一、实验目的
在已知f(x),x∈[a,b]的表达式，但函数值不便计算,或不知f(x),x∈[a,b]而又需要给出其在[a,b]上的值时，按插值原则f(xi)= yi(i= 0,1…….,n)求出简单函数P(x)(常是多项式),使其在插值基点xi,处成立P(xi)= yi(i=0,1,……,n),而在[a,b]上的其它点处成立f(x)≈P(x).
二、实验原理
三、实验内容
求f(x)=x4在[0,2]上按5个等距节点确定的Hermite插值多项式.
四、实验程序
1 import numpy as np 2 from sympy import * 3 import matplotlib.pyplot as plt 4 5 6 def f(x): 7 return x ** 4 8 9 10 def ff(x): # f[x0, x1, ..., xk] 11 ans = 0 12 for i in range(len(x)): 13 temp = 1 14 for j in range(len(x)): 15 if i != j: 16 temp *= (x[i] - x[j]) 17 ans += f(x[i]) / temp 18 return ans 19 20 21 def draw(L, newlabel= 'Lagrange插值函数'): 22 plt.rcParams['font.sans-serif'] = ['SimHei'] 23 plt.rcParams['axes.unicode_minus'] = False 24 x = np.linspace(0, 2, 100) 25 y = f(x) 26 Ly = [] 27 for xx in x: 28 Ly.append(L.subs(n, xx)) 29 plt.plot(x, y, label='原函数') 30 plt.plot(x, Ly, label=newlabel) 31 plt.xlabel('x') 32 plt.ylabel('y') 33 plt.legend() 34 35 plt.savefig('1.png') 36 plt.show() 37 38 39 def lossCal(L): 40 x = np.linspace(0, 2, 101) 41 y = f(x) 42 Ly = [] 43 for xx in x: 44 Ly.append(L.subs(n, xx)) 45 Ly = np.array(Ly) 46 temp = Ly - y 47 temp = abs(temp) 48 print(temp.mean()) 49 50 51 def calM(P, x): 52 Y = n ** 4 53 dfP = diff(P, n) 54 return solve(Y.subs(n, x[0]) - dfP.subs(n, x[0]), [m,])[0] 55 56 57 if __name__ == '__main__': 58 x = np.array(range(11)) - 5 59 y = f(x) 60 61 n, m = symbols('n m') 62 init_printing(use_unicode=True) 63 64 P = f(x[0]) 65 for i in range(len(x)): 66 if i != len(x) - 1: 67 temp = ff(x[0:i + 2]) 68 else: 69 temp = m 70 for j in x[0:i + 1]: 71 temp *= (n - j) 72 P += temp 73 P = expand(P) 74 75 P = P.subs(m, calM(P, x)) 76 draw(P, newlabel='Hermite插值多项式') 77 lossCal(P)
五、运算结果
