Solveset
========
.. module:: sympy.solvers.solveset
This is the official documentation of the ``solveset`` module in solvers.
It contains the frequently asked questions about our new module to solve
equations.
What's wrong with solve():
--------------------------
SymPy already has a pretty powerful ``solve`` function. But it has a lot of major
issues
1. It doesn't have a consistent output for various types of solutions
It needs to return a lot of types of solutions consistently:
* Single solution : `x = 1`
* Multiple solutions: `x^2 = 1`
* No Solution: `x^2 + 1 = 0 ; x \in \mathbb{R}`
* Interval of solution: `\lfloor x \rfloor = 0`
* Infinitely many solutions: `sin(x) = 0`
* Multivariate functions with point solutions: `x^2 + y^2 = 0`
* Multivariate functions with non-point solution: `x^2 + y^2 = 1`
* System of equations: `x + y = 1` and `x - y = 0`
* Relational: `x > 0`
* And the most important case: "We don't Know"
2. The input API is also a mess, there are a lot of parameters. Many of them
are not needed and they make it hard for the user and the developers to
work on solvers.
3. There are cases like finding the maxima and minima of function using
critical points where it is important to know if it has returned all the
solutions. ``solve`` does not guarantee this.
Why Solveset?
-------------
* ``solveset`` has a cleaner input and output interface: ``solveset`` returns
a set object and a set object takes care of all types of output. For
cases where it doesn't "know" all the solutions a ``ConditionSet`` with a partial
solution is returned. For input it only takes the equation, the variables
to solve for and the optional argument ``domain`` over which the equation is to
be solved.
* ``solveset`` can return infinitely many solutions. For example solving for
`\sin{(x)} = 0` returns `\{2 n \pi | n \in \mathbb{Z}\} \cup \{2 n \pi + \pi | n \in \mathbb{Z}\}`,
whereas ``solve`` only returns `[0, \pi]`.
* There is a clear code level and interface level separation between solvers
for equations in the complex domain and the real domain. For example
solving `e^x = 1` when `x` is to be solved in the complex domain, returns
the set of all solutions, that is `\{2 n i \pi | n \in \mathbb{Z}\}`, whereas
if `x` is to be solved in the real domain then only `\{0\}` is returned.
Why do we use Sets as an output type?
-------------------------------------
SymPy has a well developed sets module, which can represent most of the set
containers in Mathematics such as:
* ``FiniteSet``
Represents a finite set of discrete numbers.
* ``Interval``
Represents a real interval as a set.
* ``ProductSet``
Represents a Cartesian product of sets.
* ``ImageSet``
Represents the image of a set under a mathematical function
>>> from sympy import ImageSet, S, Lambda
>>> from sympy.abc import x
>>> squares = ImageSet(Lambda(x, x**2), S.Naturals) # {x**2 for x in N}
>>> 4 in squares
True
* ``ComplexRegion``
Represents the set of all complex numbers in a region in the Argand plane.
* ``ConditionSet``
Represents the set of elements, which satisfies a given condition.
Also, the predefined set classes such as:
* ``Naturals`` `\mathbb{N}`
Represents the natural numbers (or counting numbers), which are all
positive integers starting from 1.
* ``Naturals0`` `\mathbb{N_0}`
Represents the whole numbers, which are all the non-negative integers,
inclusive of 0.
* ``Integers`` `\mathbb{Z}`
Represents all integers: positive, negative and zero.
* ``Reals`` `\mathbb{R}`
Represents the set of all real numbers.
* ``Complexes`` `\mathbb{C}`
Represents the set of all complex numbers.
* ``EmptySet`` `\phi`
Represents the empty set.
The above six sets are available as Singletons, like ``S.Integers``.
It is capable of most of the set operations in mathematics:
* ``Union``
* ``Intersection``
* ``Complement``
* ``SymmetricDifference``
The main reason for using sets as output to solvers is that it can consistently
represent many types of solutions. For the single variable case it can represent:
* No solution (by the empty set).
* Finitely many solutions (by ``FiniteSet``).
* Infinitely many solutions, both countably and uncountably infinite solutions
(using the ``ImageSet`` module).
* ``Interval``
* There can also be bizarre solutions to equations like the set of rational
numbers.
No other Python object (list, dictionary, generator, Python sets) provides the
flexibility of mathematical sets which our sets module tries to emulate. The
second reason to use sets is that they are close to the entities which
mathematicians deal with and it makes it easier to reason about them. Set
objects conform to Pythonic conventions when possible, i.e., ``x in A`` and
``for i in A`` both work when they can be computed. Another advantage of using
objects closer to mathematical entities is that the user won't have to "learn"
our representation and she can have her expectations transferred from her
mathematical experience.
For the multivariate case we represent solutions as a set of points in a
n-dimensional space and a point is represented by a ``FiniteSet`` of ordered
tuples, which is a point in `\mathbb{R}^n` or `\mathbb{C}^n`.
Please note that, the general ``FiniteSet`` is unordered, but a ``FiniteSet``
with a tuple as its only argument becomes ordered, since a tuple is ordered.
So the order in the tuple is mapped to a pre-defined order of variables
while returning solutions.
For example:
>>> from sympy import FiniteSet
>>> FiniteSet(1, 2, 3) # Unordered
{1, 2, 3}
>>> FiniteSet((1, 2, 3)) # Ordered
{(1, 2, 3)}
Why not use dicts as output?
Dictionary are easy to deal with programmatically but mathematically they are
not very precise and use of them can quickly lead to inconsistency and a lot
of confusion. For example:
* There are a lot of cases where we don't know the complete solution and we
may like to output a partial solution, consider the equation `fg = 0`. The
solution of this equation is the union of the solution of the following
two equations: `f = 0`, `g = 0`. Let's say that we are able to solve
`f = 0` but solving `g = 0` isn't supported yet. In this case we cannot
represent partial solution of the given equation `fg = 0` using dicts.
This problem is solved with sets using a ``ConditionSet`` object:
`sol_f \cup \{x | x ∊ \mathbb{R} ∧ g = 0\}`, where `sol_f` is the solution
of the equation `f = 0`.
* Using a dict may lead to surprising results like:
- ``solve(Eq(x**2, 1), x) != solve(Eq(y**2, 1), y)``
Mathematically, this doesn't make sense. Using ``FiniteSet`` here
solves the problem.
* It also cannot represent solutions for equations like `|x| < 1`, which is
a disk of radius 1 in the Argand Plane. This problem is solved using
complex sets implemented as ``ComplexRegion``.
Input API of ``solveset``
-------------------------
``solveset`` has a cleaner input API, unlike ``solve``. It takes a maximum
of three arguments:
``solveset(equation, variable=None, domain=S.Complexes)``
* Equation(s)
The equation(s) to solve.
* Variable(s)
The variable(s) for which the equation is to be solved.
* Domain
The domain in which the equation is to be solved.
``solveset`` removes the ``flags`` argument of ``solve``, which had made
the input API messy and output API inconsistent.
What is this domain argument about?
-----------------------------------
Solveset is designed to be independent of the assumptions on the
variable being solved for and instead, uses the ``domain`` argument to
decide the solver to dispatch the equation to, namely ``solveset_real``
or ``solveset_complex``. It's unlike the old ``solve`` which considers the
assumption on the variable.
>>> from sympy import solveset, S
>>> from sympy.abc import x
>>> solveset(x**2 + 1, x) # domain=S.Complexes is default
{-I, I}
>>> solveset(x**2 + 1, x, domain=S.Reals)
EmptySet()
What are the general methods employed by solveset to solve an equation?
-----------------------------------------------------------------------
Solveset uses various methods to solve an equation, here is a brief overview
of the methodology:
* The ``domain`` argument is first considered to know the domain in which
the user is interested to get the solution.
* If the given function is a relational (``>=``, ``<=``, ``>``, ``<``), and
the domain is real, then ``solve_univariate_inequality`` and solutions are
returned. Solving for complex solutions of inequalities, like `x^2 < 0`
is not yet supported.
* Based on the ``domain``, the equation is dispatched to one of the two
functions ``solveset_real`` or ``solveset_complex``, which solves the
given equation in the complex or real domain, respectively.
* If the given expression is a product of two or more functions,
like say `gh = 0`, then the solution to the given equation is the Union
of the solution of the equations `g = 0` and `h = 0`, if and only if both
`g` and `h` are finite for a finite input. So, the solution is built up
recursively.
* If the function is trigonometric or hyperbolic, the function
``_solve_real_trig`` is called, which solves it by converting it to complex
exponential form.
* The function is now checked if there is any instance of a ``Piecewise``
expression, if it is, then it's converted to explicit expression and
set pairs and then solved recursively.
* The respective solver now tries to invert the equation using the routines
``invert_real`` and ``invert_complex``. These routines are based on the
concept of mathematical inverse (though not exactly). It reduces the
real/complex valued equation `f(x) = y` to a set of equations:
`\{g(x) = h_1(y), g(x) = h_2(y), ..., g(x) = h_n(y) \}` where `g(x)` is a
simpler function than `f(x)`. There is some work needed to be done in
this to find invert of more complex expressions.
* After the invert, the equations are checked for radical or Abs (Modulus),
then the method ``_solve_radical`` tries to simplify the radical, by
removing it using techniques like squaring, cubing etc, and ``_solve_abs``
solves nested Modulus by considering the positive and negative variants,
iteratively.
* If none of the above method is successful, then methods of polynomial is
used as follows:
- The method to solve the rational function, ``_solve_as_rational``, is
called. Based on the domain, the respective poly solver
``_solve_as_poly_real`` or ``_solve_as_poly_complex`` is called to solve
``f`` as a polynomial.
- The underlying method ``_solve_as_poly`` solves the equation using
polynomial techniques if it's already a polynomial equation or, with
a change of variables, can be made so.
* The final solution set returned by ``solveset`` is the intersection of the
set of solutions found above and the input domain.
.. Remember to change the above part when the new solver is implemented.
How do we manipulate and return an infinite solution?
-----------------------------------------------------
* In the real domain, we use our ``ImageSet`` class in the sets module to
return infinite solutions. ``ImageSet`` is an image of a set under
a mathematical function. For example, to represent the solution
of the equation `\sin{(x)} = 0`, we can use the ``ImageSet`` as:
>>> from sympy import ImageSet, Lambda, pi, S, Dummy, pprint
>>> n = Dummy('n')
>>> pprint(ImageSet(Lambda(n, 2*pi*n), S.Integers), use_unicode=True)
{2⋅n⋅π | n ∊ ℤ}
Where ``n`` is a dummy variable. It is basically the image of the
set of integers under the function `2\pi n`.
* In the complex domain, we use complex sets, which are implemented as the
``ComplexRegion`` class in the sets module, to represent infinite
solution in the Argand plane. For example to represent the solution
of the equation `|z| = 1`, which is a unit circle, we can use
the ``ComplexRegion`` as:
>>> from sympy import ComplexRegion, FiniteSet, Interval, pi, pprint
>>> pprint(ComplexRegion(FiniteSet(1)*Interval(0, 2*pi), polar=True), use_unicode=True)
{r⋅(ⅈ⋅sin(θ) + cos(θ)) | r, θ ∊ {1} × [0, 2⋅π)}
Where the ``FiniteSet`` in the ``ProductSet`` is the range of the value
of `r`, which is the radius of the circle and the ``Interval`` is the range
of `\theta`, the angle from the `x` axis representing a unit circle in the
Argand plane.
Note: We also have non-polar form notation for representing solution
in rectangular form. For example, to represent first two quadrants in the
Argand plane, we can write the ``ComplexRegion`` as:
>>> from sympy import ComplexRegion, Interval, pi, oo, pprint
>>> pprint(ComplexRegion(Interval(-oo, oo)*Interval(0, oo)), use_unicode=True)
{x + y⋅ⅈ | x, y ∊ (-∞, ∞) × [0, ∞)}
where the Intervals are the range of `x` and `y` for the set of complex
numbers `x + iy`.
How does ``solveset`` ensure that it is not returning any wrong solution?
--------------------------------------------------------------------------
Solvers in a Computer Algebra System are based on heuristic algorithms,
so it's usually very hard to ensure 100% percent correctness, in every
possible case. However there are still a lot of cases where we can ensure
correctness. Solveset tries to verify correctness wherever it can. For
example:
Consider the equation `|x| = n`. A naive method to solve this equation
would return ``{-n, n}`` as its solution, which is not correct since
``{-n, n}`` can be its solution if and only if ``n`` is positive.
Solveset returns this information as well to ensure correctness.
>>> from sympy import symbols, S, pprint, solveset
>>> x, n = symbols('x, n')
>>> pprint(solveset(abs(x) - n, x, domain=S.Reals), use_unicode=True)
([0, ∞) ∩ {n}) ∪ ((-∞, 0] ∩ {-n})
Though, there still a lot of work needs to be done in this regard.
Search based solver and step-by-step solution
---------------------------------------------
Note: This is under Development.
After the introduction of :py:class:`~sympy.sets.conditionset.ConditionSet`, the
solving of equations can be seen as set transformations. Here is an abstract
view of the things we can do to solve equations.
* Apply various set transformations on the given set.
* Define a metric of the usability of solutions, or a notion of some
solutions being better than others.
* Different transformations would be the nodes of a tree.
* Suitable searching techniques could be applied to get the best solution.
``ConditionSet`` gives us the ability to represent unevaluated equations and
inequalities in forms like `\{x|f(x)=0; x \in S\}` and `\{x|f(x)>0; x \in S\}`
but a more powerful thing about ``ConditionSet`` is that it allows us to write
the intermediate steps as set to set transformation. Some of the transformations
are:
* Composition: `\{x|f(g(x))=0;x \in S\} \Rightarrow \{x|g(x)=y; x \in S, y \in \{z|f(z)=0; z \in S\}\}`
* Polynomial Solver: `\{x | P(x) = 0;x \in S\} \Rightarrow \{x_1,x_2, ... ,x_n\} \cap S`,
where `x_i` are roots of `P(x)`.
* Invert solver: `\{x|f(x)=0;x \in S\} \Rightarrow \{g(0)| \text{ all g such that } f(g(x)) = x\}`
* logcombine: `\{x| \log(f(x)) + \log(g(x));x \in S\}`
`\Rightarrow \{x| \log(f(x).g(x)); x \in S\} \text{ if } f(x) > 0 \text{ and } g(x) > 0`
`\Rightarrow \{x| \log(f(x)) + \log(g(x));x \in S\} \text{ otherwise}`
* product solve: `\{x|f(x)g(x)=0; x \in S\}`
`\Rightarrow \{x|f(x)=0; x \in S\} U \{x|g(x)=0; x \in S\}`
`\text{ given } f(x) \text{ and } g(x) \text{ are bounded.}`
`\Rightarrow \{x|f(x)g(x)=0; x \in S\}, \text{ otherwise}`
Since the output type is same as the input type any composition of these
transformations is also a valid transformation. And our aim is to find
the right sequence of compositions (given the atoms) which transforms
the given condition set to a set which is not a condition set i.e.,
FiniteSet, Interval, Set of Integers and their Union, Intersection,
Complement or ImageSet. We can assign a cost function to each set,
such that, the more desirable that form of set is to us, the less the value
of the cost function. This way our problem is now reduced to finding the path
from the initial ConditionSet to the lowest valued set on a graph where
the atomic transformations forms the edges.
How do we deal with cases where only some of the solutions are known?
---------------------------------------------------------------------
Creating a universal equation solver, which can solve each and every equation
we encounter in mathematics is an ideal case for solvers in a Computer
Algebra System. When cases which are not solved or can only be solved
incompletely, a ``ConditionSet`` is used and acts as an unevaluated solveset
object.
Note that, mathematically, finding a complete set of solutions for an
equation is undecidable. See `Richardson's theorem
`_.
``ConditionSet`` is basically a Set of elements which satisfy a given
condition. For example, to represent the solutions of the equation in
the real domain:
.. math:: (x^2 - 4)(\sin(x) + x)
We can represent it as:
`\{-2, 2\} ∪ \{x | x \in \mathbb{R} ∧ x + \sin(x) = 0\}`
What will you do with the old solve?
------------------------------------
There are still a few things ``solveset`` can't do, which the old ``solve``
can, such as solving non linear multivariate & LambertW type equations.
Hence, it's not yet a perfect replacement for old ``solve``. The ultimate
goal is to:
* Replace ``solve`` with ``solveset`` once solveset is at least as powerful as
``solve``, i.e., ``solveset`` does everything that ``solve`` can do
currently, and
* eventually rename ``solveset`` to ``solve``.
How are symbolic parameters handled in solveset?
------------------------------------------------
Solveset is in its initial phase of development, so the symbolic parameters
aren't handled well for all the cases, but some work has been done in this
regard to depict our ideology towards symbolic parameters. As an example,
consider the solving of `|x| = n` for real `x`, where `n` is a symbolic
parameter. Solveset returns the value of `x` considering the domain of the
symbolic parameter `n` as well:
.. math:: ([0, \infty) \cap \{n\}) \cup ((-\infty, 0] \cap \{-n\}).
This simply means `n` is the solution only when it belongs to the
``Interval`` `[0, \infty)` and `-n` is the solution only when `-n` belongs to
the ``Interval`` `(- \infty, 0]`.
There are other cases to address too, like solving `2^x + (a - 2)` for `x`
where `a` is a symbolic parameter. As of now, It returns the solution as an
intersection with `\mathbb{R}`, which is trivial, as it doesn't reveal the
domain of `a` in the solution.
Recently, we have also implemented a function to find the domain of the
expression in a FiniteSet (Intersection with the interval) in which it is
not-empty. It is a useful addition for dealing with symbolic parameters.
For example:
>>> from sympy import Symbol, FiniteSet, Interval, not_empty_in, sqrt, oo
>>> from sympy.abc import x
>>> not_empty_in(FiniteSet(x/2).intersect(Interval(0, 1)), x)
Interval(0, 2)
>>> not_empty_in(FiniteSet(x, x**2).intersect(Interval(1, 2)), x)
Union(Interval(-sqrt(2), -1), Interval(1, 2))
References
----------
.. [1] https://github.com/sympy/sympy/wiki/GSoC-2015-Ideas#solvers
.. [2] https://github.com/sympy/sympy/wiki/GSoC-2014-Application-Harsh-Gupta:-Solvers
.. [3] https://github.com/sympy/sympy/wiki/GSoC-2015-Application-AMiT-Kumar--Solvers-:-Extending-Solveset
.. [5] http://iamit.in/blog/
.. [6] https://github.com/sympy/sympy/pull/2948 : Action Plan for improving solvers.
.. [7] https://github.com/sympy/sympy/issues/6659 : ``solve()`` is a giant mess
.. [8] https://github.com/sympy/sympy/pull/7523 : ``solveset`` PR
.. [9] https://groups.google.com/forum/#!topic/sympy/-SIbX0AFL3Q
.. [10] https://github.com/sympy/sympy/pull/9696
.. [11] https://en.wikipedia.org/wiki/Richardson%27s_theorem
Solveset Module Reference
-------------------------
Use :func:`solveset` to solve equations or expressions (assumed to be equal to 0) for a single variable.
Solving an equation like `x^2 == 1` can be done as follows::
>>> from sympy import solveset
>>> from sympy import Symbol, Eq
>>> x = Symbol('x')
>>> solveset(Eq(x**2, 1), x)
{-1, 1}
Or one may manually rewrite the equation as an expression equal to 0::
>>> solveset(x**2 - 1, x)
{-1, 1}
The first argument for :func:`solveset` is an expression (equal to zero) or an equation and the second argument
is the symbol that we want to solve the equation for.
.. autofunction:: sympy.solvers.solveset.solveset
.. autofunction:: sympy.solvers.solveset.solveset_real
.. autofunction:: sympy.solvers.solveset.solveset_complex
.. autofunction:: sympy.solvers.solveset.invert_real
.. autofunction:: sympy.solvers.solveset.invert_complex
.. autofunction:: sympy.solvers.solveset.domain_check
linear_eq_to_matrix
-------------------
.. autofunction:: sympy.solvers.solveset.linear_eq_to_matrix
linsolve
--------
.. autofunction:: sympy.solvers.solveset.linsolve
nonlinsolve
-----------
.. autofunction:: sympy.solvers.solveset.nonlinsolve
Diophantine Equations (DEs)
---------------------------
See :ref:`diophantine-docs`
Inequalities
------------
See :ref:`inequality-docs`
Ordinary Differential equations (ODEs)
--------------------------------------
See :ref:`ode-docs`.
Partial Differential Equations (PDEs)
-------------------------------------
See :ref:`pde-docs`.