SymPy is an

open-source Open source is source code that is made freely available for possible modification and redistribution. Products include permission to use and view the source code, design documents, or content of the product. The open source model is a decentrali ...

Python

library A library is a collection of Book, books, and possibly other Document, materials and Media (communication), media, that is accessible for use by its members and members of allied institutions. Libraries provide physical (hard copies) or electron ...

for

symbolic computation In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions ...

. 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 A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The de ...

with a relatively low barrier to entry. SymPy includes features ranging from basic symbolic

arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...

calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

discrete mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous f ...

, and

quantum physics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

. It is capable of formatting the result of the computations as

LaTeX Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latices are found in nature, but synthetic latices are common as well. In nature, latex is found as a wikt:milky, milky fluid, which is present in 10% of all floweri ...

code. SymPy is

free software Free software, libre software, libreware sometimes known as freedom-respecting software is computer software distributed open-source license, under terms that allow users to run the software for any purpose as well as to study, change, distribut ...

and is licensed under the 3-clause BSD. 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 Hyperbolic may refer to: * of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics ** Hyperbolic geometry, a non-Euclidean geometry ** Hyperbolic functions, analogues of ordinary trigonometric functions, defined u ...

, exponential,

roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusin ...

logarithms In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...

absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...

spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. The table of spherical harmonics co ...

, factorials and gamma functions, zeta functions,

polynomials In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative int ...

, hypergeometric, special functions, etc. * Substitution * Arbitrary precision

integers An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

, rationals and floats * Noncommutative symbols *

Pattern matching In computer science, pattern matching is the act of checking a given sequence of tokens for the presence of the constituents of some pattern. In contrast to pattern recognition, the match usually must be exact: "either it will or will not be a ...

Polynomials

* Basic arithmetic: division, gcd, etc. *

Factorization In mathematics, factorization (or factorisation, see American and British English spelling differences#-ise, -ize (-isation, -ization), English spelling differences) or factoring consists of writing a number or another mathematical object as a p ...

* Square-free factorization * Gröbner bases * Partial fraction decomposition *

Resultant In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over th ...

Calculus

* Limits * Differentiation * Integration: Implemented Risch–Norman heuristic *

Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...

(

Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansio ...

)

Solving equations

Systems of linear equations In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables. For example, : \begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in ...

* Systems of

algebraic equation In mathematics, an algebraic equation or polynomial equation is an equation of the form P = 0, where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For example, x^5-3x+1=0 is an algebraic equati ...

s that are solvable by radicals * Differential equations *

Difference equation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

Discrete math

Binomial coefficients In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the te ...

Summation In mathematics, summation is the addition of a sequence of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, pol ...

s * Products *

Number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

: generating

Prime numbers A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

, primality testing,

integer factorization In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a comp ...

, etc. * Logic expressions

Matrices

* Basic arithmetic *

Eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

s and their

eigenvector In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by ...

s when the

characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...

is solvable by radicals *

Determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...

s * Inversion * Solving

Geometry

Points A point is a small dot or the sharp tip of something. Point or points may refer to: Mathematics * Point (geometry), an entity that has a location in space or on a plane, but has no extent; more generally, an element of some abstract topologica ...

, lines, rays,

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

s, etc. * Intersections * Tangency * Similarity

Plotting

Note, plotting requires the external

Matplotlib Matplotlib (portmanteau of MATLAB, plot, and library) is a Plotter, plotting Library (computer science), library for the Python (programming language), Python programming language and its Numerical analysis, numerical mathematics extension NumPy. ...

Pyglet Pyglet is a Library (computing), library for the Python (programming language), Python programming language that provides an Object-oriented programming, object-oriented application programming interface for the creation of games and other multim ...

module. * Coordinate models * Plotting Geometric Entities * 2D and 3D * Interactive interface * Colors * Animations

Physics

* Units *

Classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...

Continuum mechanics Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles. Continuum mec ...

Quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

Gaussian optics Gaussian optics is a technique in geometrical optics that describes the behaviour of light rays in optical systems by using the paraxial approximation, in which only rays which make small angles with the optical axis of the system are considered. ...

* Linear control

Statistics

Normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...

s * Uniform distributions *

Probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...

Combinatorics

Permutations In mathematics, a permutation of a Set (mathematics), set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example ...

Combinations In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are t ...

* Partitions *

Subsets In mathematics, a set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subse ...

Permutation group In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to ...

: Polyhedral, Rubik,

Symmetric Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...

, etc. * Prufer sequence and Gray codes

Printing

* Pretty-printing:

ASCII ASCII ( ), an acronym for American Standard Code for Information Interchange, is a character encoding standard for representing a particular set of 95 (English language focused) printable character, printable and 33 control character, control c ...

Unicode Unicode or ''The Unicode Standard'' or TUS is a character encoding standard maintained by the Unicode Consortium designed to support the use of text in all of the world's writing systems that can be digitized. Version 16.0 defines 154,998 Char ...

pretty-printing, LaTeX * Code generation: C, Fortran, Python

Related projects

SageMath SageMath (previously Sage or SAGE, "System for Algebra and Geometry Experimentation") is a computer algebra system (CAS) with features covering many aspects of mathematics, including algebra, combinatorics, graph theory, group theory, differentia ...

: an open source alternative to

Mathematica Wolfram (previously known as Mathematica and Wolfram Mathematica) is a software system with built-in libraries for several areas of technical computing that allows machine learning, statistics, symbolic computation, data manipulation, network ...

Maple ''Acer'' is a genus of trees and shrubs commonly known as maples. The genus is placed in the soapberry family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated si ...

MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...

, and

Magma Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma (sometimes colloquially but incorrectly referred to as ''lava'') is found beneath the surface of the Earth, and evidence of magmatism has also ...

(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 In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a ''significand'' (a Sign (mathematics), signed sequence of a fixed number of digits in some Radix, base) multiplied by an integer power of that ba ...

* 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 ). * Quameon:

Quantum Monte Carlo Quantum Monte Carlo encompasses a large family of computational methods whose common aim is the study of complex quantum systems. One of the major goals of these approaches is to provide a reliable solution (or an accurate approximation) of the ...

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 is installed, 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 Matplotlib (portmanteau of MATLAB, plot, and library) is a Plotter, plotting Library (computer science), library for the Python (programming language), Python programming language and its Numerical analysis, numerical mathematics extension NumPy. ...

: If matplotlib is installed, SymPy can use it for plotting. *

: Alternative plotting package.

References

External links

*
Planet SymPy

SymPy Tutorials Collection

Code Repository on GitHub

Support and development forum

Gitter chat room
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