用于符号数学的 Python 库——sympy(一)

SymPy 是用于符号数学的 Python 库。 它旨在成为功能齐全的计算机代数系统。 SymPy 包括从基本符号算术到微积分，代数，离散数学和量子物理学的功能。 它可以在 LaTeX 中显示结果。

打印普通公式

from sympy import Symbol
x = Symbol('x')

a = sqrt(2)
a
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$\sqrt{2}$

from sympy.abc import a, b

expr = b*a*a + -4*a + b + a*b + 4*a + (a + b)*3
expr
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$a^{2}b+ab+3a+4b$

计算

公式化简

from sympy import sin, cos, simplify
from sympy.abc import x

expr = sin(x) / cos(x)
simplify(expr)
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$\tan \left(x\right)$

解普通方程

from sympy import Symbol, solve

x = Symbol('x')
sol = solve(x**2 - x, x)
sol
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$\left[0,1\right]$

求极限

from sympy import sin, limit, oo
from sympy.abc import x

l1 = limit(1/x, x, oo)
l1
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$0$

线性代数

矩阵运算

设 A = ( a 11 a 12 a 21 a 22 a 31 a 32 ) A=(a11a12a21a22a31a32)

A=⎝ ⎛​a11​a21​a31​​a12​a22​a32​​⎠ ⎞​,$B=\left(\begin{array}{cc}{b}_1& 0\\ 0& b_2\end{array}\right)$,$C=\left(\begin{array}{ccc}{c}_1& 0& 0\\ 0& c_2& 0\\ 0& 0& c_3\end{array}\right)$,求$AB$和$CA$.

import numpy as np
import sympy as sp
A = np.array([['a11','a12'],['a21','a22'],['a21','a32']])
B = np.array([['b1',0],[0,'b2']])
C = np.array([['c1',0,0],[0,'c2',0],[0,0,'c3']])

A=sp.Matrix(A)
B=sp.Matrix(B)
C=sp.Matrix(C)
A*B
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[ a 11 b 1 a 12 b 2 a 21 b 1 a 22 b 2 a 21 b 1 a 32 b 2 ] \displaystyle \left[a11b1a12b2a21b1a22b2a21b1a32b2

\right] ⎣ ⎡​a11​b1​a21​b1​a21​b1​​a12​b2​a22​b2​a32​b2​​⎦ ⎤​

解线性方程组

用消元法解下列方程组：

{ x 1 − 2 x 2 + 3 x 3 − x 4 + 2 x 5 = 2 3 x 1 − x 2 + 5 x 3 − 3 x 4 + x 5 = 6 2 x 1 + x 2 + 2 x 3 − 2 x 4 − x 5 = 8 \left \{ x1−2x2+3x3−x4+2x5=23x1−x2+5x3−3x4+x5=62x1+x2+2x3−2x4−x5=8

\right. ⎩ ⎨ ⎧​x1​−2x2​+3x3​−x4​+2x5​=23x1​−x2​+5x3​−3x4​+x5​=62x1​+x2​+2x3​−2x4​−x5​=8​

import numpy as np
import sympy as sp

x1,x2,x3,x4,x5 = sp.symbols("x1 x2 x3 x4 x5")
A = sp.Matrix([[1,-2,3,-1,2],[3,-1,4,-3,1],[2,1,2,-2,-1]])
b = sp.Matrix([[2],[6],[8]])
b0 = sp.Matrix([[0],[0],[0]])

system1 = A,b
system0 = A,b0

result0 = sp.linsolve(system0,x1,x2,x3,x4,x5)
result = sp.linsolve(system1,x1,x2,x3,x4,x5)
print('特解：')
print(result)


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特解：
{(x4 - 2, x5 + 4, 4, x4, x5)}
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print('基础解：')
result0
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基础解：
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{ ( x 4 ,   x 5 ,   0 ,   x 4 ,   x 5 ) } \displaystyle \left\{\left( x_{4}, \ x_{5}, \ 0, \ x_{4}, \ x_{5}\right)\right\} {(x4​, x5​, 0, x4​, x5​)}

绘图

import sympy
from sympy.abc import x
from sympy.plotting import plot

plot(x*x)
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其它

欧拉公式

$$e^{ix}=\cos x+i\sin x$$

from sympy import *

# 若x为复数
expand(exp(I*x), complex=True)
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$i{e}^{-\mathrm{im}\left(x\right)}\sin \left(\mathrm{re}\left(x\right)\right)+e^{-\mathrm{im}\left(x\right)}\cos \left(\mathrm{re}\left(x\right)\right)$

# 若x为实数
x = Symbol("x", real=True)
expand(exp(I*x), complex=True)
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$i\sin \left(x\right)+\cos \left(x\right)$

# 泰勒公式展开
series(exp(I*x), x, 0, 10)
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$1+ix-\frac{x^2}{2}-\frac{i{x}^3}{6}+\frac{x^4}{24}+\frac{i{x}^5}{120}-\frac{x^6}{720}-\frac{i{x}^7}{5040}+\frac{x^8}{40320}+\frac{i{x}^9}{362880}+O\left(x^{10}\right)$

分数计算

$$\frac{1}{2}+\frac{1}{3}$$

S(1)/2 + 1/S(3)
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$\frac{5}{6}$

官方例子

微分

求导$\sin (x)e^x$

from sympy import *
x, t, z, nu = symbols('x t z nu')

diff(sin(x)*exp(x), x)
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$e^x\sin \left(x\right)+e^x\cos \left(x\right)$

计算$\frac{{\partial }^7}{\partial x\partial y^2\partial z^4}{e}^{xyz}$

x,y,z = symbols('x y z')
expr = exp(x*y*z)
diff(expr, x, y, y, z, z, z, z)
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$x^3{y}^2\left(x^3{y}^3{z}^3+14x^2{y}^2{z}^2+52xyz+48\right)e^{xyz}$

积分

$\cos (x)$的积分

integrate(cos(x), x)
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$\sin \left(x\right)$

定积分
${\int }_0^{\infty }{e}^{-x}dx$

integrate(exp(-x), (x, 0, oo))
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$1$

二重积分
$${\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{e}^{-x^2-y^2}dxdy$$

integrate(exp(-x**2 - y**2), (x, -oo, oo), (y, -oo, oo))
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$\pi$

极限

$$\underset{x\to x_0}{\lim }f(x)$$

limit(sin(x)/x,x,0)
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$1$

有限差分

f = Function('f')
d2fdx2 = f(x).diff(x, 2)
h = Symbol('h')
d2fdx2.as_finite_difference([-3*h,-h,2*h])
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$\frac{f\left(-3h\right)}{5h^2}-\frac{f\left(-h\right)}{3h^2}+\frac{2f\left(2h\right)}{15h^2}$