Algebraic Equations with SymPy\n\nIntroduction | Output Formatting\n| Installation |\nTry Live | Issues or Comments |\nChange Log |\nLicense\n| GIT Repository\n| PyPi Link
\n\n\n\nIntroduction
\n\n
\n\nThis tool defines relations that all high school and college students would\nrecognize as mathematical equations. \nThey consist of a left hand side (lhs) and a right hand side (rhs) connected by\nthe relation operator \"=\". In addition, it sets some convenient defaults and \nprovides some controls of output formatting that may be useful even if\nyou do not use the Equation class (see Conveniences for\nSymPy).
\n\nThis tool applies operations to both sides of the equation simultaneously, just\nas students are taught to do when \nattempting to isolate (solve for) a variable. Thus the statement Equation/b\nyields a new equation Equation.lhs/b = Equation.rhs/b
\n\nThe intent is to allow using the mathematical tools in SymPy to rearrange\nequations and perform algebra\nin a stepwise fashion using as close to standard mathematical notation as \npossible. In this way more people can successfully perform \nalgebraic rearrangements without stumbling\nover missed details such as a negative sign.
\n\nA simple example as it would appear in a Jupyter \nnotebook is shown immediately below:
\n\n![\"screenshot](\"https://gutow.github.io/Algebra_with_Sympy/resources/simple_example.png\")
\n\nThe last cell illustrates how it is possible to substitute numbers with \nunits into the solved equation to calculate a numerical solution with \nproper units. The units(...) operation is part this package, not Sympy.
\n\nIn IPython environments (IPython, Jupyter, Google Colab, etc...) there is \nalso a shorthand syntax for entering equations provided through the IPython \npreparser. An equation can be specified as eq1 =@ a/b = c/d.
\n\n![\"screenshot](\"https://gutow.github.io/Algebra_with_Sympy/resources/short_syntax.png\")
\n\nIf no Python name is \nspecified for the equation (no eq_name to the left of =@), the equation \nwill still be defined, but will not be easily accessible for further \ncomputation. The =@ symbol combination was chosen to avoid conflicts with \nreserved python symbols while minimizing impacts on syntax highlighting \nand autoformatting.
\n\nMore examples of the capabilities of Algebra with Sympy are \nhere.
\n\nMany math packages such as SageMath \nand Maxima have similar capabilities, \nbut require more knowledge of command syntax, plus they cannot easily be \ninstalled in a generic python environment.
\n\n\n\nEven if you do not use the Equation class, there are some convenience \ntools and defaults that will probably make interactive use of SymPy in \nJupyter/IPython environments easier:
\n\n\n- By default, all numbers in Sympy expressions without decimal points are \ninterpreted as integers (e.g.
2/3*x, where x is a sympy symbol, -> \n2*x/3 not x*0.6666..., but if x is just a plain Python object then 2/3*x \n-> x*0.66666...). This can be turned off with unset_integers_as_exact(), \nwhich leads to standard Python behavior (2/3*x -> x*0.6666...) even for \nSympy expressions. Turn on with set_integers_as_exact(). When on the flag\nalgwsym_config.numerics.integers_as_exact = True. \n- Results of
solve() are wrapped in FiniteSet() to force pretty-printing \nof all of a solution set. See Controlling the Format of Interactive \nOutputs. \n- It is possible to set the default display to show both the pretty-printed \nresult and the code version simultaneously. See Controlling the Format of Interactive \nOutputs.
\n
\n\n\n\n
\n\n\n- These controls impact all Sympy objects and the
Equation class. \n- In graphical environments (Jupyter) you will get rendered Latex such as \n$\\frac{a}{b} = \\frac{c}{d}$ or $e^{\\frac{-x^2}{\\sigma^2}}$. To also see the \ncode representation (what can be copied and pasted for \nadditional computation) set
algwsym_config.output.show_code = True. \nThis will print the code version (e.g. Equation(a,b/c)) of equations \nand sympy expression in addition to the human readable version. This code \nversion can be accessed directly by calling repr() on the \nequation or expression. \n
\n\n\nIn interactive text environments (IPython and command line) The human \nreadable string version of Sympy expressions are returned (for Equations a \n= b rather than Equation(a,b)). This is equivalent to Calling print() \nor str() on an expression.
\n\n\n- To have the code version (can be copied and pasted as a \nPython statement) returned, set
algwsym_config.output.human_text = False. \n- Setting both
algwsym_config.output.human_text = True\nand algwsym_config.output.show_code = True, will return both the \ncode and human readable versions. \n
The equation label can be turned off by setting\nalgwsym_config.output.label = False.
Automatic wrapping of Equations as Latex equations can be activated \nby setting algwsym_config.output.latex_as_equations to True. The \ndefault is False. Setting this to True wraps output as LaTex equations,\nwrapping them in \\begin{equation}...\\end{equation}. Equations formatted \nthis way will not be labeled with the internal name for the equation, \nindependent of the setting of algwsym_config.output.label.
By default solutions output by solve() are returned as a SymPy \nFiniteSet() to force typesetting of the included solutions. To get Python \nlists instead you can override this for the whole session by setting\nalgwsym_config.output.solve_to_list = True. For a one-off, simply \nwrap the output of a solve in list() (e.g. list(solve(...))). One \nadvantage of list mode is that lists can be ordered. When\nalgwsym_config.output.solve_to_list = True solve() maintains the \nsolutions in the order the solve for variables were input.
\n\nSetup/Installation
\n\n
\n\n\n- Use pip to install in your python environment: \n
pip install -U Algebra-with-SymPy \n- To use in a running python session issue\nthe following command :
from algebra_with_sympy import *. \nThis will also import the SymPy tools. \n- If you want to isolate this tool from the global namespace you are \nworking with change the import statement \nto
import algebra_with_sympy as spa, where \nspa stands for \"SymPy Algebra\". Then all calls would be made to \nspa.funcname(). WARNING: Doing this makes shorthand equation input and \ncontrol of interactive output formats unavailable. To recover this \nfunctionality the following code must be run in the interactive session. \n
\n\nEquation = spa.Equation\nEqn = Equation\nalgwsym_config = spa.algwsym_config\n
\n\nTry in binder
\n\n\n![\"Binder\"](\"https://mybinder.org/badge_logo.svg\")
\n\n\n\n
\n\n\n\nChange Log
\n\n
\n\n\n- 1.1.0 (July 21, 2024)\n
\n- Setting integers as exact (
set_integers_as_exact(), the default) now \nonly sets integers as exact within Sympy and Algebra_with_Sympy \nexpressions. This increases compatibility with other packages that \ndepend on integers being Python integers. \n- Refuse to import Algebra_with_Sympy if an incompatible \nversion of Sympy is installed in the environment.
\n- Added warning explaining how to install a compatible version of Sympy.
\n
\n- 1.0.2 (July 5, 2024)\n
\n- Removed requirements for Jupyter and Jupyterlab as code will work in \nvanilla python or Google Colab.
\n- Workaround for Google Colab's inconsistent handling of mixed Latex and \nplain text strings. This impacted display of equation labels in Colab.
\n- BUG FIX: catch IPython not installed so that can run in plain vanilla \npython.
\n
\n- 1.0.1 (May 22, 2024)\n
\n- BUG FIX: equation labels that include underscore characters \"_\" are now \naccepted.
\n- BUG FIX: wrapping equations formatted as LaTex equation (ie. surrounded \nby
\\begin{equation}...\\end{equation}) in the $..$ code used to \nindicate markdown for MathJax was causing output errors in Quarto when \noutputing to .tex or .pdf. This is now fixed without negatively \nimpacting MathJax rendering. \n- BUG FIX: Singleton results of solve unnecessarily wrapped by extra list \nor finiteset. No longer double nested.
\n- BUG FIX: When returning lists make solve respect user order of solutions.
\n- BUG FIX: Equation output threw error when Algebra_with_Sympy was \nimported as a submodule. Equation labeling turned off for this type of \nimport to avoid error.
\n- BUG FIX: Equation labels are now copyable even with the newer MathJax \ncommonHTML rendering.
\n- Updates to requirements.txt.
\n- Documentation updates.
\n
\n- 1.0.0 (January 2, 2024)\n
\n- Added convenience operation
units(...) which takes a string of space \nseparated symbols to use as units. This simply declares the symbols \nto be positive, making them behave as units. This does not create units \nthat know about conversions, prefixes or systems of units. This lack \nis on purpose to provide units that require the user to worry about \nconversions (ideal in a teaching situation). To get units with built-in \nconversions see sympy.physics.units. \n- Fixed issue #23 where
cos() multiplied by a factor was not the same \ntype of object after simplify() acted on an expression. Required \nembedding the Equation type in the sympy library. Until Equation is \nincorporated into the primary Sympy repository a customized version of \nthe latest stable release will be used. \n- Fixed issue where trailing comments (ie.
# a comment at the end of a \nline) lead to input errors using compact =@ notation. \nalgwsym_config.output.latex_as_equations has a default value of False.\n Setting this to True wraps output as LaTex equations wrapping them \nin \\begin{equation}...\\end{equation}. Equations formatted this way \nwill not be labeled with the internal name for the equation.
\n- 0.12.0 (July 12, 2023)\n
\n- Now defaults to interpreting numbers without decimal points as integers. \nThis can be turned off with
unset_integers_as_exact() and on with\nset_integers_as_exact(). When on the flag\nalgwsym_config.numerics.integers_as_exact = True. \n
\n- 0.11.0 (June 5, 2023)\n
\n- Formatting of
FiniteSets overridden so that the contents always\npretty-print. This removes the necessity of special flags to get \npretty output from solve. \n- Sympy
solve() now works reliably with equations and outputs \npretty-printed solutions. \n- Added option
algwsym_config.output.solve_to_list = True which causes \nsolve() to return solutions sets as Python lists. Using this option \nprevents pretty-printing of the solutions produced by solve(). \nalgwsym_config.output.show_code and \nalgwsym_config.output.human_text now work for all sympy objects, not \njust Equation objects. This works\nin terminal, IPython terminal and Jupyter. This is achieved by hooking \ninto the python display_hook and IPython display_formatter.- Added jupyter to requirements.txt so that virtual environment builds\nwill include jupyter.
\n- The way
__version__ was handled could break pip install. Changed to\ngenerating the internal version during setup. This means the version\nis now available as algwsym_version. \n
\n- 0.10.0 (Sep. 5, 2022)\n
\n- Documentation updates and fixes.
\n- Significantly increased test coverage (~98%).
\n- Support for
Eqn.rewrite(Add) \n- Solving (e.g.
solve(Eqn,x)) now supported fully. Still experimental. \n- Bug fix: latex printing now supports custom printer.
\n- Substitution into an Equation using Equations is now \nsupported (e.g.
eq1.subs(eq2, eq3, ...)). \nalgebra_with_sympy.__version__ is now available for version checking \nwithin python.- Bug fix: preparsing for
=@ syntax no longer blocks obj? syntax for \ngetting docstrings in ipython. \n- More robust determination of equation names for labeling.
\n
\n- 0.9.4 (Aug. 11, 2022)\n
\n- Update to deal with new Sympy function
piecewise_exclusive in v1.11. \n- Added user warning if a function does not extend for use with
Equations \nas expected. This also allows the package to be used even when a function \nextension does fail. \n- Simplification of documentation preparation.
\n- Typo fixes in preparser error messages.
\n
\n- 0.9.3 (Aug. 9, 2022)\n
\n- Added check for new enough version of IPython to use the preparser.
\n- If IPython version too old, issue warning and do not accept
=@ shorthand. \n
\n- 0.9.2 (Jun. 5, 2022)\n
\n=@ shorthand syntax for defining equations in IPython compatible \nenvironments.- Fixed bug where
root() override called sqrt() on bare expressions. \n
\n- 0.9.1 (Mar. 24, 2022)\n
\n- Equations labeled with their python name, if they have one.
\n- Added flags to adjust human readable output and equation labeling.
\n- Accept equation as function argument in any position.
\n- First pass at
solve() accepting equations. \n- Added override of
root() to avoid warning messages. \n- More unit tests.
\n- First pass at documentation.
\n
\n- 0.9.0 functionality equivalent to extension of SymPy in\nPR#21333.
\n
\n\n\n\n\nThis program is free software: you can redistribute it and/or modify\n it under the terms of the GNU General Public License as published by\n the Free Software Foundation, either version 3 of the License, or\n (at your option) any later version.\n This program is distributed in the hope that it will be useful,\n but WITHOUT ANY WARRANTY; without even the implied warranty of\n MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the\n GNU General Public License for more details.
\n\nCopyright - Algebra with Sympy Contributors 2021, 2022, 2023, 2024
\n\nDevelopment Notes
\n\nGeneral | Make Docs | \nRunning Tests | \nBuild PyPi Package|
\n\nGeneral Notes
\n\n
\n\n\n- TODOs\n
\n- Test collect when there isn't an available _eval_collect (not sure how \nto get there).
\n- Test for _binary_op NotImplemented error (not sure how to get there).
\n
\n- To consider\n
\n- Include Sympy Plot Backends\nin the default setup.
\n- Change
Equation constructor to accept Equality, Set, List or \nlhs, rhs, rather than just lhs, rhs. \n- Extend
.subs to accept .subs(a=2*c, b = sin(q), ...). \n- MathLive on another web page as possible\ninput engine.
\n
\n
\n\nConstructing the Documentation
\n\n
\n\n\n- Make sure pdoc is installed and updated in the virtual environment
pip \ninstall -U pdoc. \n- Update any
.md files included in _init_.py.\n\n- Generally URLs should be absolute, not relative.
\n
\n- At the root level run pdoc \n
\npdoc --logo https://gutow.github.io/Algebra_with_Sympy/alg_w_sympy.svg \n--logo-link https://gutow.github.io/Algebra_with_Sympy/\n--footer-text \"Algebra with Sympy vX.X.X\" --math -html -o docs \n./algebra_with_sympy\n\nwhere X.X.X is the version number. \n
\n\nTasks for Documentation
\n\n
\n\n\n- Readme.md & Development Notes.md\n
\n- Use absolute path to github pages for more examples.
\n
\n
\n\nRunning Tests
\n\n
\n\n\n- Install updated pytest in the virtual environment:\n
pipenv shell\npip install -U pytest\n
\n- Run standard tests:\n
pytest --ignore='Developer Testing' --ignore-glob='*test_preparser.py'. \n- Run preparser tests:\n
ipython -m pytest tests/test_preparser.py \n- Run doctests:\n
pytest --ignore='tests' --ignore='Developer Testing' \n--ignore-glob='*old*' --doctest-modules \n
\n\nYou can run all the tests using the dotests script: ./dotests.sh.
\n\nNOTE: Some warnings about invalid escape characters are expected because \nraw strings are being passed with specialized LaTex escaped characters.
\n\nBuilding PyPi package
\n\n
\n\n\n- Make sure to update the version number in setup.py first.
\n- Install updated setuptools and twine in the virtual environment:\n
pipenv shell\npip install -U setuptools wheel twine\n
\n- Build the distribution
python -m setup sdist bdist_wheel. \n- Test it on
test.pypi.org.\n\n- Upload it (you will need an account on test.pypi.org):\n
python -m twine upload --repository testpypi dist/*. \n- Create a new virtual environment and test install into it:\n
exit # to get out of the current environment\ncd <somewhere>\nmkdir <new virtual environment>\ncd <new directory>\npipenv shell #creates the new environment and enters it.\npip install -i https://test.pypi.org/..... # copy actual link from the\n # repository on test.pypi.\n
\nThere are often install issues because sometimes only older versions of\nsome of the required packages are available on test.pypi.org. If this\nis the only problem change the version to end in rc0 for release\ncandidate and try it on the regular pypi.org as described below for\nreleasing on PyPi. \n- After install test by running a jupyter notebook in the virtual \nenvironment.
\n
\n
\n\nReleasing on PyPi
\n\n\nProceed only if testing of the build is successful.
\n\n\n- Double check the version number in version.py.
\n- Rebuild the release:
python -m setup sdist bdist_wheel. \n- Upload it:
python -m twine upload dist/* \n- Make sure it works by installing it in a clean virtual environment. This\nis the same as on test.pypi.org except without
-i https://test.pypy.... If\nit does not work, pull the release. \n
\n"}, "algebra_with_sympy.proper_sympy": {"fullname": "algebra_with_sympy.proper_sympy", "modulename": "algebra_with_sympy", "qualname": "proper_sympy", "kind": "variable", "doc": "\n", "default_value": "True"}, "algebra_with_sympy.algebraic_equation": {"fullname": "algebra_with_sympy.algebraic_equation", "modulename": "algebra_with_sympy.algebraic_equation", "kind": "module", "doc": "This package uses a special version of sympy which defines an equation \nwith a left-hand-side (lhs) and a right-\nhand-side (rhs) connected by the \"=\" operator (e.g. p*V = n*R*T).
\n\nThe intent is to allow using the mathematical tools in SymPy to rearrange\nequations and perform algebra in a stepwise fashion. In this way more people\ncan successfully perform algebraic rearrangements without stumbling over\nmissed details such as a negative sign. This mimics the capabilities available\nin SageMath and\nMaxima.
\n\nThis package also provides convenient settings for interactive use on the \ncommand line, in ipython and Jupyter notebook environments. See the \ndocumentation at https://gutow.github.io/Algebra_with_Sympy/.
\n\nExplanation
\n\nThis class defines relations that all high school and college students\nwould recognize as mathematical equations. At present only the \"=\" relation\noperator is recognized.
\n\nThis class is intended to allow using the mathematical tools in SymPy to\nrearrange equations and perform algebra in a stepwise fashion. In this\nway more people can successfully perform algebraic rearrangements without\nstumbling over missed details such as a negative sign.
\n\nCreate an equation with the call Equation(lhs,rhs), where lhs and\nrhs are any valid Sympy expression. Eqn(...) is a synonym for\nEquation(...).
\n\nParameters
\n\nlhs: sympy expression, class Expr.\nrhs: sympy expression, class Expr.\nkwargs:
\n\nExamples
\n\nNOTE: All the examples below are in vanilla python. You can get human\nreadable eqautions \"lhs = rhs\" in vanilla python by adjusting the settings\nin algwsym_config (see it's documentation). Output is human readable by\ndefault in IPython and Jupyter environments.
\n\n\n
>>> from algebra_with_sympy import *\n>>> a, b, c, x = var('a b c x')\n>>> Equation(a,b/c)\nEquation(a, b/c)\n>>> t=Eqn(a,b/c)\n>>> t\nEquation(a, b/c)\n>>> t*c\nEquation(a*c, b)\n>>> c*t\nEquation(a*c, b)\n>>> exp(t)\nEquation(exp(a), exp(b/c))\n>>> exp(log(t))\nEquation(a, b/c)\n
\n
Simplification and Expansion
\n\n\n
>>> f = Eqn(x**2 - 1, c)\n>>> f\nEquation(x**2 - 1, c)\n>>> f/(x+1)\nEquation((x**2 - 1)/(x + 1), c/(x + 1))\n>>> (f/(x+1)).simplify()\nEquation(x - 1, c/(x + 1))\n>>> simplify(f/(x+1))\nEquation(x - 1, c/(x + 1))\n>>> (f/(x+1)).expand()\nEquation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))\n>>> expand(f/(x+1))\nEquation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))\n>>> factor(f)\nEquation((x - 1)*(x + 1), c)\n>>> f.factor()\nEquation((x - 1)*(x + 1), c)\n>>> f2 = f+a*x**2+b*x +c\n>>> f2\nEquation(a*x**2 + b*x + c + x**2 - 1, a*x**2 + b*x + 2*c)\n>>> collect(f2,x)\nEquation(b*x + c + x**2*(a + 1) - 1, a*x**2 + b*x + 2*c)\n
\n
Apply operation to only one side
\n\n\n
>>> poly = Eqn(a*x**2 + b*x + c*x**2, a*x**3 + b*x**3 + c*x)\n>>> poly.applyrhs(factor,x)\nEquation(a*x**2 + b*x + c*x**2, x*(c + x**2*(a + b)))\n>>> poly.applylhs(factor)\nEquation(x*(a*x + b + c*x), a*x**3 + b*x**3 + c*x)\n>>> poly.applylhs(collect,x)\nEquation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)\n
\n
.apply... also works with user defined python functions
\n\n\n
>>> def addsquare(eqn):\n... return eqn+eqn**2\n...\n>>> t.apply(addsquare)\nEquation(a**2 + a, b**2/c**2 + b/c)\n>>> t.applyrhs(addsquare)\nEquation(a, b**2/c**2 + b/c)\n>>> t.apply(addsquare, side = 'rhs')\nEquation(a, b**2/c**2 + b/c)\n>>> t.applylhs(addsquare)\nEquation(a**2 + a, b/c)\n>>> addsquare(t)\nEquation(a**2 + a, b**2/c**2 + b/c)\n
\n
Inaddition to .apply... there is also the less general .do,\n.dolhs, .dorhs, which only works for operations defined on the\nExpr class (e.g..collect(), .factor(), .expand(), etc...).
\n\n\n
>>> poly.dolhs.collect(x)\nEquation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)\n>>> poly.dorhs.collect(x)\nEquation(a*x**2 + b*x + c*x**2, c*x + x**3*(a + b))\n>>> poly.do.collect(x)\nEquation(b*x + x**2*(a + c), c*x + x**3*(a + b))\n>>> poly.dorhs.factor()\nEquation(a*x**2 + b*x + c*x**2, x*(a*x**2 + b*x**2 + c))\n
\n
poly.do.exp() or other sympy math functions will raise an error.
\n\nRearranging an equation (simple example made complicated as illustration)
\n\n\n
>>> p, V, n, R, T = var('p V n R T')\n>>> eq1=Eqn(p*V,n*R*T)\n>>> eq1\nEquation(V*p, R*T*n)\n>>> eq2 =eq1/V\n>>> eq2\nEquation(p, R*T*n/V)\n>>> eq3 = eq2/R/T\n>>> eq3\nEquation(p/(R*T), n/V)\n>>> eq4 = eq3*R/p\n>>> eq4\nEquation(1/T, R*n/(V*p))\n>>> 1/eq4\nEquation(T, V*p/(R*n))\n>>> eq5 = 1/eq4 - T\n>>> eq5\nEquation(0, -T + V*p/(R*n))\n
\n
Substitution (#'s and units)
\n\n\n
>>> L, atm, mol, K = var('L atm mol K', positive=True, real=True) # units\n>>> eq2.subs({R:0.08206*L*atm/mol/K,T:273*K,n:1.00*mol,V:24.0*L})\nEquation(p, 0.9334325*atm)\n>>> eq2.subs({R:0.08206*L*atm/mol/K,T:273*K,n:1.00*mol,V:24.0*L}).evalf(4)\nEquation(p, 0.9334*atm)\n
\n
Substituting an equation into another equation:
\n\n\n
>>> P, P1, P2, A1, A2, E1, E2 = symbols("P, P1, P2, A1, A2, E1, E2")\n>>> eq1 = Eqn(P, P1 + P2)\n>>> eq2 = Eqn(P1 / (A1 * E1), P2 / (A2 * E2))\n>>> P1_val = (eq1 - P2).swap\n>>> P1_val\nEquation(P1, P - P2)\n>>> eq2 = eq2.subs(P1_val)\n>>> eq2\nEquation((P - P2)/(A1*E1), P2/(A2*E2))\n>>> P2_val = solve(eq2.subs(P1_val), P2).args[0]\n>>> P2_val\nEquation(P2, A2*E2*P/(A1*E1 + A2*E2))\n
\n
Combining equations (Math with equations: lhs with lhs and rhs with rhs)
\n\n\n
>>> q = Eqn(a*c, b/c**2)\n>>> q\nEquation(a*c, b/c**2)\n>>> t\nEquation(a, b/c)\n>>> q+t\nEquation(a*c + a, b/c + b/c**2)\n>>> q/t\nEquation(c, 1/c)\n>>> t**q\nEquation(a**(a*c), (b/c)**(b/c**2))\n
\n
Utility operations
\n\n\n
>>> t.reversed\nEquation(b/c, a)\n>>> t.swap\nEquation(b/c, a)\n>>> t.lhs\na\n>>> t.rhs\nb/c\n>>> t.as_Boolean()\nEq(a, b/c)\n
\n
.check() convenience method for .as_Boolean().simplify()
\n\n\n
>>> from sympy import I, pi\n>>> Equation(pi*(I+2), pi*I+2*pi).check()\nTrue\n>>> Eqn(a,a+1).check()\nFalse\n
\n
Differentiation\nDifferentiation is applied to both sides if the wrt variable appears on\nboth sides.
\n\n\n
>>> q=Eqn(a*c, b/c**2)\n>>> q\nEquation(a*c, b/c**2)\n>>> diff(q,b)\nEquation(Derivative(a*c, b), c**(-2))\n>>> diff(q,c)\nEquation(a, -2*b/c**3)\n>>> diff(log(q),b)\nEquation(Derivative(log(a*c), b), 1/b)\n>>> diff(q,c,2)\nEquation(Derivative(a, c), 6*b/c**4)\n
\n
If you specify multiple differentiation all at once the assumption\nis order of differentiation matters and the lhs will not be\nevaluated.
\n\n\n
>>> diff(q,c,b)\nEquation(Derivative(a*c, b, c), -2/c**3)\n
\n
To overcome this specify the order of operations.
\n\n\n
>>> diff(diff(q,c),b)\nEquation(Derivative(a, b), -2/c**3)\n
\n
But the reverse order returns an unevaulated lhs (a may depend on b).
\n\n\n
>>> diff(diff(q,b),c)\nEquation(Derivative(a*c, b, c), -2/c**3)\n
\n
Integration can only be performed on one side at a time.
\n\n\n
>>> q=Eqn(a*c,b/c)\n>>> integrate(q,b,side='rhs')\nb**2/(2*c)\n>>> integrate(q,b,side='lhs')\na*b*c\n
\n
Make a pretty statement of integration from an equation
\n\n\n
>>> Eqn(Integral(q.lhs,b),integrate(q,b,side='rhs'))\nEquation(Integral(a*c, b), b**2/(2*c))\n
\n
Integration of each side with respect to different variables
\n\n\n
>>> q.dorhs.integrate(b).dolhs.integrate(a)\nEquation(a**2*c/2, b**2/(2*c))\n
\n
Automatic solutions using sympy solvers. THIS IS EXPERIMENTAL. Please\nreport issues at https://github.com/gutow/Algebra_with_Sympy/issues.
\n\n\n
>>> tosolv = Eqn(a - b, c/a)\n>>> solve(tosolv,a)\nFiniteSet(Equation(a, b/2 - sqrt(b**2 + 4*c)/2), Equation(a, b/2 + sqrt(b**2 + 4*c)/2))\n>>> solve(tosolv, b)\nFiniteSet(Equation(b, (a**2 - c)/a))\n>>> solve(tosolv, c)\nFiniteSet(Equation(c, a**2 - a*b))\n
\n
This is a class to hold parameters that control behavior of\nthe algebra_with_sympy package.
\n\nSettings
\n\nPrinting
\n\nIn interactive environments the default output of an equation is a\nhuman readable string with the two sides connected by an equals\nsign or a typeset equation with the two sides connected by an equals sign.\nprint(Eqn) or str(Eqn) will return this human readable text version of\nthe equation as well. This is consistent with python standards, but not\nsympy, where str() is supposed to return something that can be\ncopy-pasted into code. If the equation has a declared name as in eq1 =\nEqn(a,b/c) the name will be displayed to the right of the equation in\nparentheses (eg. a = b/c (eq1)). Use print(repr(Eqn)) instead of\nprint(Eqn) or repr(Eqn) instead of str(Eqn) to get a code\ncompatible version of the equation.
\n\nYou can adjust this behavior using some flags that impact output:
\n\n\nalgwsym_config.output.show_code default is False.algwsym_config.output.human_text default is True.algwsym_config.output.label default is True.algwsym_config.output.latex_as_equations default is False
\n\nIn interactive environments you can get both types of output by setting\nthe algwsym_config.output.show_code flag. If this flag is true\ncalls to latex and str will also print an additional line \"code\nversion: repr(Eqn)\". Thus in Jupyter you will get a line of typeset\nmathematics output preceded by the code version that can be copy-pasted.\nDefault is False.
\n\nA second flag algwsym_config.output.human_text is useful in\ntext-based interactive environments such as command line python or\nipython. If this flag is true repr will return str. Thus the human\nreadable text will be printed as the output of a line that is an\nexpression containing an equation.\nDefault is True.
\n\nSetting both of these flags to true in a command line or ipython\nenvironment will show both the code version and the human readable text.\nThese flags impact the behavior of the print(Eqn) statement.
\n\nThe third flag algwsym_config.output.label has a default value of\nTrue. Setting this to False suppresses the labeling of an equation\nwith its python name off to the right of the equation.
\n\nThe fourth flag algwsym_config.output.latex_as_equations has\na default value of False. Setting this to True wraps\noutput as LaTex equations wrapping them in \\begin{equation}...\\end{\nequation}.
\n", "signature": "()"}, "algebra_with_sympy.algebraic_equation.algwsym_config.output": {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output", "kind": "class", "doc": "\n"}, "algebra_with_sympy.algebraic_equation.algwsym_config.output.__init__": {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output.__init__", "kind": "function", "doc": "This holds settings that impact output.
\n", "signature": "()"}, "algebra_with_sympy.algebraic_equation.algwsym_config.output.show_code": {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output.show_code", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output.show_code", "kind": "variable", "doc": "If True code versions of the equation expression will be\noutput in interactive environments. Default = False.
\n", "default_value": "False"}, "algebra_with_sympy.algebraic_equation.algwsym_config.output.human_text": {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output.human_text", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output.human_text", "kind": "variable", "doc": "If True the human readable equation expression will be\noutput in text interactive environments. Default = False.
\n", "default_value": "True"}, "algebra_with_sympy.algebraic_equation.algwsym_config.output.solve_to_list": {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output.solve_to_list", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output.solve_to_list", "kind": "variable", "doc": "If True the results of a call to solve(...) will return a\nPython list rather than a Sympy FiniteSet. This recovers\nbehavior for versions before 0.11.0.
\n\nNote: setting this True means that expressions within the\nreturned solutions will not be pretty-printed in Jupyter and\nIPython.
\n", "default_value": "False"}, "algebra_with_sympy.algebraic_equation.algwsym_config.output.latex_as_equations": {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output.latex_as_equations", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output.latex_as_equations", "kind": "variable", "doc": "If True any output that is returned as LaTex for\npretty-printing will be wrapped in the formal Latex for an\nequation. For example rather than
\n\n$\\frac{a}{b}=c$\n
\n\nthe output will be
\n\n\\begin{equation}\\frac{a}{b}=c\\end{equation}\n
\n", "default_value": "False"}, "algebra_with_sympy.algebraic_equation.algwsym_config.output.label": {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output.label", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output.label", "kind": "variable", "doc": "\n", "default_value": "True"}, "algebra_with_sympy.algebraic_equation.algwsym_config.numerics": {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.numerics", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.numerics", "kind": "class", "doc": "\n"}, "algebra_with_sympy.algebraic_equation.algwsym_config.numerics.__init__": {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.numerics.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.numerics.__init__", "kind": "function", "doc": "This class holds settings for how numerical computation and\ninputs are handled.
\n", "signature": "()"}, "algebra_with_sympy.algebraic_equation.algwsym_config.numerics.integers_as_exact": {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.numerics.integers_as_exact", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.numerics.integers_as_exact", "kind": "function", "doc": "This is a flag for informational purposes and interface\nconsistency. Changing the value will not change the behavior.
\n\nTo change the behavior call:
\n\n\nunset_integers_as_exact() to turn this feature off.set_integers_as_exact() to turn this feature on (on by\ndefault).
\n\nIf set to True (the default) and if running in an\nIPython/Jupyter environment any number input without a decimal\nwill be interpreted as a sympy integer. Thus, fractions and\nrelated expressions will not evalute to floating point numbers,\nbut be maintained as exact expressions (e.g. 2/3 -> 2/3 not the\nfloat 0.6666...).
\n", "signature": "(self):", "funcdef": "def"}, "algebra_with_sympy.algebraic_equation.ip": {"fullname": "algebra_with_sympy.algebraic_equation.ip", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "ip", "kind": "variable", "doc": "\n", "default_value": "None"}, "algebra_with_sympy.algebraic_equation.formatter": {"fullname": "algebra_with_sympy.algebraic_equation.formatter", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "formatter", "kind": "variable", "doc": "\n", "default_value": "None"}, "algebra_with_sympy.algebraic_equation.set_integers_as_exact": {"fullname": "algebra_with_sympy.algebraic_equation.set_integers_as_exact", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "set_integers_as_exact", "kind": "function", "doc": "This operation causes any number input without a decimal that is\npart of a Sympy expression to be interpreted as a sympy\ninteger, by using a custom preparser to cast integers within Sympy\nexpressions as Sympy integers (Integer()). It also sets the flag\nalgwsym_config.numerics.integers_as_exact = True This is the default\nmode of algebra_with_sympy. To turn this off call\nunset_integers_as_exact().
\n\nNOTE: 2/3 --> 0.6666... even when this is set, but 2*x/3 -->\nInteger(2)/Integer(3)*x if x is a sympy object. If x is just a Python\nobject 2*x/3 --> x*0.6666666666....
\n", "signature": "():", "funcdef": "def"}, "algebra_with_sympy.algebraic_equation.unset_integers_as_exact": {"fullname": "algebra_with_sympy.algebraic_equation.unset_integers_as_exact", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "unset_integers_as_exact", "kind": "function", "doc": "This operation disables forcing of numbers input without\ndecimals being interpreted as sympy integers. Numbers input without a\ndecimal may be interpreted as floating point if they are part of an\nexpression that undergoes python evaluation (e.g. 2/3 -> 0.6666...). It\nalso sets the flag algwsym_config.numerics.integers_as_exact = False.\nCall set_integers_as_exact() to avoid this conversion of rational\nfractions and related expressions to floating point. Algebra_with_sympy\nstarts with set_integers_as_exact() enabled (\nalgwsym_config.numerics.integers_as_exact = True).
\n", "signature": "():", "funcdef": "def"}, "algebra_with_sympy.algebraic_equation.Eqn": {"fullname": "algebra_with_sympy.algebraic_equation.Eqn", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn", "kind": "variable", "doc": "\n", "default_value": "<class 'sympy.core.equation.Equation'>"}, "algebra_with_sympy.algebraic_equation.units": {"fullname": "algebra_with_sympy.algebraic_equation.units", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "units", "kind": "function", "doc": "This operation declares the symbols to be positive values, so that sympy\nwill handle them properly when simplifying expressions containing units.\nUnits defined this way are just unit symbols. If you want units that are\naware of conversions see sympy.physics.units.
\n\nParameters
\n\n\n- string names: a string containing a space separated list of\nsymbols to be treated as units.
\n
\n\nReturns
\n\n\n calls name = symbols(name,\npositive=True) in the interactive namespace for each symbol name.
\n
\n", "signature": "(names):", "funcdef": "def"}, "algebra_with_sympy.algebraic_equation.solve": {"fullname": "algebra_with_sympy.algebraic_equation.solve", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "solve", "kind": "function", "doc": "Override of sympy solve().
\n\nIf passed an expression and variable(s) to solve for it behaves\nalmost the same as normal solve with dict = True, except that solutions\nare wrapped in a FiniteSet() to guarantee that the output will be pretty\nprinted in Jupyter like environments.
\n\nIf passed an equation or equations it returns solutions as a\nFiniteSet() of solutions, where each solution is represented by an\nequation or set of equations.
\n\nTo get a Python list of solutions (pre-0.11.0 behavior) rather than a\nFiniteSet issue the command algwsym_config.output.solve_to_list = True.\nThis also prevents pretty-printing in IPython and Jupyter.
\n\nExamples
\n\n\n
>>> a, b, c, x, y = symbols('a b c x y', real = True)\n>>> import sys\n>>> sys.displayhook = __command_line_printing__ # set by default on normal initialization.\n>>> eq1 = Eqn(abs(2*x+y),3)\n>>> eq2 = Eqn(abs(x + 2*y),3)\n>>> B = solve((eq1,eq2))\n
\n
Default human readable output on command line
\n\n\n
>>> B\n{{x = -3, y = 3}, {x = -1, y = -1}, {x = 1, y = 1}, {x = 3, y = -3}}\n
\n
To get raw output turn off by setting
\n\n\n
>>> algwsym_config.output.human_text=False\n>>> B\nFiniteSet(FiniteSet(Equation(x, -3), Equation(y, 3)), FiniteSet(Equation(x, -1), Equation(y, -1)), FiniteSet(Equation(x, 1), Equation(y, 1)), FiniteSet(Equation(x, 3), Equation(y, -3)))\n
\n
Pre-0.11.0 behavior where a python list of solutions is returned
\n\n\n
>>> algwsym_config.output.solve_to_list = True\n>>> solve((eq1,eq2))\n[[Equation(x, -3), Equation(y, 3)], [Equation(x, -1), Equation(y, -1)], [Equation(x, 1), Equation(y, 1)], [Equation(x, 3), Equation(y, -3)]]\n>>> algwsym_config.output.solve_to_list = False # reset to default\n
\n
algwsym_config.output.human_text = True with\nalgwsym_config.output.how_code=True shows both.\nIn Jupyter-like environments show_code=True yields the Raw output and\na typeset version. If show_code=False (the default) only the\ntypeset version is shown in Jupyter.
\n\n\n
>>> algwsym_config.output.show_code=True\n>>> algwsym_config.output.human_text=True\n>>> B\nCode version: FiniteSet(FiniteSet(Equation(x, -3), Equation(y, 3)), FiniteSet(Equation(x, -1), Equation(y, -1)), FiniteSet(Equation(x, 1), Equation(y, 1)), FiniteSet(Equation(x, 3), Equation(y, -3)))\n{{x = -3, y = 3}, {x = -1, y = -1}, {x = 1, y = 1}, {x = 3, y = -3}}\n
\n
Very experimental override of sympy solveset, which we hope will replace\nsolve. Much is not working. It is not clear how to input a system of\nequations unless you directly select linsolve, etc...
\n", "signature": "(f, symbols, domain=Complexes):", "funcdef": "def"}, "algebra_with_sympy.algebraic_equation.Equality": {"fullname": "algebra_with_sympy.algebraic_equation.Equality", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equality", "kind": "class", "doc": "Extension of Equality class to include the ability to convert it to an\nEquation.
\n", "bases": "sympy.core.relational.Equality"}, "algebra_with_sympy.algebraic_equation.Equality.to_Equation": {"fullname": "algebra_with_sympy.algebraic_equation.Equality.to_Equation", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equality.to_Equation", "kind": "function", "doc": "Return: recasts the Equality as an Equation.
\n", "signature": "(self):", "funcdef": "def"}, "algebra_with_sympy.algebraic_equation.Equality.to_Eqn": {"fullname": "algebra_with_sympy.algebraic_equation.Equality.to_Eqn", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equality.to_Eqn", "kind": "function", "doc": "Synonym for to_Equation.\nReturn: recasts the Equality as an Equation.
\n", "signature": "(self):", "funcdef": "def"}, "algebra_with_sympy.algebraic_equation.Equality.default_assumptions": {"fullname": "algebra_with_sympy.algebraic_equation.Equality.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equality.default_assumptions", "kind": "variable", "doc": "\n", "default_value": "{}"}, "algebra_with_sympy.algebraic_equation.Eq": {"fullname": "algebra_with_sympy.algebraic_equation.Eq", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eq", "kind": "variable", "doc": "\n", "default_value": "<class 'algebra_with_sympy.algebraic_equation.Equality'>"}, "algebra_with_sympy.algebraic_equation.abs": {"fullname": "algebra_with_sympy.algebraic_equation.abs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "abs", "kind": "variable", "doc": "\n", "default_value": "Abs"}, "algebra_with_sympy.preparser": {"fullname": "algebra_with_sympy.preparser", "modulename": "algebra_with_sympy.preparser", "kind": "module", "doc": "\n"}, "algebra_with_sympy.preparser.algebra_with_sympy_preparser": {"fullname": "algebra_with_sympy.preparser.algebra_with_sympy_preparser", "modulename": "algebra_with_sympy.preparser", "qualname": "algebra_with_sympy_preparser", "kind": "function", "doc": "In IPython compatible environments (Jupyter, IPython, etc...) this supports\na special compact input method for equations.
\n\nThe syntax supported is equation_name =@ equation.lhs = equation.rhs,\nwhere equation_name is a valid Python name that can be used to refer to\nthe equation later. equation.lhs is the left-hand side of the equation\nand equation.rhs is the right-hand side of the equation. Each side of the\nequation must parse into a valid Sympy expression.
\n\nNote: This does not support line continuation. Long equations should be\nbuilt by combining expressions using names short enough to do this on one\nline. The alternative is to use equation_name = Eqn(long ...\nexpressions ... with ... multiple ... lines).
\n\nNote: If the equation_name is omitted the equation will be formed,\nbut it will not be assigned to a name that can be used to refer to it\nlater. You may be able to access it through one of the special IPython\nunderscore names. This is not recommended.
\n\nTHIS FUNCTION IS USED BY THE IPYTHON ENVIRONMENT TO PREPARSE THE INPUT\nBEFORE IT IS PASSED TO THE PYTHON INTERPRETER. IT IS NOT MEANT TO BE USED\nDIRECTLY BY A USER
\n", "signature": "(lines):", "funcdef": "def"}, "algebra_with_sympy.preparser.toIntegerInSympyExpr": {"fullname": "algebra_with_sympy.preparser.toIntegerInSympyExpr", "modulename": "algebra_with_sympy.preparser", "qualname": "toIntegerInSympyExpr", "kind": "function", "doc": "This function takes a string of valid Python and wraps integers within Sympy expressions\nin sympy.Integer() to make them Sympy integers rather than Python Int(). The\nadvantage of this is that calculations with Integer() types can be exact. This function\nis careful not to wrap Int() types that are not part of Sympy expressions, making it\npossible for this functionality to exist with operations (e.g. array and numpy indexing)\nthat are not compatible with the Integer() type.
\n", "signature": "(string):", "funcdef": "def"}, "algebra_with_sympy.preparser.integers_as_exact": {"fullname": "algebra_with_sympy.preparser.integers_as_exact", "modulename": "algebra_with_sympy.preparser", "qualname": "integers_as_exact", "kind": "function", "doc": "This preparser uses sympy.interactive.session.int_to_Integer to\nconvert numbers without decimal points into sympy integers so that math\non them will be exact rather than defaulting to floating point. This\nshould not be called directly by the user. It is plugged into the\nIPython preparsing sequence when the feature is requested. The default for\nAlgebra_with_sympy is to use this preparser. This can be turned on and\noff using the Algebra_with_sympy functions:
\n\n\nset_integers_as_exact()unset_integers_as_exact()\nNOTE: This option does not work in plain vanilla Python sessions. You\nmust be running in an IPython environment (Jupyter, Notebook, Colab,\netc...).
\n", "signature": "(lines):", "funcdef": "def"}, "algebra_with_sympy.version": {"fullname": "algebra_with_sympy.version", "modulename": "algebra_with_sympy.version", "kind": "module", "doc": "\n"}}, "docInfo": {"algebra_with_sympy": {"qualname": 0, "fullname": 3, "annotation": 0, "default_value": 0, "signature": 0, "bases": 0, "doc": 3534}, "algebra_with_sympy.proper_sympy": {"qualname": 2, "fullname": 5, "annotation": 0, "default_value": 1, "signature": 0, "bases": 0, "doc": 3}, "algebra_with_sympy.algebraic_equation": {"qualname": 0, "fullname": 5, "annotation": 0, "default_value": 0, "signature": 0, "bases": 0, "doc": 4002}, "algebra_with_sympy.algebraic_equation.algwsym_config": {"qualname": 2, "fullname": 7, "annotation": 0, "default_value": 0, "signature": 0, "bases": 0, "doc": 3}, "algebra_with_sympy.algebraic_equation.algwsym_config.__init__": {"qualname": 4, "fullname": 9, "annotation": 0, "default_value": 0, "signature": 4, "bases": 0, "doc": 530}, "algebra_with_sympy.algebraic_equation.algwsym_config.output": {"qualname": 3, "fullname": 8, "annotation": 0, "default_value": 0, "signature": 0, "bases": 0, "doc": 3}, "algebra_with_sympy.algebraic_equation.algwsym_config.output.__init__": {"qualname": 5, "fullname": 10, "annotation": 0, "default_value": 0, "signature": 4, "bases": 0, "doc": 9}, "algebra_with_sympy.algebraic_equation.algwsym_config.output.show_code": {"qualname": 5, "fullname": 10, "annotation": 0, "default_value": 1, "signature": 0, "bases": 0, "doc": 23}, "algebra_with_sympy.algebraic_equation.algwsym_config.output.human_text": {"qualname": 5, "fullname": 10, "annotation": 0, "default_value": 1, "signature": 0, "bases": 0, "doc": 23}, "algebra_with_sympy.algebraic_equation.algwsym_config.output.solve_to_list": {"qualname": 6, "fullname": 11, "annotation": 0, "default_value": 1, "signature": 0, "bases": 0, "doc": 65}, "algebra_with_sympy.algebraic_equation.algwsym_config.output.latex_as_equations": {"qualname": 6, "fullname": 11, "annotation": 0, "default_value": 1, "signature": 0, "bases": 0, "doc": 50}, "algebra_with_sympy.algebraic_equation.algwsym_config.output.label": {"qualname": 4, "fullname": 9, "annotation": 0, "default_value": 1, "signature": 0, "bases": 0, "doc": 3}, "algebra_with_sympy.algebraic_equation.algwsym_config.numerics": {"qualname": 3, "fullname": 8, "annotation": 0, "default_value": 0, "signature": 0, "bases": 0, "doc": 3}, "algebra_with_sympy.algebraic_equation.algwsym_config.numerics.__init__": {"qualname": 5, "fullname": 10, "annotation": 0, "default_value": 0, "signature": 4, "bases": 0, "doc": 15}, "algebra_with_sympy.algebraic_equation.algwsym_config.numerics.integers_as_exact": {"qualname": 6, "fullname": 11, "annotation": 0, "default_value": 0, "signature": 11, "bases": 0, "doc": 126}, "algebra_with_sympy.algebraic_equation.ip": {"qualname": 1, "fullname": 6, "annotation": 0, "default_value": 1, "signature": 0, "bases": 0, "doc": 3}, 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\n\nIntroduction | Output Formatting\n| Installation |\nTry Live | Issues or Comments |\nChange Log |\nLicense\n| GIT Repository\n| PyPi Link
\n\n\n\nIntroduction
\n\n
\n\nThis tool defines relations that all high school and college students would\nrecognize as mathematical equations. \nThey consist of a left hand side (lhs) and a right hand side (rhs) connected by\nthe relation operator \"=\". In addition, it sets some convenient defaults and \nprovides some controls of output formatting that may be useful even if\nyou do not use the Equation class (see Conveniences for\nSymPy).
\n\nThis tool applies operations to both sides of the equation simultaneously, just\nas students are taught to do when \nattempting to isolate (solve for) a variable. Thus the statement Equation/b\nyields a new equation Equation.lhs/b = Equation.rhs/b
\n\nThe intent is to allow using the mathematical tools in SymPy to rearrange\nequations and perform algebra\nin a stepwise fashion using as close to standard mathematical notation as \npossible. In this way more people can successfully perform \nalgebraic rearrangements without stumbling\nover missed details such as a negative sign.
\n\nA simple example as it would appear in a Jupyter \nnotebook is shown immediately below:
\n\n
\n\nThe last cell illustrates how it is possible to substitute numbers with \nunits into the solved equation to calculate a numerical solution with \nproper units. The units(...) operation is part this package, not Sympy.
\n\nIn IPython environments (IPython, Jupyter, Google Colab, etc...) there is \nalso a shorthand syntax for entering equations provided through the IPython \npreparser. An equation can be specified as eq1 =@ a/b = c/d.
\n\n
\n\nIf no Python name is \nspecified for the equation (no eq_name to the left of =@), the equation \nwill still be defined, but will not be easily accessible for further \ncomputation. The =@ symbol combination was chosen to avoid conflicts with \nreserved python symbols while minimizing impacts on syntax highlighting \nand autoformatting.
\n\nMore examples of the capabilities of Algebra with Sympy are \nhere.
\n\nMany math packages such as SageMath \nand Maxima have similar capabilities, \nbut require more knowledge of command syntax, plus they cannot easily be \ninstalled in a generic python environment.
\n\n\n\nEven if you do not use the Equation class, there are some convenience \ntools and defaults that will probably make interactive use of SymPy in \nJupyter/IPython environments easier:
\n\n\n- By default, all numbers in Sympy expressions without decimal points are \ninterpreted as integers (e.g.
2/3*x, where x is a sympy symbol, -> \n2*x/3 not x*0.6666..., but if x is just a plain Python object then 2/3*x \n-> x*0.66666...). This can be turned off with unset_integers_as_exact(), \nwhich leads to standard Python behavior (2/3*x -> x*0.6666...) even for \nSympy expressions. Turn on with set_integers_as_exact(). When on the flag\nalgwsym_config.numerics.integers_as_exact = True. \n- Results of
solve() are wrapped in FiniteSet() to force pretty-printing \nof all of a solution set. See Controlling the Format of Interactive \nOutputs. \n- It is possible to set the default display to show both the pretty-printed \nresult and the code version simultaneously. See Controlling the Format of Interactive \nOutputs.
\n
\n\n\n\n
\n\n\n- These controls impact all Sympy objects and the
Equation class. \n- In graphical environments (Jupyter) you will get rendered Latex such as \n$\\frac{a}{b} = \\frac{c}{d}$ or $e^{\\frac{-x^2}{\\sigma^2}}$. To also see the \ncode representation (what can be copied and pasted for \nadditional computation) set
algwsym_config.output.show_code = True. \nThis will print the code version (e.g. Equation(a,b/c)) of equations \nand sympy expression in addition to the human readable version. This code \nversion can be accessed directly by calling repr() on the \nequation or expression. \n
\n\n\nIn interactive text environments (IPython and command line) The human \nreadable string version of Sympy expressions are returned (for Equations a \n= b rather than Equation(a,b)). This is equivalent to Calling print() \nor str() on an expression.
\n\n\n- To have the code version (can be copied and pasted as a \nPython statement) returned, set
algwsym_config.output.human_text = False. \n- Setting both
algwsym_config.output.human_text = True\nand algwsym_config.output.show_code = True, will return both the \ncode and human readable versions. \n
The equation label can be turned off by setting\nalgwsym_config.output.label = False.
Automatic wrapping of Equations as Latex equations can be activated \nby setting algwsym_config.output.latex_as_equations to True. The \ndefault is False. Setting this to True wraps output as LaTex equations,\nwrapping them in \\begin{equation}...\\end{equation}. Equations formatted \nthis way will not be labeled with the internal name for the equation, \nindependent of the setting of algwsym_config.output.label.
By default solutions output by solve() are returned as a SymPy \nFiniteSet() to force typesetting of the included solutions. To get Python \nlists instead you can override this for the whole session by setting\nalgwsym_config.output.solve_to_list = True. For a one-off, simply \nwrap the output of a solve in list() (e.g. list(solve(...))). One \nadvantage of list mode is that lists can be ordered. When\nalgwsym_config.output.solve_to_list = True solve() maintains the \nsolutions in the order the solve for variables were input.
\n\nSetup/Installation
\n\n
\n\n\n- Use pip to install in your python environment: \n
pip install -U Algebra-with-SymPy \n- To use in a running python session issue\nthe following command :
from algebra_with_sympy import *. \nThis will also import the SymPy tools. \n- If you want to isolate this tool from the global namespace you are \nworking with change the import statement \nto
import algebra_with_sympy as spa, where \nspa stands for \"SymPy Algebra\". Then all calls would be made to \nspa.funcname(). WARNING: Doing this makes shorthand equation input and \ncontrol of interactive output formats unavailable. To recover this \nfunctionality the following code must be run in the interactive session. \n
\n\nEquation = spa.Equation\nEqn = Equation\nalgwsym_config = spa.algwsym_config\n
\n\nTry in binder
\n\n\n
\n\n\n\n
\n\n\n\nChange Log
\n\n
\n\n\n- 1.1.0 (July 22, 2024)\n
\n- Setting integers as exact (
set_integers_as_exact(), the default) now \nonly sets integers as exact within Sympy and Algebra_with_Sympy \nexpressions. This increases compatibility with other packages that \ndepend on integers being Python integers. \n- Refuse to import Algebra_with_Sympy if an incompatible \nversion of Sympy is installed in the environment.
\n- Added warning explaining how to install a compatible version of Sympy.
\n
\n- 1.0.2 (July 5, 2024)\n
\n- Removed requirements for Jupyter and Jupyterlab as code will work in \nvanilla python or Google Colab.
\n- Workaround for Google Colab's inconsistent handling of mixed Latex and \nplain text strings. This impacted display of equation labels in Colab.
\n- BUG FIX: catch IPython not installed so that can run in plain vanilla \npython.
\n
\n- 1.0.1 (May 22, 2024)\n
\n- BUG FIX: equation labels that include underscore characters \"_\" are now \naccepted.
\n- BUG FIX: wrapping equations formatted as LaTex equation (ie. surrounded \nby
\\begin{equation}...\\end{equation}) in the $..$ code used to \nindicate markdown for MathJax was causing output errors in Quarto when \noutputing to .tex or .pdf. This is now fixed without negatively \nimpacting MathJax rendering. \n- BUG FIX: Singleton results of solve unnecessarily wrapped by extra list \nor finiteset. No longer double nested.
\n- BUG FIX: When returning lists make solve respect user order of solutions.
\n- BUG FIX: Equation output threw error when Algebra_with_Sympy was \nimported as a submodule. Equation labeling turned off for this type of \nimport to avoid error.
\n- BUG FIX: Equation labels are now copyable even with the newer MathJax \ncommonHTML rendering.
\n- Updates to requirements.txt.
\n- Documentation updates.
\n
\n- 1.0.0 (January 2, 2024)\n
\n- Added convenience operation
units(...) which takes a string of space \nseparated symbols to use as units. This simply declares the symbols \nto be positive, making them behave as units. This does not create units \nthat know about conversions, prefixes or systems of units. This lack \nis on purpose to provide units that require the user to worry about \nconversions (ideal in a teaching situation). To get units with built-in \nconversions see sympy.physics.units. \n- Fixed issue #23 where
cos() multiplied by a factor was not the same \ntype of object after simplify() acted on an expression. Required \nembedding the Equation type in the sympy library. Until Equation is \nincorporated into the primary Sympy repository a customized version of \nthe latest stable release will be used. \n- Fixed issue where trailing comments (ie.
# a comment at the end of a \nline) lead to input errors using compact =@ notation. \nalgwsym_config.output.latex_as_equations has a default value of False.\n Setting this to True wraps output as LaTex equations wrapping them \nin \\begin{equation}...\\end{equation}. Equations formatted this way \nwill not be labeled with the internal name for the equation.
\n- 0.12.0 (July 12, 2023)\n
\n- Now defaults to interpreting numbers without decimal points as integers. \nThis can be turned off with
unset_integers_as_exact() and on with\nset_integers_as_exact(). When on the flag\nalgwsym_config.numerics.integers_as_exact = True. \n
\n- 0.11.0 (June 5, 2023)\n
\n- Formatting of
FiniteSets overridden so that the contents always\npretty-print. This removes the necessity of special flags to get \npretty output from solve. \n- Sympy
solve() now works reliably with equations and outputs \npretty-printed solutions. \n- Added option
algwsym_config.output.solve_to_list = True which causes \nsolve() to return solutions sets as Python lists. Using this option \nprevents pretty-printing of the solutions produced by solve(). \nalgwsym_config.output.show_code and \nalgwsym_config.output.human_text now work for all sympy objects, not \njust Equation objects. This works\nin terminal, IPython terminal and Jupyter. This is achieved by hooking \ninto the python display_hook and IPython display_formatter.- Added jupyter to requirements.txt so that virtual environment builds\nwill include jupyter.
\n- The way
__version__ was handled could break pip install. Changed to\ngenerating the internal version during setup. This means the version\nis now available as algwsym_version. \n
\n- 0.10.0 (Sep. 5, 2022)\n
\n- Documentation updates and fixes.
\n- Significantly increased test coverage (~98%).
\n- Support for
Eqn.rewrite(Add) \n- Solving (e.g.
solve(Eqn,x)) now supported fully. Still experimental. \n- Bug fix: latex printing now supports custom printer.
\n- Substitution into an Equation using Equations is now \nsupported (e.g.
eq1.subs(eq2, eq3, ...)). \nalgebra_with_sympy.__version__ is now available for version checking \nwithin python.- Bug fix: preparsing for
=@ syntax no longer blocks obj? syntax for \ngetting docstrings in ipython. \n- More robust determination of equation names for labeling.
\n
\n- 0.9.4 (Aug. 11, 2022)\n
\n- Update to deal with new Sympy function
piecewise_exclusive in v1.11. \n- Added user warning if a function does not extend for use with
Equations \nas expected. This also allows the package to be used even when a function \nextension does fail. \n- Simplification of documentation preparation.
\n- Typo fixes in preparser error messages.
\n
\n- 0.9.3 (Aug. 9, 2022)\n
\n- Added check for new enough version of IPython to use the preparser.
\n- If IPython version too old, issue warning and do not accept
=@ shorthand. \n
\n- 0.9.2 (Jun. 5, 2022)\n
\n=@ shorthand syntax for defining equations in IPython compatible \nenvironments.- Fixed bug where
root() override called sqrt() on bare expressions. \n
\n- 0.9.1 (Mar. 24, 2022)\n
\n- Equations labeled with their python name, if they have one.
\n- Added flags to adjust human readable output and equation labeling.
\n- Accept equation as function argument in any position.
\n- First pass at
solve() accepting equations. \n- Added override of
root() to avoid warning messages. \n- More unit tests.
\n- First pass at documentation.
\n
\n- 0.9.0 functionality equivalent to extension of SymPy in\nPR#21333.
\n
\n\n\n\n\nThis program is free software: you can redistribute it and/or modify\n it under the terms of the GNU General Public License as published by\n the Free Software Foundation, either version 3 of the License, or\n (at your option) any later version.\n This program is distributed in the hope that it will be useful,\n but WITHOUT ANY WARRANTY; without even the implied warranty of\n MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the\n GNU General Public License for more details.
\n\nCopyright - Algebra with Sympy Contributors 2021, 2022, 2023, 2024
\n\nDevelopment Notes
\n\nGeneral | Make Docs | \nRunning Tests | \nBuild PyPi Package|
\n\nGeneral Notes
\n\n
\n\n\n- TODOs\n
\n- Test collect when there isn't an available _eval_collect (not sure how \nto get there).
\n- Test for _binary_op NotImplemented error (not sure how to get there).
\n
\n- To consider\n
\n- Include Sympy Plot Backends\nin the default setup.
\n- Change
Equation constructor to accept Equality, Set, List or \nlhs, rhs, rather than just lhs, rhs. \n- Extend
.subs to accept .subs(a=2*c, b = sin(q), ...). \n- MathLive on another web page as possible\ninput engine.
\n
\n
\n\nConstructing the Documentation
\n\n
\n\n\n- Make sure pdoc is installed and updated in the virtual environment
pip \ninstall -U pdoc. \n- Update any
.md files included in _init_.py.\n\n- Generally URLs should be absolute, not relative.
\n
\n- At the root level run pdoc \n
\npdoc --logo https://gutow.github.io/Algebra_with_Sympy/alg_w_sympy.svg \n--logo-link https://gutow.github.io/Algebra_with_Sympy/\n--footer-text \"Algebra with Sympy vX.X.X\" --math -html -o docs \n./algebra_with_sympy\n\nwhere X.X.X is the version number. \n
\n\nTasks for Documentation
\n\n
\n\n\n- Readme.md & Development Notes.md\n
\n- Use absolute path to github pages for more examples.
\n
\n
\n\nRunning Tests
\n\n
\n\n\n- Install updated pytest in the virtual environment:\n
pipenv shell\npip install -U pytest\n
\n- Run standard tests:\n
pytest --ignore='Developer Testing' --ignore-glob='*test_preparser.py'. \n- Run preparser tests:\n
ipython -m pytest tests/test_preparser.py \n- Run doctests:\n
pytest --ignore='tests' --ignore='Developer Testing' \n--ignore-glob='*old*' --doctest-modules \n
\n\nYou can run all the tests using the dotests script: ./dotests.sh.
\n\nNOTE: Some warnings about invalid escape characters are expected because \nraw strings are being passed with specialized LaTex escaped characters.
\n\nBuilding PyPi package
\n\n
\n\n\n- Make sure to update the version number in setup.py first.
\n- Install updated setuptools and twine in the virtual environment:\n
pipenv shell\npip install -U setuptools wheel twine\n
\n- Build the distribution
python -m setup sdist bdist_wheel. \n- Test it on
test.pypi.org.\n\n- Upload it (you will need an account on test.pypi.org):\n
python -m twine upload --repository testpypi dist/*. \n- Create a new virtual environment and test install into it:\n
exit # to get out of the current environment\ncd <somewhere>\nmkdir <new virtual environment>\ncd <new directory>\npipenv shell #creates the new environment and enters it.\npip install -i https://test.pypi.org/..... # copy actual link from the\n # repository on test.pypi.\n
\nThere are often install issues because sometimes only older versions of\nsome of the required packages are available on test.pypi.org. If this\nis the only problem change the version to end in rc0 for release\ncandidate and try it on the regular pypi.org as described below for\nreleasing on PyPi. \n- After install test by running a jupyter notebook in the virtual \nenvironment.
\n
\n
\n\nReleasing on PyPi
\n\n\nProceed only if testing of the build is successful.
\n\n\n- Double check the version number in version.py.
\n- Rebuild the release:
python -m setup sdist bdist_wheel. \n- Upload it:
python -m twine upload dist/* \n- Make sure it works by installing it in a clean virtual environment. This\nis the same as on test.pypi.org except without
-i https://test.pypy.... If\nit does not work, pull the release. \n
\n"}, "algebra_with_sympy.proper_sympy": {"fullname": "algebra_with_sympy.proper_sympy", "modulename": "algebra_with_sympy", "qualname": "proper_sympy", "kind": "variable", "doc": "\n", "default_value": "True"}, "algebra_with_sympy.algebraic_equation": {"fullname": "algebra_with_sympy.algebraic_equation", "modulename": "algebra_with_sympy.algebraic_equation", "kind": "module", "doc": "This package uses a special version of sympy which defines an equation \nwith a left-hand-side (lhs) and a right-\nhand-side (rhs) connected by the \"=\" operator (e.g. p*V = n*R*T).
\n\nThe intent is to allow using the mathematical tools in SymPy to rearrange\nequations and perform algebra in a stepwise fashion. In this way more people\ncan successfully perform algebraic rearrangements without stumbling over\nmissed details such as a negative sign. This mimics the capabilities available\nin SageMath and\nMaxima.
\n\nThis package also provides convenient settings for interactive use on the \ncommand line, in ipython and Jupyter notebook environments. See the \ndocumentation at https://gutow.github.io/Algebra_with_Sympy/.
\n\nExplanation
\n\nThis class defines relations that all high school and college students\nwould recognize as mathematical equations. At present only the \"=\" relation\noperator is recognized.
\n\nThis class is intended to allow using the mathematical tools in SymPy to\nrearrange equations and perform algebra in a stepwise fashion. In this\nway more people can successfully perform algebraic rearrangements without\nstumbling over missed details such as a negative sign.
\n\nCreate an equation with the call Equation(lhs,rhs), where lhs and\nrhs are any valid Sympy expression. Eqn(...) is a synonym for\nEquation(...).
\n\nParameters
\n\nlhs: sympy expression, class Expr.\nrhs: sympy expression, class Expr.\nkwargs:
\n\nExamples
\n\nNOTE: All the examples below are in vanilla python. You can get human\nreadable eqautions \"lhs = rhs\" in vanilla python by adjusting the settings\nin algwsym_config (see it's documentation). Output is human readable by\ndefault in IPython and Jupyter environments.
\n\n\n
>>> from algebra_with_sympy import *\n>>> a, b, c, x = var('a b c x')\n>>> Equation(a,b/c)\nEquation(a, b/c)\n>>> t=Eqn(a,b/c)\n>>> t\nEquation(a, b/c)\n>>> t*c\nEquation(a*c, b)\n>>> c*t\nEquation(a*c, b)\n>>> exp(t)\nEquation(exp(a), exp(b/c))\n>>> exp(log(t))\nEquation(a, b/c)\n
\n
Simplification and Expansion
\n\n\n
>>> f = Eqn(x**2 - 1, c)\n>>> f\nEquation(x**2 - 1, c)\n>>> f/(x+1)\nEquation((x**2 - 1)/(x + 1), c/(x + 1))\n>>> (f/(x+1)).simplify()\nEquation(x - 1, c/(x + 1))\n>>> simplify(f/(x+1))\nEquation(x - 1, c/(x + 1))\n>>> (f/(x+1)).expand()\nEquation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))\n>>> expand(f/(x+1))\nEquation(x**2/(x + 1) - 1/(x + 1), c/(x + 1))\n>>> factor(f)\nEquation((x - 1)*(x + 1), c)\n>>> f.factor()\nEquation((x - 1)*(x + 1), c)\n>>> f2 = f+a*x**2+b*x +c\n>>> f2\nEquation(a*x**2 + b*x + c + x**2 - 1, a*x**2 + b*x + 2*c)\n>>> collect(f2,x)\nEquation(b*x + c + x**2*(a + 1) - 1, a*x**2 + b*x + 2*c)\n
\n
Apply operation to only one side
\n\n\n
>>> poly = Eqn(a*x**2 + b*x + c*x**2, a*x**3 + b*x**3 + c*x)\n>>> poly.applyrhs(factor,x)\nEquation(a*x**2 + b*x + c*x**2, x*(c + x**2*(a + b)))\n>>> poly.applylhs(factor)\nEquation(x*(a*x + b + c*x), a*x**3 + b*x**3 + c*x)\n>>> poly.applylhs(collect,x)\nEquation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)\n
\n
.apply... also works with user defined python functions
\n\n\n
>>> def addsquare(eqn):\n... return eqn+eqn**2\n...\n>>> t.apply(addsquare)\nEquation(a**2 + a, b**2/c**2 + b/c)\n>>> t.applyrhs(addsquare)\nEquation(a, b**2/c**2 + b/c)\n>>> t.apply(addsquare, side = 'rhs')\nEquation(a, b**2/c**2 + b/c)\n>>> t.applylhs(addsquare)\nEquation(a**2 + a, b/c)\n>>> addsquare(t)\nEquation(a**2 + a, b**2/c**2 + b/c)\n
\n
Inaddition to .apply... there is also the less general .do,\n.dolhs, .dorhs, which only works for operations defined on the\nExpr class (e.g..collect(), .factor(), .expand(), etc...).
\n\n\n
>>> poly.dolhs.collect(x)\nEquation(b*x + x**2*(a + c), a*x**3 + b*x**3 + c*x)\n>>> poly.dorhs.collect(x)\nEquation(a*x**2 + b*x + c*x**2, c*x + x**3*(a + b))\n>>> poly.do.collect(x)\nEquation(b*x + x**2*(a + c), c*x + x**3*(a + b))\n>>> poly.dorhs.factor()\nEquation(a*x**2 + b*x + c*x**2, x*(a*x**2 + b*x**2 + c))\n
\n
poly.do.exp() or other sympy math functions will raise an error.
\n\nRearranging an equation (simple example made complicated as illustration)
\n\n\n
>>> p, V, n, R, T = var('p V n R T')\n>>> eq1=Eqn(p*V,n*R*T)\n>>> eq1\nEquation(V*p, R*T*n)\n>>> eq2 =eq1/V\n>>> eq2\nEquation(p, R*T*n/V)\n>>> eq3 = eq2/R/T\n>>> eq3\nEquation(p/(R*T), n/V)\n>>> eq4 = eq3*R/p\n>>> eq4\nEquation(1/T, R*n/(V*p))\n>>> 1/eq4\nEquation(T, V*p/(R*n))\n>>> eq5 = 1/eq4 - T\n>>> eq5\nEquation(0, -T + V*p/(R*n))\n
\n
Substitution (#'s and units)
\n\n\n
>>> L, atm, mol, K = var('L atm mol K', positive=True, real=True) # units\n>>> eq2.subs({R:0.08206*L*atm/mol/K,T:273*K,n:1.00*mol,V:24.0*L})\nEquation(p, 0.9334325*atm)\n>>> eq2.subs({R:0.08206*L*atm/mol/K,T:273*K,n:1.00*mol,V:24.0*L}).evalf(4)\nEquation(p, 0.9334*atm)\n
\n
Substituting an equation into another equation:
\n\n\n
>>> P, P1, P2, A1, A2, E1, E2 = symbols("P, P1, P2, A1, A2, E1, E2")\n>>> eq1 = Eqn(P, P1 + P2)\n>>> eq2 = Eqn(P1 / (A1 * E1), P2 / (A2 * E2))\n>>> P1_val = (eq1 - P2).swap\n>>> P1_val\nEquation(P1, P - P2)\n>>> eq2 = eq2.subs(P1_val)\n>>> eq2\nEquation((P - P2)/(A1*E1), P2/(A2*E2))\n>>> P2_val = solve(eq2.subs(P1_val), P2).args[0]\n>>> P2_val\nEquation(P2, A2*E2*P/(A1*E1 + A2*E2))\n
\n
Combining equations (Math with equations: lhs with lhs and rhs with rhs)
\n\n\n
>>> q = Eqn(a*c, b/c**2)\n>>> q\nEquation(a*c, b/c**2)\n>>> t\nEquation(a, b/c)\n>>> q+t\nEquation(a*c + a, b/c + b/c**2)\n>>> q/t\nEquation(c, 1/c)\n>>> t**q\nEquation(a**(a*c), (b/c)**(b/c**2))\n
\n
Utility operations
\n\n\n
>>> t.reversed\nEquation(b/c, a)\n>>> t.swap\nEquation(b/c, a)\n>>> t.lhs\na\n>>> t.rhs\nb/c\n>>> t.as_Boolean()\nEq(a, b/c)\n
\n
.check() convenience method for .as_Boolean().simplify()
\n\n\n
>>> from sympy import I, pi\n>>> Equation(pi*(I+2), pi*I+2*pi).check()\nTrue\n>>> Eqn(a,a+1).check()\nFalse\n
\n
Differentiation\nDifferentiation is applied to both sides if the wrt variable appears on\nboth sides.
\n\n\n
>>> q=Eqn(a*c, b/c**2)\n>>> q\nEquation(a*c, b/c**2)\n>>> diff(q,b)\nEquation(Derivative(a*c, b), c**(-2))\n>>> diff(q,c)\nEquation(a, -2*b/c**3)\n>>> diff(log(q),b)\nEquation(Derivative(log(a*c), b), 1/b)\n>>> diff(q,c,2)\nEquation(Derivative(a, c), 6*b/c**4)\n
\n
If you specify multiple differentiation all at once the assumption\nis order of differentiation matters and the lhs will not be\nevaluated.
\n\n\n
>>> diff(q,c,b)\nEquation(Derivative(a*c, b, c), -2/c**3)\n
\n
To overcome this specify the order of operations.
\n\n\n
>>> diff(diff(q,c),b)\nEquation(Derivative(a, b), -2/c**3)\n
\n
But the reverse order returns an unevaulated lhs (a may depend on b).
\n\n\n
>>> diff(diff(q,b),c)\nEquation(Derivative(a*c, b, c), -2/c**3)\n
\n
Integration can only be performed on one side at a time.
\n\n\n
>>> q=Eqn(a*c,b/c)\n>>> integrate(q,b,side='rhs')\nb**2/(2*c)\n>>> integrate(q,b,side='lhs')\na*b*c\n
\n
Make a pretty statement of integration from an equation
\n\n\n
>>> Eqn(Integral(q.lhs,b),integrate(q,b,side='rhs'))\nEquation(Integral(a*c, b), b**2/(2*c))\n
\n
Integration of each side with respect to different variables
\n\n\n
>>> q.dorhs.integrate(b).dolhs.integrate(a)\nEquation(a**2*c/2, b**2/(2*c))\n
\n
Automatic solutions using sympy solvers. THIS IS EXPERIMENTAL. Please\nreport issues at https://github.com/gutow/Algebra_with_Sympy/issues.
\n\n\n
>>> tosolv = Eqn(a - b, c/a)\n>>> solve(tosolv,a)\nFiniteSet(Equation(a, b/2 - sqrt(b**2 + 4*c)/2), Equation(a, b/2 + sqrt(b**2 + 4*c)/2))\n>>> solve(tosolv, b)\nFiniteSet(Equation(b, (a**2 - c)/a))\n>>> solve(tosolv, c)\nFiniteSet(Equation(c, a**2 - a*b))\n
\n
This is a class to hold parameters that control behavior of\nthe algebra_with_sympy package.
\n\nSettings
\n\nPrinting
\n\nIn interactive environments the default output of an equation is a\nhuman readable string with the two sides connected by an equals\nsign or a typeset equation with the two sides connected by an equals sign.\nprint(Eqn) or str(Eqn) will return this human readable text version of\nthe equation as well. This is consistent with python standards, but not\nsympy, where str() is supposed to return something that can be\ncopy-pasted into code. If the equation has a declared name as in eq1 =\nEqn(a,b/c) the name will be displayed to the right of the equation in\nparentheses (eg. a = b/c (eq1)). Use print(repr(Eqn)) instead of\nprint(Eqn) or repr(Eqn) instead of str(Eqn) to get a code\ncompatible version of the equation.
\n\nYou can adjust this behavior using some flags that impact output:
\n\n\nalgwsym_config.output.show_code default is False.algwsym_config.output.human_text default is True.algwsym_config.output.label default is True.algwsym_config.output.latex_as_equations default is False
\n\nIn interactive environments you can get both types of output by setting\nthe algwsym_config.output.show_code flag. If this flag is true\ncalls to latex and str will also print an additional line \"code\nversion: repr(Eqn)\". Thus in Jupyter you will get a line of typeset\nmathematics output preceded by the code version that can be copy-pasted.\nDefault is False.
\n\nA second flag algwsym_config.output.human_text is useful in\ntext-based interactive environments such as command line python or\nipython. If this flag is true repr will return str. Thus the human\nreadable text will be printed as the output of a line that is an\nexpression containing an equation.\nDefault is True.
\n\nSetting both of these flags to true in a command line or ipython\nenvironment will show both the code version and the human readable text.\nThese flags impact the behavior of the print(Eqn) statement.
\n\nThe third flag algwsym_config.output.label has a default value of\nTrue. Setting this to False suppresses the labeling of an equation\nwith its python name off to the right of the equation.
\n\nThe fourth flag algwsym_config.output.latex_as_equations has\na default value of False. Setting this to True wraps\noutput as LaTex equations wrapping them in \\begin{equation}...\\end{\nequation}.
\n", "signature": "()"}, "algebra_with_sympy.algebraic_equation.algwsym_config.output": {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output", "kind": "class", "doc": "\n"}, "algebra_with_sympy.algebraic_equation.algwsym_config.output.__init__": {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output.__init__", "kind": "function", "doc": "This holds settings that impact output.
\n", "signature": "()"}, "algebra_with_sympy.algebraic_equation.algwsym_config.output.show_code": {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output.show_code", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output.show_code", "kind": "variable", "doc": "If True code versions of the equation expression will be\noutput in interactive environments. Default = False.
\n", "default_value": "False"}, "algebra_with_sympy.algebraic_equation.algwsym_config.output.human_text": {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output.human_text", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output.human_text", "kind": "variable", "doc": "If True the human readable equation expression will be\noutput in text interactive environments. Default = False.
\n", "default_value": "True"}, "algebra_with_sympy.algebraic_equation.algwsym_config.output.solve_to_list": {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output.solve_to_list", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output.solve_to_list", "kind": "variable", "doc": "If True the results of a call to solve(...) will return a\nPython list rather than a Sympy FiniteSet. This recovers\nbehavior for versions before 0.11.0.
\n\nNote: setting this True means that expressions within the\nreturned solutions will not be pretty-printed in Jupyter and\nIPython.
\n", "default_value": "False"}, "algebra_with_sympy.algebraic_equation.algwsym_config.output.latex_as_equations": {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output.latex_as_equations", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output.latex_as_equations", "kind": "variable", "doc": "If True any output that is returned as LaTex for\npretty-printing will be wrapped in the formal Latex for an\nequation. For example rather than
\n\n$\\frac{a}{b}=c$\n
\n\nthe output will be
\n\n\\begin{equation}\\frac{a}{b}=c\\end{equation}\n
\n", "default_value": "False"}, "algebra_with_sympy.algebraic_equation.algwsym_config.output.label": {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.output.label", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.output.label", "kind": "variable", "doc": "\n", "default_value": "True"}, "algebra_with_sympy.algebraic_equation.algwsym_config.numerics": {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.numerics", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.numerics", "kind": "class", "doc": "\n"}, "algebra_with_sympy.algebraic_equation.algwsym_config.numerics.__init__": {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.numerics.__init__", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.numerics.__init__", "kind": "function", "doc": "This class holds settings for how numerical computation and\ninputs are handled.
\n", "signature": "()"}, "algebra_with_sympy.algebraic_equation.algwsym_config.numerics.integers_as_exact": {"fullname": "algebra_with_sympy.algebraic_equation.algwsym_config.numerics.integers_as_exact", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "algwsym_config.numerics.integers_as_exact", "kind": "function", "doc": "This is a flag for informational purposes and interface\nconsistency. Changing the value will not change the behavior.
\n\nTo change the behavior call:
\n\n\nunset_integers_as_exact() to turn this feature off.set_integers_as_exact() to turn this feature on (on by\ndefault).
\n\nIf set to True (the default) and if running in an\nIPython/Jupyter environment any number input without a decimal\nwill be interpreted as a sympy integer. Thus, fractions and\nrelated expressions will not evalute to floating point numbers,\nbut be maintained as exact expressions (e.g. 2/3 -> 2/3 not the\nfloat 0.6666...).
\n", "signature": "(self):", "funcdef": "def"}, "algebra_with_sympy.algebraic_equation.ip": {"fullname": "algebra_with_sympy.algebraic_equation.ip", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "ip", "kind": "variable", "doc": "\n", "default_value": "None"}, "algebra_with_sympy.algebraic_equation.formatter": {"fullname": "algebra_with_sympy.algebraic_equation.formatter", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "formatter", "kind": "variable", "doc": "\n", "default_value": "None"}, "algebra_with_sympy.algebraic_equation.set_integers_as_exact": {"fullname": "algebra_with_sympy.algebraic_equation.set_integers_as_exact", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "set_integers_as_exact", "kind": "function", "doc": "This operation causes any number input without a decimal that is\npart of a Sympy expression to be interpreted as a sympy\ninteger, by using a custom preparser to cast integers within Sympy\nexpressions as Sympy integers (Integer()). It also sets the flag\nalgwsym_config.numerics.integers_as_exact = True This is the default\nmode of algebra_with_sympy. To turn this off call\nunset_integers_as_exact().
\n\nNOTE: 2/3 --> 0.6666... even when this is set, but 2*x/3 -->\nInteger(2)/Integer(3)*x if x is a sympy object. If x is just a Python\nobject 2*x/3 --> x*0.6666666666....
\n", "signature": "():", "funcdef": "def"}, "algebra_with_sympy.algebraic_equation.unset_integers_as_exact": {"fullname": "algebra_with_sympy.algebraic_equation.unset_integers_as_exact", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "unset_integers_as_exact", "kind": "function", "doc": "This operation disables forcing of numbers input without\ndecimals being interpreted as sympy integers. Numbers input without a\ndecimal may be interpreted as floating point if they are part of an\nexpression that undergoes python evaluation (e.g. 2/3 -> 0.6666...). It\nalso sets the flag algwsym_config.numerics.integers_as_exact = False.\nCall set_integers_as_exact() to avoid this conversion of rational\nfractions and related expressions to floating point. Algebra_with_sympy\nstarts with set_integers_as_exact() enabled (\nalgwsym_config.numerics.integers_as_exact = True).
\n", "signature": "():", "funcdef": "def"}, "algebra_with_sympy.algebraic_equation.Eqn": {"fullname": "algebra_with_sympy.algebraic_equation.Eqn", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eqn", "kind": "variable", "doc": "\n", "default_value": "<class 'sympy.core.equation.Equation'>"}, "algebra_with_sympy.algebraic_equation.units": {"fullname": "algebra_with_sympy.algebraic_equation.units", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "units", "kind": "function", "doc": "This operation declares the symbols to be positive values, so that sympy\nwill handle them properly when simplifying expressions containing units.\nUnits defined this way are just unit symbols. If you want units that are\naware of conversions see sympy.physics.units.
\n\nParameters
\n\n\n- string names: a string containing a space separated list of\nsymbols to be treated as units.
\n
\n\nReturns
\n\n\n calls name = symbols(name,\npositive=True) in the interactive namespace for each symbol name.
\n
\n", "signature": "(names):", "funcdef": "def"}, "algebra_with_sympy.algebraic_equation.solve": {"fullname": "algebra_with_sympy.algebraic_equation.solve", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "solve", "kind": "function", "doc": "Override of sympy solve().
\n\nIf passed an expression and variable(s) to solve for it behaves\nalmost the same as normal solve with dict = True, except that solutions\nare wrapped in a FiniteSet() to guarantee that the output will be pretty\nprinted in Jupyter like environments.
\n\nIf passed an equation or equations it returns solutions as a\nFiniteSet() of solutions, where each solution is represented by an\nequation or set of equations.
\n\nTo get a Python list of solutions (pre-0.11.0 behavior) rather than a\nFiniteSet issue the command algwsym_config.output.solve_to_list = True.\nThis also prevents pretty-printing in IPython and Jupyter.
\n\nExamples
\n\n\n
>>> a, b, c, x, y = symbols('a b c x y', real = True)\n>>> import sys\n>>> sys.displayhook = __command_line_printing__ # set by default on normal initialization.\n>>> eq1 = Eqn(abs(2*x+y),3)\n>>> eq2 = Eqn(abs(x + 2*y),3)\n>>> B = solve((eq1,eq2))\n
\n
Default human readable output on command line
\n\n\n
>>> B\n{{x = -3, y = 3}, {x = -1, y = -1}, {x = 1, y = 1}, {x = 3, y = -3}}\n
\n
To get raw output turn off by setting
\n\n\n
>>> algwsym_config.output.human_text=False\n>>> B\nFiniteSet(FiniteSet(Equation(x, -3), Equation(y, 3)), FiniteSet(Equation(x, -1), Equation(y, -1)), FiniteSet(Equation(x, 1), Equation(y, 1)), FiniteSet(Equation(x, 3), Equation(y, -3)))\n
\n
Pre-0.11.0 behavior where a python list of solutions is returned
\n\n\n
>>> algwsym_config.output.solve_to_list = True\n>>> solve((eq1,eq2))\n[[Equation(x, -3), Equation(y, 3)], [Equation(x, -1), Equation(y, -1)], [Equation(x, 1), Equation(y, 1)], [Equation(x, 3), Equation(y, -3)]]\n>>> algwsym_config.output.solve_to_list = False # reset to default\n
\n
algwsym_config.output.human_text = True with\nalgwsym_config.output.how_code=True shows both.\nIn Jupyter-like environments show_code=True yields the Raw output and\na typeset version. If show_code=False (the default) only the\ntypeset version is shown in Jupyter.
\n\n\n
>>> algwsym_config.output.show_code=True\n>>> algwsym_config.output.human_text=True\n>>> B\nCode version: FiniteSet(FiniteSet(Equation(x, -3), Equation(y, 3)), FiniteSet(Equation(x, -1), Equation(y, -1)), FiniteSet(Equation(x, 1), Equation(y, 1)), FiniteSet(Equation(x, 3), Equation(y, -3)))\n{{x = -3, y = 3}, {x = -1, y = -1}, {x = 1, y = 1}, {x = 3, y = -3}}\n
\n
Very experimental override of sympy solveset, which we hope will replace\nsolve. Much is not working. It is not clear how to input a system of\nequations unless you directly select linsolve, etc...
\n", "signature": "(f, symbols, domain=Complexes):", "funcdef": "def"}, "algebra_with_sympy.algebraic_equation.Equality": {"fullname": "algebra_with_sympy.algebraic_equation.Equality", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equality", "kind": "class", "doc": "Extension of Equality class to include the ability to convert it to an\nEquation.
\n", "bases": "sympy.core.relational.Equality"}, "algebra_with_sympy.algebraic_equation.Equality.to_Equation": {"fullname": "algebra_with_sympy.algebraic_equation.Equality.to_Equation", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equality.to_Equation", "kind": "function", "doc": "Return: recasts the Equality as an Equation.
\n", "signature": "(self):", "funcdef": "def"}, "algebra_with_sympy.algebraic_equation.Equality.to_Eqn": {"fullname": "algebra_with_sympy.algebraic_equation.Equality.to_Eqn", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equality.to_Eqn", "kind": "function", "doc": "Synonym for to_Equation.\nReturn: recasts the Equality as an Equation.
\n", "signature": "(self):", "funcdef": "def"}, "algebra_with_sympy.algebraic_equation.Equality.default_assumptions": {"fullname": "algebra_with_sympy.algebraic_equation.Equality.default_assumptions", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Equality.default_assumptions", "kind": "variable", "doc": "\n", "default_value": "{}"}, "algebra_with_sympy.algebraic_equation.Eq": {"fullname": "algebra_with_sympy.algebraic_equation.Eq", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "Eq", "kind": "variable", "doc": "\n", "default_value": "<class 'algebra_with_sympy.algebraic_equation.Equality'>"}, "algebra_with_sympy.algebraic_equation.abs": {"fullname": "algebra_with_sympy.algebraic_equation.abs", "modulename": "algebra_with_sympy.algebraic_equation", "qualname": "abs", "kind": "variable", "doc": "\n", "default_value": "Abs"}, "algebra_with_sympy.preparser": {"fullname": "algebra_with_sympy.preparser", "modulename": "algebra_with_sympy.preparser", "kind": "module", "doc": "\n"}, "algebra_with_sympy.preparser.algebra_with_sympy_preparser": {"fullname": "algebra_with_sympy.preparser.algebra_with_sympy_preparser", "modulename": "algebra_with_sympy.preparser", "qualname": "algebra_with_sympy_preparser", "kind": "function", "doc": "In IPython compatible environments (Jupyter, IPython, etc...) this supports\na special compact input method for equations.
\n\nThe syntax supported is equation_name =@ equation.lhs = equation.rhs,\nwhere equation_name is a valid Python name that can be used to refer to\nthe equation later. equation.lhs is the left-hand side of the equation\nand equation.rhs is the right-hand side of the equation. Each side of the\nequation must parse into a valid Sympy expression.
\n\nNote: This does not support line continuation. Long equations should be\nbuilt by combining expressions using names short enough to do this on one\nline. The alternative is to use equation_name = Eqn(long ...\nexpressions ... with ... multiple ... lines).
\n\nNote: If the equation_name is omitted the equation will be formed,\nbut it will not be assigned to a name that can be used to refer to it\nlater. You may be able to access it through one of the special IPython\nunderscore names. This is not recommended.
\n\nTHIS FUNCTION IS USED BY THE IPYTHON ENVIRONMENT TO PREPARSE THE INPUT\nBEFORE IT IS PASSED TO THE PYTHON INTERPRETER. IT IS NOT MEANT TO BE USED\nDIRECTLY BY A USER
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