Matlab符号函数练习

1．用2种方式创建符号表达式f=sin(x)+e^x

方法一

clear
clc
x=sym('x');
f=sin(x)+exp(x)

结果

f =

exp(x) + sin(x)

>>

方法二

clear
clc
syms x;
f=sin(x)+exp(x)

结果

f =

exp(x) + sin(x)

>>

2．计算习题 1 中表达式在x=0、x=π/4、x=2π，以及ones(3,4)处的值

clear
clc
syms x;
f=sin(x)+exp(x)
subs(f, x, 0)
subs(f, x, pi/4)
subs(f, x, pi*2)
subs(f, x, ones(3,4))

结果

f =

exp(x) + sin(x)

ans =

1

ans =

2.9004

ans =

535.4917

ans =

3.5598    3.5598    3.5598    3.5598
3.5598    3.5598    3.5598    3.5598
3.5598    3.5598    3.5598    3.5598

>>

3．计算习题 1 中表达式在x=π/6处的值，并将结果设置为以下 5 种精度： 1 位、2 位、5 位、10位和20位有效数字。

本题在高版本的Matlab(Matlab2010b)上无法设置1位精度。（所以本题1位精度不做了）

clear
clc
syms x;
f=sin(x)+exp(x);
y=subs(f, x, pi/6);
digits(2);
vpa(y)
digits(5);
vpa(y)
digits(10);
vpa(y)
digits(20);
vpa(y)

结果

ans =

2.2

ans =

2.1881

ans =

2.188091795

ans =

2.1880917949644684839

>>

方法二：

clear
clc
syms x;
f=sin(x)+exp(x);
y=subs(f, x, pi/6);
%vpa(y,1)
vpa(y,2)
vpa(y,5)
vpa(y,10)
vpa(y,20)

结果

ans =

2.2

ans =

2.1881

ans =

2.188091795

ans =

2.1880917949644684839

>>

4．设x为符号变量，f(x)=x^4+x^2+1，g(x)=x^3+4x^2+5x+8，试进行如下运算

（1）f(x)+g(x)

（2）f(x)×g(x)

（3）对f(x)进行因式分解

（4）求g(x)的反函数

（5）求以g为f(x)自变量的复合函数

（6）用2个命令分别化简f*g的结果

clear
clc
syms x;
f=x^4+x^2+1;
g=x^4+4*x^2+5*x+8;
f+g
f*g
factor(f)
finverse(g)
compose(f,g)
simplify(f)
simple(f)

结果

 
ans =
 
2*x^4 + 5*x^2 + 5*x + 9
 
 
ans =
 
(x^4 + x^2 + 1)*(x^4 + 4*x^2 + 5*x + 8)
 
 
ans =
 
(x^2 - x + 1)*(x^2 + x + 1)
 
 
ans =
 
RootOf(X10^4 + 4*X10^2 + 5*X10 - x + 8, X10)
 
 
ans =
 
(x^4 + 4*x^2 + 5*x + 8)^2 + (x^4 + 4*x^2 + 5*x + 8)^4 + 1
 
 
ans =
 
x^4 + x^2 + 1
 
 
simplify:
 
x^4 + x^2 + 1
 
 
radsimp:
 
x^4 + x^2 + 1
 
 
simplify(100):
 
x^4 + x^2 + 1
 
 
combine(sincos):
 
x^4 + x^2 + 1
 
 
combine(sinhcosh):
 
x^4 + x^2 + 1
 
 
combine(ln):
 
x^4 + x^2 + 1
 
 
factor:
 
(x^2 - x + 1)*(x^2 + x + 1)
 
 
expand:
 
x^4 + x^2 + 1
 
 
combine:
 
x^4 + x^2 + 1
 
 
rewrite(exp):
 
x^4 + x^2 + 1
 
 
rewrite(sincos):
 
x^4 + x^2 + 1
 
 
rewrite(sinhcosh):
 
x^4 + x^2 + 1
 
 
rewrite(tan):
 
x^4 + x^2 + 1
 
 
mwcos2sin:
 
x^4 + x^2 + 1
 
 
collect(x):
 
x^4 + x^2 + 1
 
 
ans =
 
x^4 + x^2 + 1
 
>>

5．合并同类项

（1）3x-2x^2+5+3x^2-2x-5

（2）3x^2*y-4xy^2-3+5x^2*y+2xy^2+5（分别对x和y）

（3）2x^2-3xy+y^2-2xy-2x^2+5xy-2y+1（分别对x和y）

clear
clc
syms x y;
collect(3*x-2*x^2+5+3*x^2-2*x-5)
collect(3*x^2*y-4*x*y^2-3+5*x^2*y+2*x*y^2+5,x)
collect(3*x^2*y-4*x*y^2-3+5*x^2*y+2*x*y^2+5,y)
collect(2*x^2-3*x*y+y^2-2*x*y-2*x^2+5*x*y-2*y+1,x)
collect(2*x^2-3*x*y+y^2-2*x*y-2*x^2+5*x*y-2*y+1,y)

结果

ans =

x^2 + x

ans =

(8*y)*x^2 + (-2*y^2)*x + 2

ans =

(-2*x)*y^2 + (8*x^2)*y + 2

ans =

y^2 - 2*y + 1

ans =

y^2 - 2*y + 1

>>

改进

clear
clc
syms x y;
f1=3*x-2*x^2+5+3*x^2-2*x-5;
f2=3*x^2*y-4*x*y^2-3+5*x^2*y+2*x*y^2+5;
f3=2*x^2-3*x*y+y^2-2*x*y-2*x^2+5*x*y-2*y+1;
collect(f1)
collect(f2,x)
collect(f2,y)
collect(f3,x)
collect(f3,y)

结果

ans =

x^2 + x

ans =

(8*y)*x^2 + (-2*y^2)*x + 2

ans =

(-2*x)*y^2 + (8*x^2)*y + 2

ans =

y^2 - 2*y + 1

ans =

y^2 - 2*y + 1

>> 

6．因式分解

（1）将 7798666 进行因数分解，分解为素数乘积的形式

（2）-2m^8+512

（3）3a^2*(x-y)^3-4b^2*(y-x)^2

(1)

clear
clc
factor(7798666)

结果

ans =

2          67       58199

>>

(2)

clear
clc
syms m;
factor(-2*m^8+512)

结果

ans =

-2*(m - 2)*(m + 2)*(m^2 + 4)*(m^4 + 16)

>>

(3)

clear
clc
syms a b x y;
factor(3*a^2*(x-y)^3-4*b^2*(y-x)^2)

结果

ans =

(x - y)^2*(3*a^2*x - 3*a^2*y - 4*b^2)

>>

7．计算下列各式

求2x^2-3xy+y^2-2xy-2x^2+5xy-2y+1在（2,5）的值；将（x,y）替换为{a,rand(2,3)}；将（x,y）替换为{a+bt,sin(2t)}

注：本题修改为2x^2+3xy+y^2-2xy-2x^2+5xy-2y+1，第二题矩阵维数不对，这里不做

clear
clc
syms x y;
syms a b t;
f=2*x^2+3*x*y+y^2-2*x*y-2*x^2+5*x*y-2*y+1;
subs(f,{x,y},{2,5})
%subs(f,{x,y},[a,rand(2,3)])  %不能换
subs(f,{x,y},[a+b*t,sin(2*t)])

结果

ans =

76

ans =

6*sin(2*t)*(a + b*t) - 2*sin(2*t) + sin(2*t)^2 + 1

>>

8．符号矩阵

R=[cos(a)*cos(c)-sin(a)*sin(b)*sin(c) -sin(a)*cos(b) cos(a)*sin(c)+sin(a)*sin(b)*cos(c);

sin(a)*cos(c)+cos(a)*sin(b)*sin(c) cos(a)*cos(b) sin(a)*sin(c)-cos(a)*sin(b)*cos(c);

-cos(b)*sin(c) sin(b) cos(b)*cos(c)]

，求该矩阵的秩，逆及行列式,

clear
clc
syms a b c;
R=[cos(a)*cos(c)-sin(a)*sin(b)*sin(c) -sin(a)*cos(b) cos(a)*sin(c)+sin(a)*sin(b)*cos(c);
sin(a)*cos(c)+cos(a)*sin(b)*sin(c) cos(a)*cos(b) sin(a)*sin(c)-cos(a)*sin(b)*cos(c);
-cos(b)*sin(c) sin(b) cos(b)*cos(c)]
rank(R)
inv(R)
det(R)

结果

 
R =
 
[ cos(a)*cos(c) - sin(a)*sin(b)*sin(c), -cos(b)*sin(a), cos(a)*sin(c) + cos(c)*sin(a)*sin(b)]
[ cos(c)*sin(a) + cos(a)*sin(b)*sin(c),  cos(a)*cos(b), sin(a)*sin(c) - cos(a)*cos(c)*sin(b)]
[                       -cos(b)*sin(c),         sin(b),                        cos(b)*cos(c)]
 
 
ans =
 
3
 
 
ans =
 
[ cos(a)*cos(c) - sin(a)*sin(b)*sin(c), cos(c)*sin(a) + cos(a)*sin(b)*sin(c), -cos(b)*sin(c)]
[                       -cos(b)*sin(a),                        cos(a)*cos(b),         sin(b)]
[ cos(a)*sin(c) + cos(c)*sin(a)*sin(b), sin(a)*sin(c) - cos(a)*cos(c)*sin(b),  cos(b)*cos(c)]
 
 
ans =
 
cos(a)^2*cos(b)^2*cos(c)^2 + cos(a)^2*cos(b)^2*sin(c)^2 + cos(a)^2*cos(c)^2*sin(b)^2 + cos(a)^2*sin(b)^2*sin(c)^2 + cos(b)^2*cos(c)^2*sin(a)^2 + cos(b)^2*sin(a)^2*sin(c)^2 + cos(c)^2*sin(a)^2*sin(b)^2 + sin(a)^2*sin(b)^2*sin(c)^2
 
>>