「学习记录」《数值分析》第三章计算实习题（Python语言）

第三题暂缺，之后补充。

import matplotlib.pyplot as plt
import numpy as np
import scipy.optimize as so
import sympy as sp

x = sp.symbols('x')

def calculate(expr_i, expr_j, expr_value,expr_omega):
    ans=0
    for cnt,v in enumerate(expr_value):
        if isinstance(expr_i,(type(x),type(x*x))):
            actual_expr_i=expr_i.subs(x, v[0])
        elif expr_i==1: # which means 1 or 0
            actual_expr_i=1
        else:
            actual_expr_i=v[1]
        if isinstance(expr_j,(type(x),type(x*x))):
            actual_expr_j=expr_j.subs(x, v[0])
        else: # which means 1
            actual_expr_j=1
        if type(expr_omega)==type(1):
            ans=ans+expr_omega*actual_expr_i*actual_expr_j
        else:
            ans=ans+expr_omega[cnt]*actual_expr_i*actual_expr_j

    return ans

def least_squares(_psi,_value,_omega):
    G=np.empty((len(_psi),len(_psi)))
    d=np.empty(len(_psi))
    for i,expr_i in enumerate(_psi):
        for j,expr_j in enumerate(_psi):
            G[i][j]=calculate(expr_i,expr_j,_value,_omega)
        d[i]=calculate(0,_psi[i],_value,_omega)
    a=np.linalg.solve(G, d.T) # Oh, I love solve()!
    ls_f_x=0
    for i,element in enumerate(a):
        ls_f_x=ls_f_x+element*_psi[i]
    return ls_f_x    

psi=[1,x,x*x,x*x*x]
#psi=[1,x]
value=np.loadtxt('input.txt')
omega=1
#omega=[2,1,3,1,1]
ls_f_x=least_squares(psi,value,omega)
print(ls_f_x)
f_x=1/(1+25*x*x)
# p1=sp.plot(f_x,ls_f_x,(x,-1,1))

"""
# using system functions
def func(p,x):
    a0,a1,a2,a3 = p
    return a0+a1*x+a2*x*x+a3*x*x*x
def err(p,x,y):
    return func(p,x)-y
arg_f=(np.array([x[0] for x in value[:,:1]]),np.array([y[0] for y in value[:,1:2]]))
coef_ls = so.leastsq(err, [1,1,1,1], args=arg_f)
print(coef_ls)
system_ls_f_x=0
for i,element in enumerate(coef_ls[0]):
    system_ls_f_x=system_ls_f_x+element*psi[i]
print(system_ls_f_x)
p1=sp.plot(f_x,ls_f_x,system_ls_f_x,(x,-1,1),show=False)
p1[1].line_color='r'
p1[2].line_color='g'
p1.show()
"""

# problem 2:
fig=plt.figure()

value=np.array([[0,1],[0.1,0.41],[0.2,0.50],[0.3,0.61], [0.5,0.91],[0.8,2.02],[1.0,2.46]])

ls_f_x=least_squares(psi,value,omega)
print_f=sp.lambdify(x,ls_f_x,"numpy")
print_x=np.linspace(-1,1,100)
print_y=print_f(print_x)
plt.plot(print_x,print_y,label='x^3')

psi=[1,x,x**2,x**3,x**4]
ls_f_x=least_squares(psi,value,omega)
print_f=sp.lambdify(x,ls_f_x,"numpy")
print_y=print_f(print_x)
plt.plot(print_x,print_y,label='x^4')

psi=[1,x]
ls_f_x=least_squares(psi,value,omega)
print_f=sp.lambdify(x,ls_f_x,"numpy")
print_y=print_f(print_x)
plt.plot(print_x,print_y,label='kx+b')

plt.scatter(np.array([x[0] for x in value[:,:1]]),np.array([y[0] for y in value[:,1:2]]),marker='x',label='data')
plt.legend(loc='best')
plt.savefig('output.svg')