【数值分析】多项式二次样条插值 -Python实现

本篇为数值分析课程代码实现-二次样条插值的实现

仅供参考

配置环境

• Python3.6
• matplotlib
• numpy

• # -*- coding: utf-8 -*-
  """
  Created on Tue Dec 11 21:07:59 2018
  样条插值（二次）
  @author: hhuaf
  """
  import matplotlib.pyplot as plt
  import numpy as np
  
  #可以显示中文
  plt.rcParams["font.sans-serif"] = ["SimHei"]
  plt.rcParams['axes.unicode_minus'] = False
  
  # 设置风格
  plt.style.use('ggplot')
  
  class QuadraticSplineinter():
  
      def __init__(self):
          self.x_train=None#训练数据
          self.X = None#系数矩阵
          self.Y = None#f(x)
          self.W = None#系数求值
          self.N = None#区间个数
      def fit(self, x, y):#0-n
          self.x_train = x
          self.N = len(x)-1
          self.X = np.zeros([3*(self.N)-1,3*(self.N)])
          self.Y = np.zeros([3*(self.N)-1,1])
          j=0
  
          #1-n-1 前2n-2个
          for i in range(self.N-1):
              self.Y[2*i]=y[i+1]
              self.Y[2*i+1]=y[i+1]
  #            label_y=i*2+1
              if j>1:#两个多项式相邻时候使用使对齐
                  j-=1
              for p in range(2):
                  self.X[2*i+p,3*j] = x[i+1]**2
                  self.X[2*i+p,3*j+1] = x[i+1]
                  self.X[2*i+p,3*j+2] = 1
                  j+=1
          #填充外点2个
          self.Y[2*self.N-2] = y[0]
          self.X[2*self.N-2,0] = x[0]**2
          self.X[2*self.N-2,1] = x[0]
          self.X[2*self.N-2,2] = 1  
  
          self.Y[2*self.N-1] = y[-1]
          self.X[2*self.N-1,-3] = x[-1]**2
          self.X[2*self.N-1,-2] = x[-1]
          self.X[2*self.N-1,-1] = 1       
          #填充后n-1个，一介导0-2n-1已经填充
          #Y在这一段为0
          j=0
          for i in range(self.N-1):
              self.X[i+2*self.N,3*j+0]=x[i+1]*2
              self.X[i+2*self.N,3*j+1]=1
           #   X[i+2*N,3*j+2]=0
              self.X[i+2*self.N,3*j+3]=-x[i+1]*2
              self.X[i+2*self.N,3*j+4]=-1
              j+=1
  
          self.X = np.mat(self.X[:,1:])
          self.Y = np.mat(self.Y)
          self.W = np.array(self.X.I*self.Y).reshape(-1)
      
      def predict(self, X_pre):
          if type(X_pre) != np.ndarray:
              X_pre = np.array(X_pre)
  
          def fun(x):
              for i in range(self.N):
                  if self.x_train[i] <= x and x <= self.x_train[i+1]:
                      if i==0:#第一条直线
                          return self.W[0]*x+self.W[1]
                      else:#之后为二次
  
                          return self.W[i*3-1]*x**2+self.W[i*3]*x+self.W[i*3+1]
          return np.array([fun(i) for i in X_pre])
  
  if __name__=="__main__":   
      # 原始数据
      x=[3,4.5,6,7,9,12]
      
      y=[2.5,1,2.5,1,0.5,2.1]
      # X_pre要求必须在训练的x范围内
      X_pre=np.linspace(x[0],x[-1],100)
      
      model = QuadraticSplineinter()
      model.fit(x,y)
      y_pre = model.predict(X_pre)
      
      fig = plt.figure(figsize = (6, 8))
      plt.subplot(2,1,1)
      plt.xlabel('X')
      plt.ylabel('Y')
      plt.title('原散点图像')
      plt.scatter(x,y)
      
      print("参数")
      print(model.W)
      plt.subplot(2,1,2)
      plt.xlabel('X')
      plt.ylabel('Y')
      plt.title('二次样条插值图像')
      plt.scatter(x,y,c='black')
      plt.plot(X_pre,y_pre)
      plt.show()

  欢迎大家批评指正！