Software:SymPy

Short description: Python library for symbolic computation

SymPy

Developer(s)	SymPy Development Team
Initial release	2007 (2007)
Stable release	1.12 / 10 May 2023 (2023-05-10)
Written in	Python
Operating system	Cross-platform
Type	Computer algebra system
License	New BSD License

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 or SymPy Gamma. SymPy is simple to install and to inspect because it is written entirely in Python with few dependencies. 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.

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.

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.

Features

The SymPy library is split into a core with many optional modules.

Currently, the core of SymPy has around 260,000 lines of code (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:

Core capabilities

• Basic arithmetic: *, /, +, -, **
• Simplification
• Expansion
• Functions: trigonometric, hyperbolic, exponential, roots, logarithms, absolute value, spherical harmonics, factorials and gamma functions, zeta functions, polynomials, hypergeometric, special functions, etc.
• Substitution
• Arbitrary precision integers, rationals and floats
• Noncommutative symbols
• Pattern matching

Polynomials

• Basic arithmetic: division, gcd, etc.
• Factorization
• Square-free factorization
• Gröbner bases
• Partial fraction decomposition
• Resultants

Calculus

• Limits
• Differentiation
• Integration: Implemented Risch–Norman heuristic
• Taylor series (Laurent series)

Solving equations

• Systems of linear equations
• Systems of algebraic equations that are solvable by radicals
• Differential equations
• Difference equations

Discrete math

• Binomial coefficients
• Summations
• Products
• Number theory: generating Prime numbers, primality testing, integer factorization, etc.
• Logic expressions

Matrices

• Basic arithmetic
• Eigenvalues and their eigenvectors when the characteristic polynomial is solvable by radicals
• Determinants
• Inversion
• Solving

Geometry

• Points, lines, rays, ellipses, circles, polygons, etc.
• Intersections
• Tangency
• Similarity

Plotting

Note, plotting requires the external Matplotlib or Pyglet module.

• Coordinate models
• Plotting Geometric Entities
• 2D and 3D
• Interactive interface
• Colors
• Animations

Physics

• Units
• Classical mechanics
• Continuum mechanics
• Quantum mechanics
• Gaussian optics
• Pauli algebra
• Linear control

Statistics

• Normal distributions
• Uniform distributions
• Probability

Combinatorics

• Permutations
• Combinations
• Partitions
• Subsets
• Permutation group: Polyhedral, Rubik, Symmetric, etc.
• Prufer sequence and Gray codes

Printing

• Pretty-printing: ASCII/Unicode pretty-printing, LaTeX
• Code generation: C, Fortran, Python

Related projects

• SageMath: an open source alternative to Mathematica, Maple, MATLAB, and Magma (SymPy is included in Sage)
• SymEngine: a rewriting of SymPy's core in C++, in order to increase its performance. Work is currently in progress to make SymEngine the underlying engine of Sage too.
• mpmath: a Python library for arbitrary-precision floating-point arithmetic
• SympyCore: another Python computer algebra system
• SfePy: Software for solving systems of coupled partial differential equations (PDEs) by the finite element method in 1D, 2D and 3D.
• GAlgebra: Geometric algebra module (previously sympy.galgebra).
• Quameon: Quantum Monte Carlo in Python.
• Lcapy: Experimental Python package for teaching linear circuit analysis.
• LaTeX Expression project: Easy LaTeX typesetting of algebraic expressions in symbolic form with automatic substitution and result computation.
• Symbolic statistical modeling: Adding statistical operations to complex physical models.
• Diofant: a fork of SymPy, started by Sergey B Kirpichev

Dependencies

Since version 1.0, SymPy has the mpmath package as a dependency.

There are several optional dependencies that can enhance its capabilities:

• : If gmpy is polynomial module will automatically use it for faster ground types. This can provide a several times boost in performance of certain operations.
• matplotlib: If matplotlib is installed, SymPy can use it for plotting.
• Pyglet: Alternative plotting package.

Usage examples

Pretty-printing

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

Expansion

>>> 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

Arbitrary-precision example

>>> from sympy import Rational, pprint
>>> e = 2**50 / Rational(10) ** 50
>>> pprint(e)
1/88817841970012523233890533447265625

Differentiation

>>> 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

Plotting

Output of the plotting example

>>> 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>

Limits

>>> 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

Differential equations

>>> 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

Integration

>>> 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

Series

>>> 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

Logical reasoning

Example 1

>>> 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

Example 2

>>> from sympy import *
>>> x = Symbol("x")
>>> # Assumption about x
>>> fact = [Q.prime(x)]
>>> with assuming(*fact):
...     print(ask(Q.rational(1 / x)))
True

See also

• Comparison of computer algebra systems

External links

• Planet SymPy
• SymPy Tutorials Collection
• Code Repository on GitHub
• Support and development forum
• Gitter chat room