SymPy

SymPy

Developer(s)	Independent group of people
Initial release	2007, 4–5 years ago
Stable release	0.7.1 / July 29, 2011 (2011-07-29)
Development status	Active
Written in	Python
Operating system	Cross-platform
Type	Computer algebra system
License	New BSD license
Website	sympy.org

Not to be confused with SimPy, a discrete-event simulation language.

SymPy is a Python library for symbolic computation. The stated goals of the library are to become a full-featured computer algebra system and to keep a simple code base to promote extensibility and comprehensibility. SymPy is written in Python.

SymPy is free software. The lead developers are Ondřej Čertík and Aaron Meurer.

Features

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

Currently, the core of SymPy has around 13,000 lines of code (including comments and docstrings) 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, ...

• Substitution
• Arbitrary precision integers, rationals and floats
• Noncommutative symbols
• Pattern matching

Polynomials

• Basic arithmetic: division, gcd, ...
• Factorization

• Square-free decomposition

• Gröbner bases

• Partial fraction decomposition

• Resultants

Calculus

• Limits

• Differentiation

• Integration: Implemented Risch-Norman heuristic

• Taylor (Laurent) Series

Solving Equations

• Polynomials

• Systems of equations

• Algebraic equations

• Differential equations

• Difference equations

Discrete math

• Binomial coefficients

• Summations

• Products

• Number theory: generating prime numbers, primality testing, integer factorization, ...

• Logic expressions

Matrices

• Basic arithmetic

• Eigenvalues/eigenvectors

• Determinants

• Inversion

• Solving

Geometry

• Points, lines, rays, segments, ellipses, circles, polygons, ...

• Intersections

• Tangency

• Similarity

Plotting

Note, plotting requires the external Pyglet module.

• Coordinate models

• Plotting Geometric Entities

• 2D and 3D

• Interactive interface

• Colors

Physics

• Units

• Mechanics

• Quantum

• Gaussian optics

• Paulli Algebra

Statistics

• Normal distributions

• Uniform distributions

• Probability

Printing

• Pretty printing: ASCII/Unicode pretty printing, LaTeX

• Code generation: C, FORTRAN, Python

The complete set of SymPy modules are over 300,000 lines including documentation and tests as of version 0.7.1.

SymPy also includes a comprehensive set of self-tests (over 56,000 lines in 210 files as of version 0.7.1).

Related projects

• NCLab: web framework for collaborative programming, computer modeling and scientific computing (Sympy is included in NCLab).
• Sage: an open source alternative to Mathematica, Maple, Matlab and Magma (SymPy is included in Sage)
• mpmath: a Python library for arbitrary-precision floating-point arithmetic (included in SymPy)
• sympycore: another Python CAS
• symbide: GUI for SymPy in PyGTK
• symfe: Lightweight symbolic finite element calculations in Python

Optional dependencies

SymPy requires no dependencies other than Python to run, but there are several optional dependencies that can enhance its capabilities:

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

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)))
 x
 ─
 4
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)**5
>>> e
       5
(a + b) 
>>> e.expand()
 5      4        3  2       2  3       4    5
a  + 5⋅a ⋅b + 10⋅a ⋅b  + 10⋅a ⋅b  + 5⋅a⋅b  + b

Arbitrary Precision Example

>>> from sympy import Rational, pprint
>>>
>>> e = Rational(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, Plot, cos
>>> x,y = symbols('x y')
>>> Plot(cos(x*3)*cos(y*5)-y)
[0]: -y + cos(3*x)*cos(5*y), 'mode=cartesian'

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) = ───────────
             3    
            x

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

See also

• Comparison of computer algebra systems

External links

• SymPy Project Home
• Planet SymPy
• Code Repository on Github