt.scatter(x,y,label='raw data')
plt.plot(x1,y_linear,label='linear',color='red')
plt.legend()

plt.subplot(2,3,2)
plt.scatter(x,y,label='raw data')
plt.plot(x1,y_nearest,label='nearest',color='red')
plt.legend()

plt.subplot(2,3,3)
plt.scatter(x,y,label='raw data')
plt.plot(x1,y_nearest_up,label='nearest_up',color='red')
plt.legend()

plt.subplot(2,3,4)
plt.scatter(x,y,label='raw data')
plt.plot(x1,y_zero,label='zero',color='red')
plt.legend()

plt.subplot(2,3,5)
plt.scatter(x,y,label='raw data')
plt.plot(x1,y_quadratic,label='quadratic',color='red')
plt.legend()

plt.subplot(2,3,6)
plt.scatter(x,y,label='raw data')
plt.plot(x1,y_cubic,label='cubic',color='red')
plt.legend()

plt.savefig('images/sci0202.png')

图像为：

---

import matplotlib.pyplot as plt # 导入matplotlib库中的pyplot模块，并命名为plt
%matplotlib inline
import numpy as np

x = [1,2,3,4,5,6]
y = [300,500,800,1300,3000,5000]

plt.scatter(x,y) # 绘制散点图
plt.savefig('images/sci0201.png')

#help(interp1d)

from scipy.interpolate import interp1d

func_linear = interp1d(x,y,'linear')
func_nearest = interp1d(x,y,'nearest')
func_nearest_up = interp1d(x,y,'nearest-up')
func_zero = interp1d(x,y,'zero')
func_quadratic = interp1d(x,y,'quadratic')
func_cubic = interp1d(x,y,'cubic')

import numpy as np
x1 = np.linspace(1,6,100)
y_linear = func_linear(x1)
y_nearest = func_nearest(x1)
y_nearest_up = func_nearest_up(x1)
y_zero = func_zero(x1)
y_quadratic = func_quadratic(x1)
y_cubic = func_cubic(x1)

plt.figure(figsize=(20,10))
plt.subplot(2,3,1)
plt.scatter(x,y,label='raw data')
plt.plot(x1,y_linear,label='linear',color='red')
plt.legend()

plt.subplot(2,3,2)
plt.scatter(x,y,label='raw data')
plt.plot(x1,y_nearest,label='nearest',color='red')
plt.legend()

plt.subplot(2,3,3)
plt.scatter(x,y,label='raw data')
plt.plot(x1,y_nearest_up,label='nearest_up',color='red')
plt.legend()

plt.subplot(2,3,4)
plt.scatter(x,y,label='raw data')
plt.plot(x1,y_zero,label='zero',color='red')
plt.legend()

plt.subplot(2,3,5)
plt.scatter(x,y,label='raw data')
plt.plot(x1,y_quadratic,label='quadratic',color='red')
plt.legend()

plt.subplot(2,3,6)
plt.scatter(x,y,label='raw data')
plt.plot(x1,y_cubic,label='cubic',color='red')
plt.legend()

plt.savefig('images/sci0202.png')

---