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Python library for symbolic computation From Wikipedia, the free encyclopedia
SymPy is an open-source Python library for symbolic computation. It provides computer algebra capabilities either as a standalone application, as a library to other applications, or live on the web as SymPy Live[2] or SymPy Gamma.[3] SymPy is simple to install and to inspect because it is written entirely in Python with few dependencies.[4][5][6] This ease of access combined with a simple and extensible code base in a well known language make SymPy a computer algebra system with a relatively low barrier to entry.
Developer(s) | SymPy Development Team |
---|---|
Initial release | 2007 |
Stable release | 1.12[1]
/ 10 May 2023 |
Repository | |
Written in | Python |
Operating system | Cross-platform |
Type | Computer algebra system |
License | New BSD License |
Website | www |
SymPy includes features ranging from basic symbolic arithmetic to calculus, algebra, discrete mathematics, and quantum physics. It is capable of formatting the result of the computations as LaTeX code.[4][5]
SymPy is free software and is licensed under the New BSD license. The lead developers are Ondřej Čertík and Aaron Meurer. It was started in 2005 by Ondřej Čertík.[7]
The SymPy library is split into a core with many optional modules.
Currently, the core of SymPy has around 260,000 lines of code[8] (it also includes a comprehensive set of self-tests: over 100,000 lines in 350 files as of version 0.7.5), and its capabilities include:[4][5][9][10][11]
Note, plotting requires the external Matplotlib or Pyglet module.
Since version 1.0, SymPy has the mpmath package as a dependency.
There are several optional dependencies that can enhance its capabilities:
Sympy allows outputs to be formatted into a more appealing format through the pprint
function. Alternatively, the init_printing()
method will enable pretty-printing, so pprint
need not be called. Pretty-printing will use unicode symbols when available in the current environment, otherwise it will fall back to ASCII characters.
>>> from sympy import pprint, init_printing, Symbol, sin, cos, exp, sqrt, series, Integral, Function
>>>
>>> x = Symbol("x")
>>> y = Symbol("y")
>>> f = Function("f")
>>> # pprint will default to unicode if available
>>> pprint(x ** exp(x))
⎛ x⎞
⎝ℯ ⎠
x
>>> # An output without unicode
>>> pprint(Integral(f(x), x), use_unicode=False)
/
|
| f(x) dx
|
/
>>> # Compare with same expression but this time unicode is enabled
>>> pprint(Integral(f(x), x), use_unicode=True)
⌠
⎮ f(x) dx
⌡
>>> # Alternatively, you can call init_printing() once and pretty-print without the pprint function.
>>> init_printing()
>>> sqrt(sqrt(exp(x)))
____
4 ╱ x
╲╱ ℯ
>>> (1/cos(x)).series(x, 0, 10)
2 4 6 8
x 5⋅x 61⋅x 277⋅x ⎛ 10⎞
1 + ── + ──── + ───── + ────── + O⎝x ⎠
2 24 720 8064
>>> from sympy import init_printing, Symbol, expand
>>> init_printing()
>>>
>>> a = Symbol("a")
>>> b = Symbol("b")
>>> e = (a + b) ** 3
>>> e
(a + b)³
>>> e.expand()
a³ + 3⋅a²⋅b + 3⋅a⋅b² + b³
>>> from sympy import Rational, pprint
>>> e = 2**50 / Rational(10) ** 50
>>> pprint(e)
1/88817841970012523233890533447265625
>>> from sympy import init_printing, symbols, ln, diff
>>> init_printing()
>>> x, y = symbols("x y")
>>> f = x**2 / y + 2 * x - ln(y)
>>> diff(f, x)
2⋅x
─── + 2
y
>>> diff(f, y)
2
x 1
- ── - ─
2 y
y
>>> diff(diff(f, x), y)
-2⋅x
────
2
y
>>> from sympy import symbols, cos
>>> from sympy.plotting import plot3d
>>> x, y = symbols("x y")
>>> plot3d(cos(x * 3) * cos(y * 5) - y, (x, -1, 1), (y, -1, 1))
<sympy.plotting.plot.Plot object at 0x3b6d0d0>
>>> from sympy import init_printing, Symbol, limit, sqrt, oo
>>> init_printing()
>>>
>>> x = Symbol("x")
>>> limit(sqrt(x**2 - 5 * x + 6) - x, x, oo)
-5/2
>>> limit(x * (sqrt(x**2 + 1) - x), x, oo)
1/2
>>> limit(1 / x**2, x, 0)
∞
>>> limit(((x - 1) / (x + 1)) ** x, x, oo)
-2
ℯ
>>> from sympy import init_printing, Symbol, Function, Eq, dsolve, sin, diff
>>> init_printing()
>>>
>>> x = Symbol("x")
>>> f = Function("f")
>>>
>>> eq = Eq(f(x).diff(x), f(x))
>>> eq
d
──(f(x)) = f(x)
dx
>>>
>>> dsolve(eq, f(x))
x
f(x) = C₁⋅ℯ
>>>
>>> eq = Eq(x**2 * f(x).diff(x), -3 * x * f(x) + sin(x) / x)
>>> eq
2 d sin(x)
x ⋅──(f(x)) = -3⋅x⋅f(x) + ──────
dx x
>>>
>>> dsolve(eq, f(x))
C₁ - cos(x)
f(x) = ───────────
x³
>>> from sympy import init_printing, integrate, Symbol, exp, cos, erf
>>> init_printing()
>>> x = Symbol("x")
>>> # Polynomial Function
>>> f = x**2 + x + 1
>>> f
2
x + x + 1
>>> integrate(f, x)
3 2
x x
── + ── + x
3 2
>>> # Rational Function
>>> f = x / (x**2 + 2 * x + 1)
>>> f
x
────────────
2
x + 2⋅x + 1
>>> integrate(f, x)
1
log(x + 1) + ─────
x + 1
>>> # Exponential-polynomial functions
>>> f = x**2 * exp(x) * cos(x)
>>> f
2 x
x ⋅ℯ ⋅cos(x)
>>> integrate(f, x)
2 x 2 x x x
x ⋅ℯ ⋅sin(x) x ⋅ℯ ⋅cos(x) x ℯ ⋅sin(x) ℯ ⋅cos(x)
──────────── + ──────────── - x⋅ℯ ⋅sin(x) + ───────── - ─────────
2 2 2 2
>>> # A non-elementary integral
>>> f = exp(-(x**2)) * erf(x)
>>> f
2
-x
ℯ ⋅erf(x)
>>> integrate(f, x)
___ 2
╲╱ π ⋅erf (x)
─────────────
4
>>> from sympy import Symbol, cos, sin, pprint
>>> x = Symbol("x")
>>> e = 1 / cos(x)
>>> pprint(e)
1
──────
cos(x)
>>> pprint(e.series(x, 0, 10))
2 4 6 8
x 5⋅x 61⋅x 277⋅x ⎛ 10⎞
1 + ── + ──── + ───── + ────── + O⎝x ⎠
2 24 720 8064
>>> e = 1/sin(x)
>>> pprint(e)
1
──────
sin(x)
>>> pprint(e.series(x, 0, 4))
3
1 x 7⋅x ⎛ 4⎞
─ + ─ + ──── + O⎝x ⎠
x 6 360
>>> from sympy import *
>>> x = Symbol("x")
>>> y = Symbol("y")
>>> facts = Q.positive(x), Q.positive(y)
>>> with assuming(*facts):
... print(ask(Q.positive(2 * x + y)))
True
>>> from sympy import *
>>> x = Symbol("x")
>>> # Assumption about x
>>> fact = [Q.prime(x)]
>>> with assuming(*fact):
... print(ask(Q.rational(1 / x)))
True