Python 数学符号计算模块 sympy

Python 是一种广泛使用的解释型、高级和通用的编程语言。除了 numpy 提供科学计算外，还有 sympy 用于数学符号计算。

安装

sympy 的安装非常简单，使用 pip install sympy 即可完成。更详细的安装教程请参考 sympy Installation 官网。

下面直接进入使用介绍

使用

命令行

如果您已经安装了 sympy，那么在命令行就可以直接使用，就行使用 python 或 ipython 一样，方法如下：

$ isympy
No event loop hook running.
IPython console for SymPy 1.10.1 (Python 3.9.17-64-bit) (ground types: python)

These commands were executed:
>>> from sympy import *
>>> x, y, z, t = symbols('x y z t')
>>> k, m, n = symbols('k m n', integer=True)
>>> f, g, h = symbols('f g h', cls=Function)
>>> init_printing()

Documentation can be found at https://docs.sympy.org/1.10.1/


In [1]: 

jupyter

建议使用 jupyterlab 进行更方便的交互式使用 sympy，同时可以撰写 markdown 说明文档。

import sympy

计算数值

sympy 作为一种数学符合计算工具，它同时也能够进行科学计算。

pi = sympy.pi
pi

${\displaystyle \pi }$

# 显示为实数
pi.evalf(n=100)

${\displaystyle 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068}$

# 无穷的表示
infty = sympy.oo
infty

${\displaystyle \mathrm{\infty }}$

infty > 9999999

${\displaystyle \text{True}}$

infty + 1

${\displaystyle \mathrm{\infty }}$

rational_a = sympy.Rational(1, 2)
rational_a

${\displaystyle \frac{1}{2}}$

rational_b = sympy.Rational(1, 3)
rational_b

${\displaystyle \frac{1}{3}}$

rational_a + rational_b

${\displaystyle \frac{5}{6}}$

符号表示

x = sympy.Symbol("x")
x

${\displaystyle x}$

x, y = sympy.symbols("x y")
x

${\displaystyle x}$

y

${\displaystyle y}$

# abc 模块下记录有所有的 拉丁字母和希腊字母
from sympy import abc

x = abc.x
x

${\displaystyle x}$

# 注意 Python 关键字冲突
lambda_var = abc.lamda
lambda_var

${\displaystyle \lambda }$

sympy.I

${\displaystyle i}$

xi = sympy.symbols("xi", real=True)
xi

${\displaystyle \xi }$

sympy.expand_complex(sympy.exp(sympy.I * xi))

${\displaystyle i\sin \left(\xi \right)+\cos \left(\xi \right)}$

x + y * 2 + x - y * 2

${\displaystyle 2x}$

输出格式

sympy 允许很多种形式的输出，可以通过如下方式设置

sympy.init_printing(use_unicode=False, wrap_line=True)

该函数有很多自定义参数，可以使用 sympy.init_printing? 在 jupyter 中查看。

# latex
sympy.print_latex(1 / x + 1 / y)

\frac{1}{y} + \frac{1}{x}

sympy.latex(1 / x)

'\\frac{1}{x}'

# dotprint
sympy.printing.dot.dotprint(1 / x)

'digraph{\n\n# Graph style\n"ordering"="out"\n"rankdir"="TD"\n\n#########\n# Nodes #\n#########\n\n"Pow(Symbol(\'x\'), Integer(-1))_()" ["color"="black", "label"="Pow", "shape"="ellipse"];\n"Symbol(\'x\')_(0,)" ["color"="black", "label"="x", "shape"="ellipse"];\n"Integer(-1)_(1,)" ["color"="black", "label"="-1", "shape"="ellipse"];\n\n#########\n# Edges #\n#########\n\n"Pow(Symbol(\'x\'), Integer(-1))_()" -> "Symbol(\'x\')_(0,)";\n"Pow(Symbol(\'x\'), Integer(-1))_()" -> "Integer(-1)_(1,)";\n}'

# mathml
sympy.print_mathml(1 / x)

<apply>
    <power/>
    <ci>x</ci>
    <cn>-1</cn>
</apply>

sympy.print_python(1 / x)

x = Symbol('x')
e = 1/x

sympy.print_ccode(1 / x)

1.0/x

sympy.print_rcode(1 / x)

1.0/x

更多打印相关的知识请参考：sympy文档 print

代数

展开

sympy.expand((x + y) ** 5)

${\displaystyle x^5+5x^{4}y+10x^3{y}^2+10x^2{y}^3+5x{y}^4+y^5}$

sympy.expand(sympy.cos(x + y), trig=True)

${\displaystyle -\sin \left(x\right)\sin \left(y\right)+\cos \left(x\right)\cos \left(y\right)}$

sympy.expand(sympy.cos(x - y))

${\displaystyle \cos (x-y)}$

sympy.expand(x + y, complex=True)

${\displaystyle \mathrm{re}\left(x\right)+\mathrm{re}\left(y\right)+i\mathrm{im}\left(x\right)+i\mathrm{im}\left(y\right)}$

化简

sympy.simplify((x**2 + x * y + 3) / x)

${\displaystyle x+y+\frac{3}{x}}$

sympy.simplify((x + y) / (x - y) + (x - y) / (x + y))

${\displaystyle \frac{{(x-y)}^2+{(x+y)}^2}{(x-y)(x+y)}}$

矩阵

sympy.Matrix([[1, 2], [1, 2]])

${\displaystyle \left[\begin{array}{cc}1& 2\\ 1& 2\end{array}\right]}$

sympy.Matrix([[1, 2], [1, 2]]).eigenvals()

${\displaystyle 0:1,3:1}$

sympy.Matrix([[1, x], [y, 1]])

${\displaystyle \left[\begin{array}{cc}1& x\\ y& 1\end{array}\right]}$

微分方程

f, g = sympy.symbols("f g", cls=sympy.Function)

f(x).diff(x, x) + f(x)

${\displaystyle f\left(x\right)+\frac{d^2}{d{x}^2}f\left(x\right)}$

sympy.dsolve(f(x).diff(x, x) + f(x), f(x))

${\displaystyle f\left(x\right)=C_1\sin \left(x\right)+C_2\cos \left(x\right)}$

sympy.dsolve(
    sympy.sin(x) * sympy.cos(f(x)) + sympy.cos(x) * sympy.sin(f(x)) * f(x).diff(x),
    f(x),
    hint="separable",
)

${\displaystyle [f\left(x\right)=-\mathrm{acos}\left(\frac{C_1}{\cos \left(x\right)}\right)+2\pi ,f\left(x\right)=\mathrm{acos}\left(\frac{C_1}{\cos \left(x\right)}\right)]}$

微积分

极限

sympy.limit(sympy.sin(x) / x, x, 0, dir="+-")

${\displaystyle 1}$

sympy.limit(1 / x**2, x, sympy.oo)

${\displaystyle 0}$

func = sympy.Limit(sympy.sin(x) / x, x, 0, dir="+-")
func

${\displaystyle \underset{x\to 0}{lim}\left(\frac{\sin \left(x\right)}{x}\right)}$

sympy.Eq(func, func.doit())

${\displaystyle \underset{x\to 0}{lim}\left(\frac{\sin \left(x\right)}{x}\right)=1}$

求导

sympy.diff(sympy.sin(x), x)

${\displaystyle \cos \left(x\right)}$

sympy.diff(sympy.exp(1 + x**2), x)

${\displaystyle 2x{e}^{x^2+1}}$

sympy.Eq(sympy.exp(1 + x**2), sympy.diff(sympy.exp(1 + x**2), x))

${\displaystyle e^{x^2+1}=2x{e}^{x^2+1}}$

级数

sympy.series(sympy.sin(x), x)

${\displaystyle x-\frac{x^3}{6}+\frac{x^5}{120}+O\left(x^6\right)}$

sympy.series(sympy.ln(1 + x), x)

${\displaystyle x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\frac{x^5}{5}+O\left(x^6\right)}$

积分

sympy.integrate(3 * x**2, x)

${\displaystyle x^3}$

sympy.integrate(sympy.sin(x), x)

${\displaystyle -\cos \left(x\right)}$

sympy.integrate(sympy.exp(-(x**2)) * sympy.erf(x), x)

${\displaystyle \frac{\sqrt{\pi }{\mathrm{erf}}^2\left(x\right)}{4}}$

sympy.integrate(sympy.cos(x), (x, 0, sympy.pi / 2))

${\displaystyle 1}$

sympy.integrate(sympy.exp(-(x**2)), (x, -sympy.oo, sympy.oo))

${\displaystyle \sqrt{\pi }}$

func = sympy.Integral(sympy.exp(-(x**2)), (x, -sympy.oo, sympy.oo))
func

${\displaystyle \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{-x^2}dx}$

sympy.Eq(func, func.doit())

${\displaystyle \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{-x^2}dx=\sqrt{\pi }}$

解方程

sympy.solveset(x**3 - 1, x)

${\displaystyle 1,-\frac{1}{2}-\frac{\sqrt{3}i}{2},-\frac{1}{2}+\frac{\sqrt{3}i}{2}}$

sympy.solve((x + 2 * y - 2, -3 * x + 3 * y - 5), (x, y))

${\displaystyle x:-\frac{4}{9},y:\frac{11}{9}}$

因式分解

func = x**4 - 3 * x**2 + 1
func

${\displaystyle x^4-3x^2+1}$

sympy.factor(func)

${\displaystyle (x^2-x-1)(x^2+x-1)}$

sympy.Eq(func, sympy.factor(func))

${\displaystyle x^4-3x^2+1=(x^2-x-1)(x^2+x-1)}$

sympy.factor(func, modulus=5)

${\displaystyle {(x-2)}^2{(x+2)}^2}$

参考文献

1. sympy 文档 print
2. Sympy : Symbolic Mathematics in Python