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.

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 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:

\[(x^2 - 4)(\sin(x) + x)\]

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:

\[([0, \infty) \cap \{n\}) \cup ((-\infty, 0] \cap \{-n\}).\]

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))

Solveset Module Reference

Use 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 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.

sympy.solvers.solveset.solveset(f, symbol=None, domain=S.Complexes)[source]

Solves a given inequality or equation with set as output

Parameters

f : Expr or a relational.

The target equation or inequality

symbol : Symbol

The variable for which the equation is solved

domain : Set

The domain over which the equation is solved

Returns

Set

A set of values for \(symbol\) for which \(f\) is True or is equal to zero. An \(EmptySet\) is returned if \(f\) is False or nonzero. A \(ConditionSet\) is returned as unsolved object if algorithms to evaluate complete solution are not yet implemented.

\(solveset\) claims to be complete in the solution set that it returns.

Raises

NotImplementedError

The algorithms to solve inequalities in complex domain are not yet implemented.

ValueError

The input is not valid.

RuntimeError

It is a bug, please report to the github issue tracker.

See also

solveset_real

solver for real domain

solveset_complex

solver for complex domain

Notes

Python interprets 0 and 1 as False and True, respectively, but in this function they refer to solutions of an expression. So 0 and 1 return the Domain and EmptySet, respectively, while True and False return the opposite (as they are assumed to be solutions of relational expressions).

Examples

>>> from sympy import exp, sin, Symbol, pprint, S
>>> from sympy.solvers.solveset import solveset, solveset_real

  • The default domain is complex. Not specifying a domain will lead to the solving of the equation in the complex domain (and this is not affected by the assumptions on the symbol):

>>> x = Symbol('x')
>>> pprint(solveset(exp(x) - 1, x), use_unicode=False)
{2*n*I*pi | n in S.Integers}

>>> x = Symbol('x', real=True)
>>> pprint(solveset(exp(x) - 1, x), use_unicode=False)
{2*n*I*pi | n in S.Integers}

  • If you want to use \(solveset\) to solve the equation in the real domain, provide a real domain. (Using \(solveset\_real\) does this automatically.)

>>> R = S.Reals
>>> x = Symbol('x')
>>> solveset(exp(x) - 1, x, R)
{0}
>>> solveset_real(exp(x) - 1, x)
{0}

The solution is mostly unaffected by assumptions on the symbol, but there may be some slight difference:

>>> pprint(solveset(sin(x)/x,x), use_unicode=False)
({2*n*pi | n in S.Integers} \ {0}) U ({2*n*pi + pi | n in S.Integers} \ {0})

>>> p = Symbol('p', positive=True)
>>> pprint(solveset(sin(p)/p, p), use_unicode=False)
{2*n*pi | n in S.Integers} U {2*n*pi + pi | n in S.Integers}

  • Inequalities can be solved over the real domain only. Use of a complex domain leads to a NotImplementedError.

>>> solveset(exp(x) > 1, x, R)
Interval.open(0, oo)
sympy.solvers.solveset.solveset_real(f, symbol)[source]
sympy.solvers.solveset.solveset_complex(f, symbol)[source]
sympy.solvers.solveset.invert_real(f_x, y, x, domain=S.Reals)[source]

Inverts a real-valued function. Same as _invert, but sets the domain to S.Reals before inverting.

sympy.solvers.solveset.invert_complex(f_x, y, x, domain=S.Complexes)

Reduce the 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). The return value is a tuple (g(x), set_h), where g(x) is a function of x and set_h is the set of function {h_1(y), h_2(y), ..., h_n(y)}. Here, y is not necessarily a symbol.

The set_h contains the functions, along with the information about the domain in which they are valid, through set operations. For instance, if y = Abs(x) - n is inverted in the real domain, then set_h is not simply \({-n, n}\) as the nature of \(n\) is unknown; rather, it is: \(Intersection([0, oo) {n}) U Intersection((-oo, 0], {-n})\)

By default, the complex domain is used which means that inverting even seemingly simple functions like exp(x) will give very different results from those obtained in the real domain. (In the case of exp(x), the inversion via log is multi-valued in the complex domain, having infinitely many branches.)

If you are working with real values only (or you are not sure which function to use) you should probably set the domain to S.Reals (or use \(invert\_real\) which does that automatically).

See also

invert_real, invert_complex

Examples

>>> from sympy.solvers.solveset import invert_complex, invert_real
>>> from sympy.abc import x, y
>>> from sympy import exp, log

When does exp(x) == y?

>>> invert_complex(exp(x), y, x)
(x, ImageSet(Lambda(_n, I*(2*_n*pi + arg(y)) + log(Abs(y))), S.Integers))
>>> invert_real(exp(x), y, x)
(x, Intersection(S.Reals, {log(y)}))

When does exp(x) == 1?

>>> invert_complex(exp(x), 1, x)
(x, ImageSet(Lambda(_n, 2*_n*I*pi), S.Integers))
>>> invert_real(exp(x), 1, x)
(x, {0})
sympy.solvers.solveset.domain_check(f, symbol, p)[source]

Returns False if point p is infinite or any subexpression of f is infinite or becomes so after replacing symbol with p. If none of these conditions is met then True will be returned.

Examples

>>> from sympy import Mul, oo
>>> from sympy.abc import x
>>> from sympy.solvers.solveset import domain_check
>>> g = 1/(1 + (1/(x + 1))**2)
>>> domain_check(g, x, -1)
False
>>> domain_check(x**2, x, 0)
True
>>> domain_check(1/x, x, oo)
False

  • The function relies on the assumption that the original form of the equation has not been changed by automatic simplification.

>>> domain_check(x/x, x, 0) # x/x is automatically simplified to 1
True

  • To deal with automatic evaluations use evaluate=False:

>>> domain_check(Mul(x, 1/x, evaluate=False), x, 0)
False

linear_eq_to_matrix

sympy.solvers.solveset.linear_eq_to_matrix(equations, *symbols)[source]

Converts a given System of Equations into Matrix form. Here \(equations\) must be a linear system of equations in \(symbols\). The order of symbols in input \(symbols\) will determine the order of coefficients in the returned Matrix.

The Matrix form corresponds to the augmented matrix form. For example:

\[4x + 2y + 3z = 1\]
\[3x + y + z = -6\]
\[2x + 4y + 9z = 2\]

This system would return \(A\) & \(b\) as given below:

    [ 4  2  3 ]          [ 1 ]
A = [ 3  1  1 ]   b  =   [-6 ]
    [ 2  4  9 ]          [ 2 ]

Examples

>>> from sympy import linear_eq_to_matrix, symbols
>>> x, y, z = symbols('x, y, z')
>>> eqns = [x + 2*y + 3*z - 1, 3*x + y + z + 6, 2*x + 4*y + 9*z - 2]
>>> A, b = linear_eq_to_matrix(eqns, [x, y, z])
>>> A
Matrix([
[1, 2, 3],
[3, 1, 1],
[2, 4, 9]])
>>> b
Matrix([
[ 1],
[-6],
[ 2]])
>>> eqns = [x + z - 1, y + z, x - y]
>>> A, b = linear_eq_to_matrix(eqns, [x, y, z])
>>> A
Matrix([
[1,  0, 1],
[0,  1, 1],
[1, -1, 0]])
>>> b
Matrix([
[1],
[0],
[0]])

  • Symbolic coefficients are also supported

>>> a, b, c, d, e, f = symbols('a, b, c, d, e, f')
>>> eqns = [a*x + b*y - c, d*x + e*y - f]
>>> A, B = linear_eq_to_matrix(eqns, x, y)
>>> A
Matrix([
[a, b],
[d, e]])
>>> B
Matrix([
[c],
[f]])

linsolve

sympy.solvers.solveset.linsolve(system, *symbols)[source]

Solve system of N linear equations with M variables, which means both under - and overdetermined systems are supported. The possible number of solutions is zero, one or infinite. Zero solutions throws a ValueError, where as infinite solutions are represented parametrically in terms of given symbols. For unique solution a FiniteSet of ordered tuple is returned.

All Standard input formats are supported: For the given set of Equations, the respective input types are given below:

\[3x + 2y - z = 1\]
\[2x - 2y + 4z = -2\]
\[2x - y + 2z = 0\]

  • Augmented Matrix Form, \(system\) given below:

         [3   2  -1  1]
system = [2  -2   4 -2]
         [2  -1   2  0]

  • List Of Equations Form

\(system = [3x + 2y - z - 1, 2x - 2y + 4z + 2, 2x - y + 2z]\)

  • Input A & b Matrix Form (from Ax = b) are given as below:

    [3   2  -1 ]         [  1 ]
A = [2  -2   4 ]    b =  [ -2 ]
    [2  -1   2 ]         [  0 ]

\(system = (A, b)\)

Symbols to solve for should be given as input in all the cases either in an iterable or as comma separated arguments. This is done to maintain consistency in returning solutions in the form of variable input by the user.

The algorithm used here is Gauss-Jordan elimination, which results, after elimination, in an row echelon form matrix.

Returns

A FiniteSet of ordered tuple of values of \(symbols\) for which

the \(system\) has solution.

Please note that general FiniteSet is unordered, the solution

returned here is not simply a FiniteSet of solutions, rather

it is a FiniteSet of ordered tuple, i.e. the first & only

argument to FiniteSet is a tuple of solutions, which is ordered,

& hence the returned solution is ordered.

Also note that solution could also have been returned as an

ordered tuple, FiniteSet is just a wrapper \({}\) around

the tuple. It has no other significance except for

the fact it is just used to maintain a consistent output

format throughout the solveset.

Returns EmptySet(), if the linear system is inconsistent.

Raises

ValueError

The input is not valid. The symbols are not given.

Examples

>>> from sympy import Matrix, S, linsolve, symbols
>>> x, y, z = symbols("x, y, z")
>>> A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 10]])
>>> b = Matrix([3, 6, 9])
>>> A
Matrix([
[1, 2,  3],
[4, 5,  6],
[7, 8, 10]])
>>> b
Matrix([
[3],
[6],
[9]])
>>> linsolve((A, b), [x, y, z])
{(-1, 2, 0)}

  • Parametric Solution: In case the system is under determined, the function will return parametric solution in terms of the given symbols. Free symbols in the system are returned as it is. For e.g. in the system below, \(z\) is returned as the solution for variable z, which means z is a free symbol, i.e. it can take arbitrary values.

>>> A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
>>> b = Matrix([3, 6, 9])
>>> linsolve((A, b), [x, y, z])
{(z - 1, -2*z + 2, z)}

  • List of Equations as input

>>> Eqns = [3*x + 2*y - z - 1, 2*x - 2*y + 4*z + 2, - x + S(1)/2*y - z]
>>> linsolve(Eqns, x, y, z)
{(1, -2, -2)}

  • Augmented Matrix as input

>>> aug = Matrix([[2, 1, 3, 1], [2, 6, 8, 3], [6, 8, 18, 5]])
>>> aug
Matrix([
[2, 1,  3, 1],
[2, 6,  8, 3],
[6, 8, 18, 5]])
>>> linsolve(aug, x, y, z)
{(3/10, 2/5, 0)}

  • Solve for symbolic coefficients

>>> a, b, c, d, e, f = symbols('a, b, c, d, e, f')
>>> eqns = [a*x + b*y - c, d*x + e*y - f]
>>> linsolve(eqns, x, y)
{((-b*f + c*e)/(a*e - b*d), (a*f - c*d)/(a*e - b*d))}

  • A degenerate system returns solution as set of given symbols.

>>> system = Matrix(([0,0,0], [0,0,0], [0,0,0]))
>>> linsolve(system, x, y)
{(x, y)}

  • For an empty system linsolve returns empty set

>>> linsolve([ ], x)
EmptySet()

nonlinsolve

sympy.solvers.solveset.nonlinsolve(system, *symbols)[source]

Solve system of N non linear equations with M variables, which means both under and overdetermined systems are supported. Positive dimensional system is also supported (A system with infinitely many solutions is said to be positive-dimensional). In Positive dimensional system solution will be dependent on at least one symbol. Returns both real solution and complex solution(If system have). The possible number of solutions is zero, one or infinite.

Parameters

system : list of equations

The target system of equations

symbols : list of Symbols

symbols should be given as a sequence eg. list

Returns

A FiniteSet of ordered tuple of values of \(symbols\) for which the \(system\)

has solution. Order of values in the tuple is same as symbols present in

the parameter \(symbols\).

Please note that general FiniteSet is unordered, the solution returned

here is not simply a FiniteSet of solutions, rather it is a FiniteSet of

ordered tuple, i.e. the first & only argument to FiniteSet is a tuple of

solutions, which is ordered, & hence the returned solution is ordered.

Also note that solution could also have been returned as an ordered tuple,

FiniteSet is just a wrapper \({}\) around the tuple. It has no other

significance except for the fact it is just used to maintain a consistent

output format throughout the solveset.

For the given set of Equations, the respective input types

are given below:

\[x*y - 1 = 0\]
\[4*x**2 + y**2 - 5 = 0\]

\(system = [x*y - 1, 4*x**2 + y**2 - 5]\)

\(symbols = [x, y]\)

Raises

ValueError

The input is not valid. The symbols are not given.

AttributeError

The input symbols are not \(Symbol\) type.

Examples

>>> from sympy.core.symbol import symbols
>>> from sympy.solvers.solveset import nonlinsolve
>>> x, y, z = symbols('x, y, z', real=True)
>>> nonlinsolve([x*y - 1, 4*x**2 + y**2 - 5], [x, y])
{(-1, -1), (-1/2, -2), (1/2, 2), (1, 1)}

  1. Positive dimensional system and complements:

>>> from sympy import pprint
>>> from sympy.polys.polytools import is_zero_dimensional
>>> a, b, c, d = symbols('a, b, c, d', real=True)
>>> eq1 =  a + b + c + d
>>> eq2 = a*b + b*c + c*d + d*a
>>> eq3 = a*b*c + b*c*d + c*d*a + d*a*b
>>> eq4 = a*b*c*d - 1
>>> system = [eq1, eq2, eq3, eq4]
>>> is_zero_dimensional(system)
False
>>> pprint(nonlinsolve(system, [a, b, c, d]), use_unicode=False)
  -1       1               1      -1
{(---, -d, -, {d} \ {0}), (-, -d, ---, {d} \ {0})}
   d       d               d       d
>>> nonlinsolve([(x+y)**2 - 4, x + y - 2], [x, y])
{(-y + 2, y)}

2. If some of the equations are non polynomial equation then \(nonlinsolve\) will call \(substitution\) function and returns real and complex solutions, if present.

>>> from sympy import exp, sin
>>> nonlinsolve([exp(x) - sin(y), y**2 - 4], [x, y])
{(log(sin(2)), 2), (ImageSet(Lambda(_n, I*(2*_n*pi + pi) +
    log(sin(2))), S.Integers), -2), (ImageSet(Lambda(_n, 2*_n*I*pi +
    Mod(log(sin(2)), 2*I*pi)), S.Integers), 2)}

3. If system is Non linear polynomial zero dimensional then it returns both solution (real and complex solutions, if present using \(solve_poly_system\)):

>>> from sympy import sqrt
>>> nonlinsolve([x**2 - 2*y**2 -2, x*y - 2], [x, y])
{(-2, -1), (2, 1), (-sqrt(2)*I, sqrt(2)*I), (sqrt(2)*I, -sqrt(2)*I)}

4. \(nonlinsolve\) can solve some linear(zero or positive dimensional) system (because it is using \(groebner\) function to get the groebner basis and then \(substitution\) function basis as the new \(system\)). But it is not recommended to solve linear system using \(nonlinsolve\), because \(linsolve\) is better for all kind of linear system.

>>> nonlinsolve([x + 2*y -z - 3, x - y - 4*z + 9 , y + z - 4], [x, y, z])
{(3*z - 5, -z + 4, z)}

5. System having polynomial equations and only real solution is present (will be solved using \(solve_poly_system\)):

>>> e1 = sqrt(x**2 + y**2) - 10
>>> e2 = sqrt(y**2 + (-x + 10)**2) - 3
>>> nonlinsolve((e1, e2), (x, y))
{(191/20, -3*sqrt(391)/20), (191/20, 3*sqrt(391)/20)}
>>> nonlinsolve([x**2 + 2/y - 2, x + y - 3], [x, y])
{(1, 2), (1 + sqrt(5), -sqrt(5) + 2), (-sqrt(5) + 1, 2 + sqrt(5))}
>>> nonlinsolve([x**2 + 2/y - 2, x + y - 3], [y, x])
{(2, 1), (2 + sqrt(5), -sqrt(5) + 1), (-sqrt(5) + 2, 1 + sqrt(5))}

6. It is better to use symbols instead of Trigonometric Function or Function (e.g. replace \(sin(x)\) with symbol, replace \(f(x)\) with symbol and so on. Get soln from \(nonlinsolve\) and then using \(solveset\) get the value of \(x\))

How Nonlinsolve Is Better Than Old Solver \(_solve_system\) :

1. A positive dimensional system solver : nonlinsolve can return solution for positive dimensional system. It finds the Groebner Basis of the positive dimensional system(calling it as basis) then we can start solving equation(having least number of variable first in the basis) using solveset and substituting that solved solutions into other equation(of basis) to get solution in terms of minimum variables. Here the important thing is how we are substituting the known values and in which equations.

2. Real and Complex both solutions : nonlinsolve returns both real and complex solution. If all the equations in the system are polynomial then using \(solve_poly_system\) both real and complex solution is returned. If all the equations in the system are not polynomial equation then goes to \(substitution\) method with this polynomial and non polynomial equation(s), to solve for unsolved variables. Here to solve for particular variable solveset_real and solveset_complex is used. For both real and complex solution function \(_solve_using_know_values\) is used inside \(substitution\) function.(\(substitution\) function will be called when there is any non polynomial equation(s) is present). When solution is valid then add its general solution in the final result.

3. Complement and Intersection will be added if any : nonlinsolve maintains dict for complements and Intersections. If solveset find complements or/and Intersection with any Interval or set during the execution of \(substitution\) function ,then complement or/and Intersection for that variable is added before returning final solution.

Diophantine Equations (DEs)

See Diophantine

Inequalities

See Inequality Solvers

Ordinary Differential equations (ODEs)

See ODE.

Partial Differential Equations (PDEs)

See PDE.