The MATLAB Notebook v1

目录

创建符号对象sym，syms (1)

符号替换成数值subs (2)

表达式化简 (2)

用各种恒等式化简simplify (2)

将(多项式、三角函数、指数函数)乘积转换为和式expand (3)

将多项式转换为乘积形式factor (3)

合并同类项collect (3)

将多项式转换为嵌套形式horner (3)

表达式替换subexpr，subs (5)

极限limit (6)

微积分diff，int (6)

代数方程、方程组求解solve (7)

微分方程、方程组求解dsolve (8)

创建符号对象SYM，SYMS

a = sym('a'),

b = sym('b'), x = sym('x')

f = a^x+b*sin(x)

f = sym('a^x+b*sin(x)')

a =

a

b =

b

x =

x

f =

a^x + b*sin(x)

f =

a^x+b*sin(x)

syms a b

f = a+b

f =

a + b

符号替换成数值SUBS f = sym('x+sin(x)')

subs(f , pi/4)

f =

x+sin(x)

ans =

1.4925

f = sym('x+sin(y)')

subs(f ,x, 2)

subs(f, y, pi/3)

f =

x+sin(y)

ans =

sin(y) + 2

ans =

x + sin(y)

表达式化简

用各种恒等式化简SIMPLIFY

将(多项式、三角函数、指数函数)乘积转换为和式EXPAND

将多项式转换为乘积形式FACTOR

合并同类项COLLECT

将多项式转换为嵌套形式HORNER

S = sym('x^2*y+x^2+2*x*y+x+x*y^2+y^2+y')

s1 = collect(S)

s2 = collect(S,'y')

S =

x^2*y+x^2+2*x*y+x+x*y^2+y^2+y

s1 =

y + x^2*(y + 1) + y^2 + x*(y^2 + 2*y + 1)

s2 =

x + y^2*(x + 1) + x^2 + y*(x^2 + 2*x + 1)

f = sym('(x+y)^3')

expand(f)

f =

(x+y)^3

ans =

x^3 + 3*x^2*y + 3*x*y^2 + y^3

f = sym('4*x^3+x^4+8*x+5*x^2+6')

h = factor(f)

f =

4*x^3+x^4+8*x+5*x^2+6

h =

(x + 3)*(x + 1)*(x^2 + 2)

factor(84)

factor(sym('84'))

ans =

2 2

3 7

ans =

2^2*3*7

f = sym('sin(x)^2+cos(x)^2') simplify(f)

simple(f)

f =

sin(x)^2+cos(x)^2

ans =

1

simplify:

1

radsimp:

cos(x)^2 + sin(x)^2 simplify(100):

1

combine(sincos):

1

combine(sinhcosh):

cos(x)^2 + sin(x)^2 combine(ln):

cos(x)^2 + sin(x)^2 factor:

cos(x)^2 + sin(x)^2 expand:

cos(x)^2 + sin(x)^2 combine:

cos(x)^2 + sin(x)^2 rewrite(exp):

((i*exp(i*x))/2 - i/(2*exp(i*x)))^2 + (exp(i*x)/2 +

1/(2*exp(i*x)))^2

rewrite(sincos):

cos(x)^2 + sin(x)^2

rewrite(sinhcosh):

cosh(-i*x)^2 - sinh(-i*x)^2

rewrite(tan):

(tan(x/2)^2 - 1)^2/(tan(x/2)^2 + 1)^2 + (4*tan(x/2)^2)/(tan(x/2)^2 + 1)^2

collect(x):

cos(x)^2 + sin(x)^2

mwcos2sin:

1

ans =

1

表达式替换SUBEXPR，SUBS

syms a x

s = solve('x^3+a*x+1')

r = subexpr(s)

s =

((a^3/27 + 1/4)^(1/2) - 1/2)^(1/3) - a/(3*((a^3/27 + 1/4)^(1/2) - 1/2)^(1/3))

a/(6*((a^3/27 + 1/4)^(1/2) - 1/2)^(1/3)) - ((a^3/27 + 1/4)^(1/2) - 1/2)^(1/3)/2 - (3^(1/2)*i*(a/(3*((a^3/27 + 1/4)^(1/2) - 1/2)^(1/3)) + ((a^3/27 + 1/4)^(1/2) - 1/2)^(1/3)))/2

a/(6*((a^3/27 + 1/4)^(1/2) - 1/2)^(1/3)) - ((a^3/27 + 1/4)^(1/2) - 1/2)^(1/3)/2 + (3^(1/2)*i*(a/(3*((a^3/27 + 1/4)^(1/2) - 1/2)^(1/3)) + ((a^3/27 + 1/4)^(1/2) - 1/2)^(1/3)))/2

sigma =

(a^3/27 + 1/4)^(1/2) - 1/2

r =

sigma^(1/3) - a/(3*sigma^(1/3))

a/(6*sigma^(1/3)) - sigma^(1/3)/2 - (3^(1/2)*i*(a/(3*sigma^(1/3)) + sigma^(1/3)))/2

a/(6*sigma^(1/3)) - sigma^(1/3)/2 + (3^(1/2)*i*(a/(3*sigma^(1/3)) + sigma^(1/3)))/2

syms x y

f = sym('x+sin(x)')

subs(f , x, y)

f =

x+sin(x)

ans =

y + sin(y)

极限LIMIT

syms h x

limit((sin(x+h)-sin(x))/h, h, 0)

ans =

cos(x)

微积分DIFF，INT

syms x y

f = y*sin(x)+2*x*cos(y)

df = diff(f, x)

df2 = diff(f, y, 2)

f =

2*x*cos(y) + y*sin(x)

df =

2*cos(y) + y*cos(x)

df2 =

(-2)*x*cos(y)

syms x a

f =( a*sin(x)+2*x)*exp(-x)

F = int(f, x)

F2 = int(f, x, 0, inf)

f =

(2*x + a*sin(x))/exp(x)

F =

-(4*x + a*cos(x) + a*sin(x) + 4)/(2*exp(x))

F2 =

a/2 + 2

代数方程、方程组求解SOLVE p = solve('a*x^2+b*x+c')

q = solve('a*x^2+b*x+c', 'b')

p =

-(b + (b^2 - 4*a*c)^(1/2))/(2*a)

-(b - (b^2 - 4*a*c)^(1/2))/(2*a)

q =

-(a*x^2 + c)/x

A = solve('a*u^2+v^2', 'u-v=1', 'a^2-5*a+6')

A.a,A.u,A.v

A =

a: [4x1 sym]

u: [4x1 sym]

v: [4x1 sym]

ans =

3

2

2

3

ans =

(3^(1/2)*i)/4 + 1/4

(2^(1/2)*i)/3 + 1/3

1/3 - (2^(1/2)*i)/3

1/4 - (3^(1/2)*i)/4

ans =

(3^(1/2)*i)/4 - 3/4

(2^(1/2)*i)/3 - 2/3

- (2^(1/2)*i)/3 - 2/3

- (3^(1/2)*i)/4 - 3/4

微分方程、方程组求解DSOLVE

p = dsolve('D2y-Dy=t^2')

[x,y] = dsolve('Du=u+v, Dv=u-v')

q = dsolve('D2y=-a^2*y', 'y(0)=1', 'Dy(pi/a)=0')

p =

C3 - 2*t + C4*exp(t) - t^2 - t^3/3 - 2

x =

C5/exp(2^(1/2)*t) + C6*exp(2^(1/2)*t) - (2^(1/2)*C5)/exp(2^(1/2)*t) + 2^(1/2)*C6*exp(2^(1/2)*t)

y =

C5/exp(2^(1/2)*t) + C6*exp(2^(1/2)*t)

q =

exp(a*i*t)/2 + 1/(2*exp(a*i*t))