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interpolation.py
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# -*- coding:utf-8 -*-
"""
@author : zhangyifan
@date : 2020-04-05 18:50
@brief : 利用拉格朗日插值、分段线性插值和三�?�样条插值构造插值函�?
"""
import numpy as np
from matplotlib import pyplot as plt
from sympy import *
from matplotlib.font_manager import FontProperties
def function(x):
return 1/(1+15*pow(x,2))
# return abs(x)
class Lagrange:
def __init__(self, interval, n):
self.interval = interval #插值区间
self.num = n #次数
self.h = 2/n #插值点间隔
def cal_wn(self):
xi = np.arange(self.interval[0],self.interval[1]+self.h/2, self.h)
x = symbols('x')
wn = 1
for i in xi:
wn *= (x-i)
return wn
def cal_diff_wn(self):
x = symbols('x')
return diff(self.cal_wn(),x)
def lagrange_expression(self):
Lx = 0
x = symbols('x')
xi = np.arange(self.interval[0],self.interval[1]+self.h/2, self.h)
diff_wn = self.cal_diff_wn()
wn = self.cal_wn()
# print(xi) num个区间 num+1个点
for i in range(self.num+1):
num = wn/(x-xi[i])
li = num/diff_wn.subs('x',xi[i])
Lx += li*function(xi[i])
# print(Lx)
return Lx
def show_result(self):
x = np.linspace(self.interval[0],self.interval[1], 100)
expression = self.lagrange_expression()
print(expression)
y =np.zeros((100,))
for i in range(len(x)):
y[i] = expression.subs('x',x[i])
ax = plt.subplot(222)
ax.set_title("拉格朗日插值n=10",fontproperties=font_set)
plt.plot(x, y)
class PieceLinear:
def __init__(self, interval, n):
self.interval = interval #插值区间
self.num = n #区间个数
self.h = 2/n #插值间隔
def create_point(self):# 给出插值点
x = np.arange(self.interval[0], self.interval[1]+self.h/2, self.h)
y = function(x)
return np.stack((x,y),axis=1)
def subfun(self): #每个区间的子函数
x = symbols('x')
subfuns = []
points = self.create_point()
for i in range(self.num):
# print(i)
subfun_i =simplify(((x-points[i+1,0])*points[i,1]/(points[i,0]-points[i+1,0])+(x-points[i,0])*points[i+1,1]/(points[i+1,0]-points[i,0])))
subfuns.append(subfun_i)
return subfuns
def show_result(self):
points = self.create_point()
# print(points)
draw_y=np.zeros((20,))
subfuns = self.subfun()
for i in range(self.num):
draw_x = np.linspace(points[i, 0], points[i+1, 0],20)
for j in range(len(draw_x)):
draw_y[j] = subfuns[i].subs('x',draw_x[j])
ax1 = plt.subplot(223)
ax1.set_title("分段线性插值n=10",fontproperties=font_set)
plt.plot(draw_x, draw_y)
ax2 = plt.subplot(221)
ax2.set_title("原函数y=f(x)",fontproperties=font_set)
standard_x = np.linspace(-1, 1, 500)
standard_y = function(standard_x)
plt.plot(standard_x, standard_y,'--')
class CubicSpline:
def __init__(self, interval, n):
self.interval = interval #插值区间
self.num = n #区间数
self.h = 2/n #插值间隔
def create_point(self):# 给出插值点
x = np.arange(self.interval[0], self.interval[1]+self.h/2, self.h)
y = function(x)
return np.stack((x,y),axis=1)
def cal_mu_and_lamda(self):#计算出lamda和mu
return 1/2 # 所有的h均相通
def cal_diff2(self, xi): #计算出二阶导
x = Symbol('x')
expr = function(x)
dexpr = diff(expr, x)
return diff(dexpr, x).subs('x', xi)
def end_M(self):
# M0 = self.cal_diff2(self.interval[0])
# Mn = self.cal_diff2(self.interval[1])
# return M0, Mn #第二类补充条件
return 0,0 #自然边界条件
def cal_d(self): #计算结果向量
points = self.create_point()
d = np.zeros((self.num-1, 1))
# 计算g1至gn-1
for i in range(1, self.num):
d[i-1] = 6/(2*self.h*self.h)*(points[i+1, 1]+points[i-1, 1]-2*points[i, 1])
#已知m0和m1，所以修正方程组
M0, Mn=self.end_M()
#print(self.cal_mu_and_lamda()*M0)
d[0] = d[0] - self.cal_mu_and_lamda()*M0
d[-1] = d[-1] - self.cal_mu_and_lamda()*Mn
return d
def cal_M_coff(self): #计算系数矩阵
M_coff = np.zeros((self.num-1, self.num-1))
mu = lamda = self.cal_mu_and_lamda()
for i in range(1, self.num-2):
# M_coff[i, i-1 ,i+2] = [mu, 2, lamda]
M_coff[i,i-1] = self.cal_mu_and_lamda()
M_coff[i,i+1] = self.cal_mu_and_lamda()
M_coff[i,i] = 2
M_coff[0, 0]=M_coff[-1,-1] = 2
M_coff[0, 1] = M_coff[-1, -2] = mu
return M_coff
def cal_M(self):#
M_coff = self.cal_M_coff()
d = self.cal_d()
M = np.linalg.solve(M_coff, d)
# print(M)
M0 = self.end_M()[0]
Mn = self.end_M()[1]
M = np.vstack((M0,M))
M = np.vstack((M,Mn))
return M
def cal_diff2_sx(self):
sx0 = self.end_M()[0]
sxn = self.end_M()[1]
x = Symbol('x')
diff2_sx = [None]*self.num
M = self.cal_M() #0-num-2 对应 M1-Mnum-1
points = self.create_point()
for i in range(0, self.num):
diff2_sx[i] = M[i, 0]*(points[i,0]-x)/(self.h)+M[i+1, 0]*(x-points[i-1,0])/(self.h)
#print(diff2_sx)
return diff2_sx
def cal_sx(self):
x = Symbol('x')
sx = []
M = self.cal_M()
points = self.create_point()
for i in range(1, len(points)): #1-num-1
Ai = 1/self.h*(points[i-1,1]-1/6*M[i-1]*pow(self.h,2))
Bi = 1/self.h*(points[i,1]-1/6*M[i]*pow(self.h,2))
sxi = (M[i-1]*pow(points[i,0]-x, 3)+M[i]*pow(x-points[i-1,0],3))/(6*self.h)+Ai*(points[i,0]-x)+Bi*(x-points[i-1,0])
sx.append(sxi)
print(sx)
return sx
def show_result(self):
sx = self.cal_sx()
points = self.create_point()
draw_y=np.zeros((500,))
for i in range(len(sx)):
draw_x = np.linspace(points[i, 0], points[i+1, 0],500)
for j in range(len(draw_x)):
draw_y[j] = sx[i][0].subs('x',draw_x[j])
ax = plt.subplot(224)
ax.set_title("三次样条插值n=10",fontproperties=font_set)
plt.plot(draw_x, draw_y)
# standard_x = np.linspace(-1, 1, 500)
# standard_y = function(standard_x)
# plt.plot(standard_x, standard_y)
plt.show()
if __name__ == "__main__":
font_set = FontProperties(fname=r"c:\windows\fonts\simsun.ttc", size=12)
fig = plt.figure(figsize=(8,8))
test1 = Lagrange([-1,1], 10)
test1.show_result()
test2 = PieceLinear([-1,1], 10)
test2.show_result()
test3 = CubicSpline([-1,1], 10)
test3.show_result()
plt.show()