sympy/source/doc/src/tutorials/intro-tutorial/solvers.rst

=========
 Solvers
=========

.. note::

   For a beginner-friendly guide focused on solving common types of equations,
   refer to :ref:`solving-guide`.

>>> from sympy import *
>>> x, y, z = symbols('x y z')
>>> init_printing(use_unicode=True)

A Note about Equations
======================

Recall from the :ref:`gotchas <tutorial_gotchas_equals>` section of this
tutorial that symbolic equations in SymPy are not represented by ``=`` or
``==``, but by ``Eq``.


    >>> Eq(x, y)
    x = y


However, there is an even easier way.  In SymPy, any expression not in an
``Eq`` is automatically assumed to equal 0 by the solving functions.  Since `a
= b` if and only if `a - b = 0`, this means that instead of using ``x == y``,
you can just use ``x - y``.  For example

    >>> solveset(Eq(x**2, 1), x)
    {-1, 1}
    >>> solveset(Eq(x**2 - 1, 0), x)
    {-1, 1}
    >>> solveset(x**2 - 1, x)
    {-1, 1}

This is particularly useful if the equation you wish to solve is already equal
to 0. Instead of typing ``solveset(Eq(expr, 0), x)``, you can just use
``solveset(expr, x)``.

Solving Equations Algebraically
===============================

The main function for solving algebraic equations is ``solveset``.
The syntax for ``solveset`` is ``solveset(equation, variable=None, domain=S.Complexes)``
Where ``equations`` may be in the form of ``Eq`` instances or expressions
that are assumed to be equal to zero.

Please note that there is another function called ``solve`` which
can also be used to solve equations. The syntax is ``solve(equations, variables)``
However, it is recommended to use ``solveset`` instead.

When solving a single equation, the output of ``solveset`` is a ``FiniteSet`` or
an ``Interval`` or ``ImageSet`` of the solutions.

    >>> solveset(x**2 - x, x)
    {0, 1}
    >>> solveset(x - x, x, domain=S.Reals)
    ℝ
    >>> solveset(sin(x) - 1, x, domain=S.Reals)
    ⎧        π │      ⎫
    ⎨2⋅n⋅π + ─ │ n ∊ ℤ⎬
    ⎩        2 │      ⎭


If there are no solutions, an ``EmptySet`` is returned and if it
is not able to find solutions then a ``ConditionSet`` is returned.

    >>> solveset(exp(x), x)     # No solution exists
    ∅
    >>> solveset(cos(x) - x, x)  # Not able to find solution
    {x │ x ∊ ℂ ∧ (-x + cos(x) = 0)}


In the ``solveset`` module, the linear system of equations is solved using ``linsolve``.
In future we would be able to use linsolve directly from ``solveset``. Following
is an example of the syntax of ``linsolve``.

* List of Equations Form:

    >>> linsolve([x + y + z - 1, x + y + 2*z - 3 ], (x, y, z))
    {(-y - 1, y, 2)}

* Augmented Matrix Form:

    >>> linsolve(Matrix(([1, 1, 1, 1], [1, 1, 2, 3])), (x, y, z))
    {(-y - 1, y, 2)}

* A*x = b Form

    >>> M = Matrix(((1, 1, 1, 1), (1, 1, 2, 3)))
    >>> system = A, b = M[:, :-1], M[:, -1]
    >>> linsolve(system, x, y, z)
    {(-y - 1, y, 2)}

.. note::

   The order of solution corresponds the order of given symbols.


In the ``solveset`` module, the non linear system of equations is solved using
``nonlinsolve``. Following are examples of ``nonlinsolve``.

1. When only real solution is present:

    >>> a, b, c, d = symbols('a, b, c, d', real=True)
    >>> nonlinsolve([a**2 + a, a - b], [a, b])
    {(-1, -1), (0, 0)}
    >>> nonlinsolve([x*y - 1, x - 2], x, y)
    {(2, 1/2)}

2. When only complex solution is present:

    >>> nonlinsolve([x**2 + 1, y**2 + 1], [x, y])
    {(-ⅈ, -ⅈ), (-ⅈ, ⅈ), (ⅈ, -ⅈ), (ⅈ, ⅈ)}

3. When both real and complex solution are present:

    >>> from sympy import sqrt
    >>> system = [x**2 - 2*y**2 -2, x*y - 2]
    >>> vars = [x, y]
    >>> nonlinsolve(system, vars)
    {(-2, -1), (2, 1), (-√2⋅ⅈ, √2⋅ⅈ), (√2⋅ⅈ, -√2⋅ⅈ)}

    >>> system = [exp(x) - sin(y), 1/y - 3]
    >>> nonlinsolve(system, vars)
    {({2⋅n⋅ⅈ⋅π + log(sin(1/3)) │ n ∊ ℤ}, 1/3)}

4. When the system is positive-dimensional system (has infinitely many solutions):

    >>> nonlinsolve([x*y, x*y - x], [x, y])
    {(0, y)}

    >>> system = [a**2 + a*c, a - b]
    >>> nonlinsolve(system, [a, b])
    {(0, 0), (-c, -c)}


.. note::

   1. The order of solution corresponds the order of given symbols.

   2. Currently ``nonlinsolve`` doesn't return solution in form of ``LambertW`` (if there
   is solution present in the form of ``LambertW``).

   ``solve`` can be used for such cases:

   >>> solve([x**2 - y**2/exp(x)], [x, y], dict=True)
   ⎡⎧         ____⎫  ⎧        ____⎫⎤
   ⎢⎨        ╱  x ⎬  ⎨       ╱  x ⎬⎥
   ⎣⎩y: -x⋅╲╱  ℯ  ⎭, ⎩y: x⋅╲╱  ℯ  ⎭⎦
   >>> solve(x**2 - y**2/exp(x), x, dict=True)
   ⎡⎧      ⎛-y ⎞⎫  ⎧      ⎛y⎞⎫⎤
   ⎢⎨x: 2⋅W⎜───⎟⎬, ⎨x: 2⋅W⎜─⎟⎬⎥
   ⎣⎩      ⎝ 2 ⎠⎭  ⎩      ⎝2⎠⎭⎦

   3. Currently ``nonlinsolve`` is not properly capable of solving the system of equations
   having trigonometric functions.

   ``solve`` can be used for such cases (but does not give all solution):

   >>> solve([sin(x + y), cos(x - y)], [x, y])
   ⎡⎛-3⋅π   3⋅π⎞  ⎛-π   π⎞  ⎛π  3⋅π⎞  ⎛3⋅π  π⎞⎤
   ⎢⎜─────, ───⎟, ⎜───, ─⎟, ⎜─, ───⎟, ⎜───, ─⎟⎥
   ⎣⎝  4     4 ⎠  ⎝ 4   4⎠  ⎝4   4 ⎠  ⎝ 4   4⎠⎦


.. _tutorial-roots:

``solveset`` reports each solution only once.  To get the solutions of a
polynomial including multiplicity use ``roots``.

    >>> solveset(x**3 - 6*x**2 + 9*x, x)
    {0, 3}
    >>> roots(x**3 - 6*x**2 + 9*x, x)
    {0: 1, 3: 2}

The output ``{0: 1, 3: 2}`` of ``roots`` means that ``0`` is a root of
multiplicity 1 and ``3`` is a root of multiplicity 2.

.. note::

   Currently ``solveset`` is not capable of solving the following types of equations:

   * Equations solvable by LambertW (Transcendental equation solver).

   ``solve`` can be used for such cases:

   >>> solve(x*exp(x) - 1, x )
   [W(1)]


.. _tutorial-dsolve:

Solving Differential Equations
==============================

To solve differential equations, use ``dsolve``.  First, create an undefined
function by passing ``cls=Function`` to the ``symbols`` function.


    >>> f, g = symbols('f g', cls=Function)

``f`` and ``g`` are now undefined functions.  We can call ``f(x)``, and it
will represent an unknown function.

    >>> f(x)
    f(x)

Derivatives of ``f(x)`` are unevaluated.

    >>> f(x).diff(x)
    d
    ──(f(x))
    dx

(see the :ref:`Derivatives <tutorial-derivatives>` section for more on
derivatives).

To represent the differential equation `f''(x) - 2f'(x) + f(x) = \sin(x)`, we
would thus use

    >>> diffeq = Eq(f(x).diff(x, x) - 2*f(x).diff(x) + f(x), sin(x))
    >>> diffeq
                          2
             d           d
    f(x) - 2⋅──(f(x)) + ───(f(x)) = sin(x)
             dx           2
                        dx

To solve the ODE, pass it and the function to solve for to ``dsolve``.

    >>> dsolve(diffeq, f(x))
                        x   cos(x)
    f(x) = (C₁ + C₂⋅x)⋅ℯ  + ──────
                              2

``dsolve`` returns an instance of ``Eq``.  This is because, in general,
solutions to differential equations cannot be solved explicitly for the
function.

    >>> dsolve(f(x).diff(x)*(1 - sin(f(x))) - 1, f(x))
    -x + f(x) + cos(f(x)) = C₁

The arbitrary constants in the solutions from dsolve are symbols of the form
``C1``, ``C2``, ``C3``, and so on.