达永编程网

在使用MATLAB的solve函数求解非线性方程时，有可能会在求解的结果中出现z（或者z1）和root，无法得出数值结果。今天主要是以实际的例子，用三种方法解决这个问题。

1.实例1

程序

clc;
clear all;
syms x y z u v w;
eqn=[-5*x+20*y+y*z==0,-x-5*y+2*x*z-v==0,-z+2*u-2*x*y==0,-z-5*u+w==0,-y+v-w==0,v-u==0];
var=[x,y,z,u,v,w];
[x,y,z,u,v,w]=solve(eqn,var)

运行结果

 
x =
 
                                                                                                                                                                                                                                                                                                                                                                                                    0
 (1253323104*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 1)^2)/10108417235 - (143782272*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 1)^3)/10108417235 - (64966543683*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 1))/20216834470 - 6753430183/8086733788
 (1253323104*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 2)^2)/10108417235 - (143782272*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 2)^3)/10108417235 - (64966543683*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 2))/20216834470 - 6753430183/8086733788
 (1253323104*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 3)^2)/10108417235 - (143782272*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 3)^3)/10108417235 - (64966543683*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 3))/20216834470 - 6753430183/8086733788
 (1253323104*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 4)^2)/10108417235 - (143782272*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 4)^3)/10108417235 - (64966543683*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 4))/20216834470 - 6753430183/8086733788
 
 
y =
 
                                                                                                                                                                                                                                                                                                                                                                                                   0
 - (132488256*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 1)^2)/10108417235 - (6267264*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 1)^3)/10108417235 - (15779928907*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 1))/20216834470 - 1718896477/8086733788
 - (132488256*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 2)^2)/10108417235 - (6267264*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 2)^3)/10108417235 - (15779928907*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 2))/20216834470 - 1718896477/8086733788
 - (132488256*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 3)^2)/10108417235 - (6267264*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 3)^3)/10108417235 - (15779928907*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 3))/20216834470 - 1718896477/8086733788
 - (132488256*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 4)^2)/10108417235 - (6267264*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 4)^3)/10108417235 - (15779928907*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 4))/20216834470 - 1718896477/8086733788
 
 
z =
 
                                                                                                                                                                                                                                                                                                                                                                                            0
 (132488256*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 1)^2)/2021683447 + (6267264*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 1)^3)/2021683447 - (393538669*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 1))/4043366894 + 8594482385/8086733788
 (132488256*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 2)^2)/2021683447 + (6267264*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 2)^3)/2021683447 - (393538669*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 2))/4043366894 + 8594482385/8086733788
 (132488256*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 3)^2)/2021683447 + (6267264*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 3)^3)/2021683447 - (393538669*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 3))/4043366894 + 8594482385/8086733788
 (132488256*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 4)^2)/2021683447 + (6267264*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 4)^3)/2021683447 - (393538669*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 4))/4043366894 + 8594482385/8086733788
 
 
u =
 
                                                                                                                                                                                                                                                                                                                                                                                                0
 (4436905563*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 1))/20216834470 - (6267264*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 1)^3)/10108417235 - (132488256*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 1)^2)/10108417235 - 1718896477/8086733788
 (4436905563*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 2))/20216834470 - (6267264*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 2)^3)/10108417235 - (132488256*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 2)^2)/10108417235 - 1718896477/8086733788
 (4436905563*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 3))/20216834470 - (6267264*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 3)^3)/10108417235 - (132488256*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 3)^2)/10108417235 - 1718896477/8086733788
 (4436905563*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 4))/20216834470 - (6267264*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 4)^3)/10108417235 - (132488256*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 4)^2)/10108417235 - 1718896477/8086733788
 
 
v =
                                                                                                                                                                                                                                                                                                                                                                                            0
 (4436905563*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 1))/20216834470 - (6267264*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 1)^3)/10108417235 - (132488256*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 1)^2)/10108417235 - 1718896477/8086733788
 (4436905563*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 2))/20216834470 - (6267264*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 2)^3)/10108417235 - (132488256*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 2)^2)/10108417235 - 1718896477/8086733788
 (4436905563*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 3))/20216834470 - (6267264*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 3)^3)/10108417235 - (132488256*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 3)^2)/10108417235 - 1718896477/8086733788
 (4436905563*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 4))/20216834470 - (6267264*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 4)^3)/10108417235 - (132488256*root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 4)^2)/10108417235 - 1718896477/8086733788
 
w =
                                                                                          0
 root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 1)
 root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 2)
 root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 3)
 root(z1^4 - (159*z1^3)/19 + (3349633*z1^2)/14592 + (1397285*z1)/14592 + 4898525/58368, z1, 4)
 

在官方的帮助文档中有找到出现z和root的原因

Numerically Approximating Symbolic Solutions That Contain root
When solving polynomials, solve might return solutions containing root. 
To numerically approximate these solutions, use vpa. Consider the following 
equation and solution.
在求解多项式或者高阶非线性方程时，求解可能返回含有根的解。要用数字近似这些解，请使用vpa()。

方法1：我们可以使用double函数对结果进行数字近似化。

 double 是 MATLAB 中的默认数值数据类型（类），它可为大多数计算任务
 提供足够的精度。数值变量自动存储为 64 位（8 字节）双精度浮点值。
  Y = double(X)

程序

clc;
clear all;
syms x y z u v w;
eqn=[-5*x+20*y+y*z==0,-x-5*y+2*x*z-v==0,-z+2*u-2*x*y==0,-z-5*u+w==0,-y+v-w==0,v-u==0];
var=[x,y,z,u,v,w];
[x,y,z,u,v,w]=solve(eqn,var);
x = double(x)
y = double(y)
z = double(z)
u = double(u)
v = double(v)
w = double(w)

运行结果

x =


   0.0000 + 0.0000i
  -0.1889 + 1.8323i
  -0.1889 - 1.8323i
  -0.1861 - 1.3171i
  -0.1861 + 1.3171i




y =


   0.0000 + 0.0000i
  -0.0434 + 0.4350i
  -0.0434 - 0.4350i
   0.5960 +11.6818i
   0.5960 -11.6818i




z =


   0.0000 + 0.0000i
   1.0663 + 0.0706i
   1.0663 - 0.0706i
 -20.5663 + 0.0507i
 -20.5663 - 0.0507i




u =


   0.0000 + 0.0000i
  -0.2557 - 0.1264i
  -0.2557 + 0.1264i
   4.9926 - 2.9331i
   4.9926 + 2.9331i




v =


   0.0000 + 0.0000i
  -0.2557 - 0.1264i
  -0.2557 + 0.1264i
   4.9926 - 2.9331i
   4.9926 + 2.9331i




w =


   0.0000 + 0.0000i
  -0.2124 - 0.5614i
  -0.2124 + 0.5614i
   4.3966 -14.6150i
   4.3966 +14.6150i

2.实例2

程序

clc;
clear all;
syms x y
[x,y]=solve((x-2)^2+(y+2*x-3)^2==5,2*(x-3)^2+(y/ 3)^2==4 )

运行结果

 
x =
 
 (6449*root(z^4 + (4824*z^3)/457 + (11844*z^2)/457 - (14256*z)/457 - 54108/457, z, 1)^2)/7380 + (457*root(z^4 + (4824*z^3)/457 + (11844*z^2)/457 - (14256*z)/457 - 54108/457, z, 1)^3)/3690 - (6*root(z^4 + (4824*z^3)/457 + (11844*z^2)/457 - (14256*z)/457 - 54108/457, z, 1))/205 - 927/410
 (6449*root(z^4 + (4824*z^3)/457 + (11844*z^2)/457 - (14256*z)/457 - 54108/457, z, 2)^2)/7380 + (457*root(z^4 + (4824*z^3)/457 + (11844*z^2)/457 - (14256*z)/457 - 54108/457, z, 2)^3)/3690 - (6*root(z^4 + (4824*z^3)/457 + (11844*z^2)/457 - (14256*z)/457 - 54108/457, z, 2))/205 - 927/410
 (6449*root(z^4 + (4824*z^3)/457 + (11844*z^2)/457 - (14256*z)/457 - 54108/457, z, 3)^2)/7380 + (457*root(z^4 + (4824*z^3)/457 + (11844*z^2)/457 - (14256*z)/457 - 54108/457, z, 3)^3)/3690 - (6*root(z^4 + (4824*z^3)/457 + (11844*z^2)/457 - (14256*z)/457 - 54108/457, z, 3))/205 - 927/410
 (6449*root(z^4 + (4824*z^3)/457 + (11844*z^2)/457 - (14256*z)/457 - 54108/457, z, 4)^2)/7380 + (457*root(z^4 + (4824*z^3)/457 + (11844*z^2)/457 - (14256*z)/457 - 54108/457, z, 4)^3)/3690 - (6*root(z^4 + (4824*z^3)/457 + (11844*z^2)/457 - (14256*z)/457 - 54108/457, z, 4))/205 - 927/410
 
 
y =
 
 root(z^4 + (4824*z^3)/457 + (11844*z^2)/457 - (14256*z)/457 - 54108/457, z, 1)
 root(z^4 + (4824*z^3)/457 + (11844*z^2)/457 - (14256*z)/457 - 54108/457, z, 2)
 root(z^4 + (4824*z^3)/457 + (11844*z^2)/457 - (14256*z)/457 - 54108/457, z, 3)
 root(z^4 + (4824*z^3)/457 + (11844*z^2)/457 - (14256*z)/457 - 54108/457, z, 4)

方法2 使用vpa函数求近似数值解。

程序

clc;
clear all;
syms x y
[x,y]=solve((x-2)^2+(y+2*x-3)^2==5,2*(x-3)^2+(y/ 3)^2==4 );
x = vpa(x)
y = vpa(y)


x = double(x)
y = double(y)

运行结果

 
x =
 
 1.6580664770347998069049390497594
 1.7362259004399598338121197151769
 4.0287335406907803557776183983678
 3.4828821781145475308576204296936
 
 
y =
 
  1.8936365963298548025994430021814
 -2.6929074352940121705044040780427
 -4.1171266000258712039597783906018
  -5.639401248099686964240293356294
 


x =


    1.6581
    1.7362
    4.0287
    3.4829




y =


    1.8936
   -2.6929
   -4.1171
   -5.6394

实例3

程序

clc;
clear all;
syms x a
eqn = x^3 + x^2 + a == 0;
solve(eqn, x)

运行结果

 
ans =
 
 root(z^3 + z^2 + a, z, 1)
 root(z^3 + z^2 + a, z, 2)
 root(z^3 + z^2 + a, z, 3)

方法3 修改参数设置。

程序

clc;
clear all;
syms x a
eqn = x^3 + x^2 + a == 0;
solve(eqn, x)


%通过使用“最大度数”调用求解器，尝试获得此类方程的显式解。
% 该选项指定求解程序尝试返回显式解的多项式的最大次数。默认值为2。
% 增加这个值，可以得到高阶多项式的显式解。
% 通过将“MaxDegree”的值增加到3来求解相同的方程以获得显式解。
S = solve(eqn, x, 'MaxDegree', 3)

运行结果

S =
 
                                                                                                                                         1/(9*(((a/2 + 1/27)^2 - 1/729)^(1/2) - a/2 - 1/27)^(1/3)) + (((a/2 + 1/27)^2 - 1/729)^(1/2) - a/2 - 1/27)^(1/3) - 1/3
 - (3^(1/2)*(1/(9*(((a/2 + 1/27)^2 - 1/729)^(1/2) - a/2 - 1/27)^(1/3)) - (((a/2 + 1/27)^2 - 1/729)^(1/2) - a/2 - 1/27)^(1/3))*1i)/2 - 1/(18*(((a/2 + 1/27)^2 - 1/729)^(1/2) - a/2 - 1/27)^(1/3)) - (((a/2 + 1/27)^2 - 1/729)^(1/2) - a/2 - 1/27)^(1/3)/2 - 1/3
   (3^(1/2)*(1/(9*(((a/2 + 1/27)^2 - 1/729)^(1/2) - a/2 - 1/27)^(1/3)) - (((a/2 + 1/27)^2 - 1/729)^(1/2) - a/2 - 1/27)^(1/3))*1i)/2 - 1/(18*(((a/2 + 1/27)^2 - 1/729)^(1/2) - a/2 - 1/27)^(1/3)) - (((a/2 + 1/27)^2 - 1/729)^(1/2) - a/2 - 1/27)^(1/3)/2 - 1/3
 

实例4 solve的使用实例

程序

clc;
clear all;
syms a b c x
eqn = a*x^2 + b*x + c == 0
S = solve(eqn)
Sa = solve(eqn,a)


syms x
eqn = x^5 == 3125;
S = solve(eqn,x)
S = solve(eqn,x,'Real',true)


syms x
eqn = sin(x) == x^2 - 1;
S = solve(eqn,x)
fplot([lhs(eqn) rhs(eqn)], [-2 2])
%通过直接调用数值求解器vpasolve并指定间隔来查找另一个解。
V = vpasolve(eqn,x,[0 2])


syms u v
eqns = [2*u + v == 0, u - v == 1];
S = solve(eqns,[u v])
S.u
S.v
expr1 = u^2;
e1 = subs(expr1,S)


expr2 = 3*v + u;
e2 = subs(expr2,S)


eqns = [3*u+2, 3*u+1];
S = solve(eqns,u)

运行结果

 
eqn =
 
a*x^2 + b*x + c == 0
 
 
S =
 
 -(b + (b^2 - 4*a*c)^(1/2))/(2*a)
 -(b - (b^2 - 4*a*c)^(1/2))/(2*a)
 
 
Sa =
 
-(c + b*x)/x^2
 
 
S =
 
                                                          5
 - (2^(1/2)*(5 - 5^(1/2))^(1/2)*5i)/4 - (5*5^(1/2))/4 - 5/4
   (2^(1/2)*(5 - 5^(1/2))^(1/2)*5i)/4 - (5*5^(1/2))/4 - 5/4
   (5*5^(1/2))/4 - (2^(1/2)*(5^(1/2) + 5)^(1/2)*5i)/4 - 5/4
   (5*5^(1/2))/4 + (2^(1/2)*(5^(1/2) + 5)^(1/2)*5i)/4 - 5/4
 
 
S =
 
5
 
警告: Unable to solve symbolically. Returning a numeric solution using vpasolve. 
> In sym/solve (line 304)
  In guo_20230421_16 (line 15) 
 
S =
 
-0.63673265080528201088799090383828
 
 
V =
 
1.4096240040025962492355939705895
 


S = 


  包含以下字段的 struct:


    u: [1×1 sym]
    v: [1×1 sym]


 
ans =
 
1/3
 
 
ans =
 
-2/3
 
 
e1 =
 
1/9
 
 
e2 =
 
-5/3
 
 
S =
 
Empty sym: 0-by-1
 
>>

实例5 solve函数可以求解不等式并返回满足不等式的解

程序

clc;
clear all;
%solve函数可以求解不等式并返回满足不等式的解。解下列不等式。
syms x y
eqn1 = x > 0
eqn2 = y > 0
eqn3 = x^2 + y^2 + x*y < 1
eqns = [eqn1 eqn2 eqn3]
%将"ReturnConditions”设置为true以返回解决方案中的任何参数和解决方案的条件。
S = solve(eqns,[x y],'ReturnConditions',true);
S.x
S.y
S.parameters
S.conditions
condWithValues = subs(S.conditions, S.parameters, [7/2,1/2]);
%isAlways返回逻辑1 (true ),表示这些值满足条件。将这些参数值代入S.x和S.y，找到x和y的解。
isAlways(condWithValues)


xSol = subs(S.x, S.parameters, [7/2,1/2])
ySol = subs(S.y, S.parameters, [7/2,1/2])

运行结果

3.参考内容

[1https://ww2.mathworks.cn/help/symbolic/sym.solve.html;jsessionid=476bd2cb441b759c3d4ebd88747b

[2] CSDN博主有些时候甚至幼稚的文章《Matlab 使用solve求解方程，出现未知数z和root》，文章链接为：

https://blog.csdn.net/jyfan0806/article/details/86613224

本文内容来源于网络，仅供参考学习，如内容、图片有任何版权问题，请联系处理，24小时内删除。

作 者 | 郭志龙

编 辑 | 郭志龙
校 对 | 郭志龙