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.