SymPy
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| Sympy logo. | |
| Developer(s) | Independent group of people |
|---|---|
| Initial release | 2007 |
| Stable release | 1.0 / March 8, 2016 |
| Development status | Active |
| Written in | Python |
| Operating system | Cross-platform |
| Type | Computer algebra system |
| License | New BSD License |
| Website | sympy |
SymPy is a 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 trivial to install and to inspect because is written entirely in Python with few dependencies.[1][2] 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.[1][2]
SymPy is free software and is licensed under New BSD License. The lead developers are Ondřej Čertík and Aaron Meurer.
Contents
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[3] (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:[1][2][4][5][6]
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 factorization
- 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 matplotlib or Pyglet module.
- Coordinate models
- Plotting Geometric Entities
- 2D and 3D
- Interactive interface
- Colors
Physics
- Units
- Mechanics
- Quantum
- Gaussian optics
- Pauli Algebra
Statistics
Combinatorics
- Permutations
- Combinations
- Partitions
- Subsets
- Permutation group: Polyhedral, Rubik, Symmetric, ...
- 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.
Dependencies
Since version 1.0, SymPy has the mpmath package as a dependency.
There are several optional dependencies that can enhance its capabilities:
- 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.
- 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)**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
>>> from sympy import symbols, plot3d, 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) = ───────────
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
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References
- ↑ 1.0 1.1 1.2 Lua error in package.lua at line 80: module 'strict' not found.
- ↑ 2.0 2.1 2.2 Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
External links
Categories:
- Pages with syntax highlighting errors
- Pages with broken file links
- Computer algebra system software for Linux
- Computer algebra system software for OS X
- Computer algebra system software for Windows
- Free computer algebra systems
- Free mathematics software
- Free software programmed in Python
- Python scientific libraries
