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

$\sqrt{2}$

from sympy.abc import a, b

expr = b*a*a + -4*a + b + a*b + 4*a + (a + b)*3
expr

$a^{2}b+ab+3a+4b$

计算

公式化简

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

expr = sin(x) / cos(x)
simplify(expr)

$\tan \left(x\right)$

解普通方程

from sympy import Symbol, solve

x = Symbol('x')
sol = solve(x**2 - x, x)
sol

$\left[0,1\right]$

求极限

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

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

$0$

线性代数

矩阵运算

设$A=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\\ a_{31}& a_{32}\end{array}\right)$,$B=\left(\begin{array}{cc}{b}_1& 0\\ 0& b_2\end{array}\right)$,$C=\left(\begin{array}{ccc}{c}_1& 0& 0\\ 0& c_2& 0\\ 0& 0& c_3\end{array}\right)$,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

$\left[\begin{array}{cc}{a}_{11}{b}_1& a_{12}{b}_2\\ a_{21}{b}_1& a_{22}{b}_2\\ a_{21}{b}_1& a_{32}{b}_2\end{array}\right]$

解线性方程组

用消元法解下列方程组：

$$\{\begin{array}{c}{x}_1-2x_2+3x_3-x_4+2x_5=2\\ 3x_1-x_2+5x_3-3x_4+x_5=6\\ 2x_1+x_2+2x_3-2x_4-x_5=8\end{array}$$

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)

特解：
{(x4 - 2, x5 + 4, 4, x4, x5)}

print('基础解：')
result0

基础解：

$\left\{\left(x_4,x_5,0,x_4,x_5\right)\right\}$

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)

$i{e}^{-\mathrm{im}\left(x\right)}\sin \left(\mathrm{re}\left(x\right)\right)+e^{-\mathrm{im}\left(x\right)}\cos \left(\mathrm{re}\left(x\right)\right)$

# 若x为实数
x = Symbol("x", real=True)
expand(exp(I*x), complex=True)

$i\sin \left(x\right)+\cos \left(x\right)$

# 泰勒公式展开
series(exp(I*x), x, 0, 10)

$1+ix-\frac{x^2}{2}-\frac{i{x}^3}{6}+\frac{x^4}{24}+\frac{i{x}^5}{120}-\frac{x^6}{720}-\frac{i{x}^7}{5040}+\frac{x^8}{40320}+\frac{i{x}^9}{362880}+O\left(x^{10}\right)$

分数计算

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

S(1)/2 + 1/S(3)

$\frac{5}{6}$

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)

$e^x\sin \left(x\right)+e^x\cos \left(x\right)$

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)

$x^3{y}^2\left(x^3{y}^3{z}^3+14x^2{y}^2{z}^2+52xyz+48\right)e^{xyz}$

积分

$\cos (x)$的积分

integrate(cos(x), x)

$\sin \left(x\right)$

定积分
${\int }_0^{\infty }{e}^{-x}dx$

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

$1$

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))

$\pi$

极限

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

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

$1$

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])

$\frac{f\left(-3h\right)}{5h^2}-\frac{f\left(-h\right)}{3h^2}+\frac{2f\left(2h\right)}{15h^2}$