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SymPy
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<h2 id="features">Features</h2> <p>The SymPy library is split into a core with many optional modules. </p><p>Currently, the core of SymPy has around 260,000 lines of code<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> (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:<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a><a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a><a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a><a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p> <h3>Core capabilities</h3> <ul><li>Basic arithmetic: *, /, +, -, **</li> <li>Simplification</li> <li>Expansion</li> <li><a href="/facts/Function_(mathematics)/CdReEKCh">Functions</a>: <a href="/facts/Trigonometric_functions/czej7ur0">trigonometric</a>, <a href="/facts/Hyperbolic_functions/Y4Cb5S35">hyperbolic</a>, <a href="/facts/Exponential_function/ewlYxvR2">exponential</a>, <a href="/facts/Zero_of_a_function/mlK3dhoT">roots</a>, <a href="/facts/Logarithm/QeXaV97q">logarithms</a>, <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a>, <a href="/facts/Spherical_harmonics/bn98Gv2L">spherical harmonics</a>, <a href="/facts/Factorial/kkfy1Bwe">factorials</a> and <a href="/facts/Gamma_function/SjGQcnyw">gamma functions</a>, <a href="/facts/Zeta_function_(operator)/WFYFgfun">zeta functions</a>, <a href="/facts/Polynomial/Lzak8VVx">polynomials</a>, <a href="/facts/Hypergeometric_function/yTEC6NBJ">hypergeometric</a>, special functions, etc.</li> <li><a href="/facts/Substitution_(algebra)/GwWdQf0m">Substitution</a></li> <li><a href="/facts/Arbitrary-precision_arithmetic/cXxc8yB2">Arbitrary precision</a> <a href="/facts/Integer/PUwwrawx">integers</a>, <a href="/facts/Rational_number/VJQmyY05">rationals</a> and <a href="/facts/Floating-point_arithmetic/eIckahxe">floats</a></li> <li><a href="/facts/Noncommutative_logic/Kh8KQxYf">Noncommutative symbols</a></li> <li><a href="/facts/Pattern_matching/CEwptMs9">Pattern matching</a></li></ul> <h3>Polynomials</h3> <ul><li>Basic arithmetic: <a href="/facts/Polynomial_division/MgJlJ2G7">division</a>, <a href="/facts/Greatest_common_divisor/kNFvBuKl">gcd</a>, etc.</li> <li><a href="/facts/Factorization/tXrBXFh1">Factorization</a></li> <li><a href="/facts/Square-free_factorization/G5j2B4hM">Square-free factorization</a></li> <li><a href="/facts/Gr%C3%B6bner_bases/efMrjM7w">Gröbner bases</a></li> <li><a href="/facts/Partial_fraction_decomposition/KLRmM3Hw">Partial fraction decomposition</a></li> <li><a href="/facts/Resultant/A5g2MdhT">Resultants</a></li></ul> <h3>Calculus</h3> <ul><li><a href="/facts/Mathematical_limit/j8JBhj4e">Limits</a></li> <li><a href="/facts/Differentiation_(mathematics)/7Ito70Dh">Differentiation</a></li> <li><a href="/facts/Integral/V7lyV997">Integration</a>: Implemented <a href="/facts/Risch%E2%80%93Norman_algorithm/UcT2qrPl">Risch–Norman</a> heuristic</li> <li><a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> (<a href="/facts/Laurent_series/sSMevFU8">Laurent series</a>)</li></ul> <h3>Solving equations</h3> <ul><li><a href="/facts/Systems_of_equations/KzP6TJ4u">Systems of linear equations</a></li> <li>Systems of <a href="/facts/Algebraic_equation/1QauIGxs">algebraic equations</a> that are <a href="/facts/Solvable_by_radicals/54wscJfj">solvable by radicals</a></li> <li><a href="/facts/Differential_equation/9MGbE3Ca">Differential equations</a></li> <li><a href="/facts/Difference_equation/JolXC71M">Difference equations</a></li></ul> <h3>Discrete math</h3> <ul><li><a href="/facts/Binomial_coefficients/Tsx7eY5h">Binomial coefficients</a></li> <li><a href="/facts/Summation/V2WUxoKn">Summations</a></li> <li><a href="/facts/Product_(mathematics)/aXY11nGS">Products</a></li> <li><a href="/facts/Number_theory/zhoa7zrc">Number theory</a>: generating <a href="/facts/Prime_number/L20FLkgn">Prime numbers</a>, primality testing, <a href="/facts/Integer_factorization/EjMCTz2t">integer factorization</a>, etc.</li> <li>Logic expressions<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a></li></ul> <h3>Matrices</h3> <ul><li>Basic arithmetic</li> <li><a href="/facts/Eigenvalue/8TjEoT8u">Eigenvalues</a> and their <a href="/facts/Eigenvector/8TjEoT8u">eigenvectors</a> when the <a href="/facts/Characteristic_polynomial/7m8roDqo">characteristic polynomial</a> is solvable by radicals</li> <li><a href="/facts/Determinant/eGYqJ2TF">Determinants</a></li> <li><a href="/facts/Inversion_(discrete_mathematics)/TR7RxJ07">Inversion</a></li> <li>Solving</li></ul> <h3>Geometry</h3> <ul><li><a href="/facts/Point_(geometry)/6YPAvX2a">Points</a>, <a href="/facts/Line_geometry/Hk9UeN0t">lines</a>, <a href="/facts/Line_(geometry)/gJqsr4Gj">rays</a>, <a href="/facts/Ellipse/1wUDnGxY">ellipses</a>, <a href="/facts/Circle/9fy5ggKn">circles</a>, <a href="/facts/Polygon/yAq7RjPz">polygons</a>, etc.</li> <li><a href="/facts/Intersection/hVW6QFWD">Intersections</a></li> <li><a href="/facts/Tangent/sF0jH3EI">Tangency</a></li> <li><a href="/facts/Similarity_(geometry)/Sbd6hEAN">Similarity</a></li></ul> <h3>Plotting</h3> <p>Note, plotting requires the external <a href="/facts/Matplotlib/JQSu8CBj">Matplotlib</a> or <a href="/facts/Pyglet/os0Jzzf7">Pyglet</a> module. </p> <ul><li>Coordinate models</li> <li>Plotting Geometric Entities</li> <li>2D and 3D</li> <li>Interactive interface</li> <li>Colors</li> <li><a href="/facts/Animation/093f9nUg">Animations</a></li></ul> <h3>Physics</h3> <ul><li>Units</li> <li><a href="/facts/Classical_mechanics/ykG8upsN">Classical mechanics</a></li> <li><a href="/facts/Continuum_mechanics/vI7LpFDk">Continuum mechanics</a><a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a></li> <li><a href="/facts/Quantum_mechanics/BRc3Mzgr">Quantum mechanics</a></li> <li><a href="/facts/Gaussian_optics/Uc9IBe1v">Gaussian optics</a></li> <li><a href="/facts/Linear_control/8CEjIDlS">Linear control</a></li></ul> <h3>Statistics</h3> <ul><li><a href="/facts/Normal_distribution/UapjjPyQ">Normal distributions</a></li> <li><a href="/facts/Uniform_distribution_(continuous)/XbnlVljT">Uniform distributions</a></li> <li><a href="/facts/Probability/fgYvMhwb">Probability</a></li></ul> <h3>Combinatorics</h3> <ul><li><a href="/facts/Permutations/JSw9ukmv">Permutations</a></li> <li><a href="/facts/Combinations/nmvrhMm1">Combinations</a></li> <li><a href="/facts/Infinitary_combinatorics/USrMLPmU">Partitions</a></li> <li><a href="/facts/Subset/SJGERmbA">Subsets</a></li> <li><a href="/facts/Permutation_group/vRsNNwwx">Permutation group</a>: <a href="/facts/Polyhedral_combinatorics/oVFESPjJ">Polyhedral</a>, Rubik, <a href="/facts/Symmetric_group/RUksK5l5">Symmetric</a>, etc.</li> <li><a href="/facts/Prufer_sequence/0tmI33Mm">Prufer sequence</a> and <a href="/facts/Gray_codes/HtbVlM7T">Gray codes</a></li></ul> <h3>Printing</h3> <ul><li><a href="/facts/Pretty-printing/GNPcN6eW">Pretty-printing</a>: <a href="/facts/ASCII/vGUI33Qu">ASCII</a>/<a href="/facts/Unicode/DDu3MfPV">Unicode</a> pretty-printing, LaTeX</li> <li><a href="/facts/Code_generation_(compiler)/6Ce9gjPs">Code generation</a>: C, <a href="/facts/Fortran/m1KqjcMU">Fortran</a>, Python</li></ul> <h2 id="related-projects">Related projects</h2> <ul><li><a href="/facts/SageMath/5mzh5TlB">SageMath</a>: an open source alternative to <a href="/facts/Wolfram_Mathematica/dRwoGFG2">Mathematica</a>, <a href="/facts/Maple_(software)/L87oPlmy">Maple</a>, <a href="/facts/MATLAB/qPjLISCk">MATLAB</a>, and <a href="/facts/Magma_(computer_algebra_system)/0kJ0CyPo">Magma</a> (SymPy is included in Sage)</li> <li>SymEngine: a rewriting of SymPy's core in <a href="/facts/C%2B%2B/Et0F2qn9">C++</a>, in order to increase its performance. Work is currently in progress[<i>as of?</i>] to make SymEngine the underlying engine of Sage too.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a></li> <li>mpmath: a Python library for <a href="/facts/Arbitrary-precision_arithmetic/cXxc8yB2">arbitrary-precision</a> <a href="/facts/Floating-point_arithmetic/eIckahxe">floating-point arithmetic</a><a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a></li> <li>SympyCore: another Python computer algebra system<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a></li> <li>SfePy: Software for solving systems of coupled partial differential equations (PDEs) by the finite element method in 1D, 2D and 3D.<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a></li> <li>GAlgebra: <a href="/facts/Geometric_algebra/yykKMNYx">Geometric algebra</a> module (previously sympy.galgebra).<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a></li> <li>Quameon: <a href="/facts/Quantum_Monte_Carlo/qenhfw9u">Quantum Monte Carlo</a> in Python.<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a></li> <li>Lcapy: Experimental Python package for teaching <a href="/facts/Network_analysis_(electrical_circuits)/IjmVqbG8">linear circuit analysis</a>.<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a></li> <li>LaTeX Expression project: Easy LaTeX typesetting of algebraic expressions in symbolic form with automatic substitution and result computation.<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a></li> <li>Symbolic statistical modeling: Adding statistical operations to complex physical models.<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a></li> <li>Diofant: a fork of SymPy, started by Sergey B Kirpichev<a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a></li></ul> <h2 id="dependencies">Dependencies</h2> <p>Since version 1.0, SymPy has the mpmath package as a dependency. </p><p>There are several optional dependencies that can enhance its capabilities: </p> <ul><li>gmpy: If gmpy 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.</li> <li><a href="/facts/Matplotlib/JQSu8CBj">matplotlib</a>: If matplotlib is installed, SymPy can use it for plotting.</li> <li><a href="/facts/Pyglet/os0Jzzf7">Pyglet</a>: Alternative plotting package.</li></ul> <h2 id="see-also">See also</h2> <ul> <li>Free and open-source software portal</li><li>Mathematics portal</li></ul> <ul><li><a href="/facts/Comparison_of_computer_algebra_systems/e5m8nyK2">Comparison of computer algebra systems</a></li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://www.sympy.org/">Official website</a> </li> <li><a href="http://planet.sympy.org/">Planet SymPy</a></li> <li><a href="https://coderslegacy.com/python/sympy-tutorial/">SymPy Tutorials Collection</a></li> <li><a href="https://github.com/sympy/sympy/">Code Repository on GitHub</a></li> <li><a href="https://groups.google.com/forum/#!forum/sympy">Support and development forum</a></li> <li><a href="https://gitter.im/sympy/sympy">Gitter chat room</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"SymPy Live". live.sympy.org. Retrieved 2021-08-25. <a href="https://live.sympy.org/" target="_blank">https://live.sympy.org/</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>"SymPy Gamma". www.sympygamma.com. Retrieved 2021-08-25. <a href="https://www.sympygamma.com/" target="_blank">https://www.sympygamma.com/</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>"SymPy homepage". Retrieved 2014-10-13. <a href="http://sympy.org" target="_blank">http://sympy.org</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Joyner, David; Čertík, Ondřej; Meurer, Aaron; Granger, Brian E. (2012). "Open source computer algebra systems: SymPy". ACM Communications in Computer Algebra. 45 (3/4): 225–234. doi:10.1145/2110170.2110185. S2CID 44862851. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Meurer, Aaron; Smith, Christopher P.; Paprocki, Mateusz; Čertík, Ondřej; Kirpichev, Sergey B.; Rocklin, Matthew; Kumar, AMiT; Ivanov, Sergiu; Moore, Jason K. (2017-01-02). "SymPy: symbolic computing in Python" (PDF). PeerJ Computer Science. 3: e103. doi:10.7717/peerj-cs.103. ISSN 2376-5992. <a href="http://dspace5.zcu.cz/bitstream/11025/29246/1/peerj-cs-103-1.pdf" target="_blank">http://dspace5.zcu.cz/bitstream/11025/29246/1/peerj-cs-103-1.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>"SymPy homepage". Retrieved 2014-10-13. <a href="http://sympy.org" target="_blank">http://sympy.org</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Joyner, David; Čertík, Ondřej; Meurer, Aaron; Granger, Brian E. (2012). "Open source computer algebra systems: SymPy". ACM Communications in Computer Algebra. 45 (3/4): 225–234. doi:10.1145/2110170.2110185. S2CID 44862851. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>"SymPy vs. Mathematica · sympy/Sympy Wiki". GitHub. <a href="https://github.com/sympy/sympy/wiki/SymPy-vs.-Mathematica" target="_blank">https://github.com/sympy/sympy/wiki/SymPy-vs.-Mathematica</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>"Sympy project statistics on Open HUB". Retrieved 2014-10-13. <a href="https://www.openhub.net/p/sympy" target="_blank">https://www.openhub.net/p/sympy</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>"SymPy homepage". Retrieved 2014-10-13. <a href="http://sympy.org" target="_blank">http://sympy.org</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Joyner, David; Čertík, Ondřej; Meurer, Aaron; Granger, Brian E. (2012). "Open source computer algebra systems: SymPy". ACM Communications in Computer Algebra. 45 (3/4): 225–234. doi:10.1145/2110170.2110185. S2CID 44862851. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Gede, Gilbert; Peterson, Dale L.; Nanjangud, Angadh; Moore, Jason K.; Hubbard, Mont (2013). Constrained multibody dynamics with Python: From symbolic equation generation to publication. ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers. pp. V07BT10A051. doi:10.1115/DETC2013-13470. ISBN 978-0-7918-5597-3. <a href="978-0-7918-5597-3" target="_blank">978-0-7918-5597-3</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Rocklin, Matthew; Terrel, Andy (2012). "Symbolic Statistics with SymPy". Computing in Science & Engineering. 14 (3): 88–93. Bibcode:2012CSE....14c..88R. doi:10.1109/MCSE.2012.56. S2CID 18307629. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Asif, Mushtaq; Olaussen, Kåre (2014). "Automatic code generator for higher order integrators". Computer Physics Communications. 185 (5): 1461–1472. arXiv:1310.2111. Bibcode:2014CoPhC.185.1461M. doi:10.1016/j.cpc.2014.01.012. S2CID 42041635. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>"Assumptions Module — SymPy 1.4 documentation". docs.sympy.org. Retrieved 2019-07-05. <a href="https://docs.sympy.org/latest/modules/assumptions/index.html" target="_blank">https://docs.sympy.org/latest/modules/assumptions/index.html</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>"Continuum Mechanics — SymPy 1.4 documentation". docs.sympy.org. Retrieved 2019-07-05. <a href="https://docs.sympy.org/latest/modules/physics/continuum_mechanics/index.html" target="_blank">https://docs.sympy.org/latest/modules/physics/continuum_mechanics/index.html</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>"GitHub - symengine/symengine: SymEngine is a fast symbolic manipulation library, written in C++". GitHub. 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Retrieved 2021-08-25. <a href="https://web.archive.org/web/20210116185108/http://lcapy.elec.canterbury.ac.nz/" target="_blank">https://web.archive.org/web/20210116185108/http://lcapy.elec.canterbury.ac.nz/</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> <li id="fn:24"><p>"LaTeX Expression project documentation — LaTeX Expression 0.3.dev documentation". mech.fsv.cvut.cz. Retrieved 2021-08-25. <a href="http://mech.fsv.cvut.cz/~stransky/software/latexexpr/doc/" target="_blank">http://mech.fsv.cvut.cz/~stransky/software/latexexpr/doc/</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li> <li id="fn:25"><p>"Symbolic Statistics with SymPy". ResearchGate. 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