Source code for sympy.solvers.solveset

"""
This module contains functions to:

    - solve a single equation for a single variable, in any domain either real or complex.

    - solve a system of linear equations with N variables and M equations.

    - solve a system of Non Linear Equations with N variables and M equations
"""
from __future__ import print_function, division

from sympy.core.sympify import sympify
from sympy.core import S, Pow, Dummy, pi, Expr, Wild, Mul, Equality
from sympy.core.numbers import I, Number, Rational, oo
from sympy.core.function import (Lambda, expand_complex)
from sympy.core.relational import Eq
from sympy.simplify.simplify import simplify, fraction, trigsimp
from sympy.functions import (log, Abs, tan, cot, sin, cos, sec, csc, exp,
                             acos, asin, acsc, asec, arg,
                             piecewise_fold)
from sympy.functions.elementary.trigonometric import (TrigonometricFunction,
                                                      HyperbolicFunction)
from sympy.functions.elementary.miscellaneous import real_root
from sympy.sets import (FiniteSet, EmptySet, imageset, Interval, Intersection,
                        Union, ConditionSet, ImageSet, Complement)
from sympy.matrices import Matrix
from sympy.polys import (roots, Poly, degree, together, PolynomialError,
                         RootOf)
from sympy.solvers.solvers import checksol, denoms, unrad, _simple_dens
from sympy.solvers.polysys import solve_poly_system
from sympy.solvers.inequalities import solve_univariate_inequality
from sympy.utilities import filldedent
from sympy.calculus.util import periodicity, continuous_domain
from sympy.core.compatibility import ordered, default_sort_key


def _invert(f_x, y, x, domain=S.Complexes):
    r"""
    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).


    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})

    See Also
    ========
    invert_real, invert_complex
    """
    x = sympify(x)
    if not x.is_Symbol:
        raise ValueError("x must be a symbol")
    f_x = sympify(f_x)
    if not f_x.has(x):
        raise ValueError("Inverse of constant function doesn't exist")
    y = sympify(y)
    if y.has(x):
        raise ValueError("y should be independent of x ")

    if domain.is_subset(S.Reals):
        x1, s = _invert_real(f_x, FiniteSet(y), x)
    else:
        x1, s = _invert_complex(f_x, FiniteSet(y), x)

    if not isinstance(s, FiniteSet) or x1 == f_x:
        return x1, s

    return x1, s.intersection(domain)


invert_complex = _invert


[docs]def invert_real(f_x, y, x, domain=S.Reals): """ Inverts a real-valued function. Same as _invert, but sets the domain to ``S.Reals`` before inverting. """ return _invert(f_x, y, x, domain)
def _invert_real(f, g_ys, symbol): """Helper function for _invert.""" if f == symbol: return (f, g_ys) n = Dummy('n', real=True) if hasattr(f, 'inverse') and not isinstance(f, ( TrigonometricFunction, HyperbolicFunction, )): if len(f.args) > 1: raise ValueError("Only functions with one argument are supported.") return _invert_real(f.args[0], imageset(Lambda(n, f.inverse()(n)), g_ys), symbol) if isinstance(f, Abs): pos = Interval(0, S.Infinity) neg = Interval(S.NegativeInfinity, 0) return _invert_real(f.args[0], Union(imageset(Lambda(n, n), g_ys).intersect(pos), imageset(Lambda(n, -n), g_ys).intersect(neg)), symbol) if f.is_Add: # f = g + h g, h = f.as_independent(symbol) if g is not S.Zero: return _invert_real(h, imageset(Lambda(n, n - g), g_ys), symbol) if f.is_Mul: # f = g*h g, h = f.as_independent(symbol) if g is not S.One: return _invert_real(h, imageset(Lambda(n, n/g), g_ys), symbol) if f.is_Pow: base, expo = f.args base_has_sym = base.has(symbol) expo_has_sym = expo.has(symbol) if not expo_has_sym: res = imageset(Lambda(n, real_root(n, expo)), g_ys) if expo.is_rational: numer, denom = expo.as_numer_denom() if numer is S.One or numer is S.NegativeOne: if denom % 2 == 0: base_positive = solveset(base >= 0, symbol, S.Reals) res = imageset(Lambda(n, real_root(n, expo) ), g_ys.intersect( Interval.Ropen(S.Zero, S.Infinity))) _inv, _set = _invert_real(base, res, symbol) return (_inv, _set.intersect(base_positive)) else: return _invert_real(base, res, symbol) elif numer % 2 == 0: n = Dummy('n') neg_res = imageset(Lambda(n, -n), res) return _invert_real(base, res + neg_res, symbol) else: return _invert_real(base, res, symbol) else: if not base.is_positive: raise ValueError("x**w where w is irrational is not " "defined for negative x") return _invert_real(base, res, symbol) if not base_has_sym: return _invert_real(expo, imageset(Lambda(n, log(n)/log(base)), g_ys), symbol) if isinstance(f, TrigonometricFunction): if isinstance(g_ys, FiniteSet): def inv(trig): if isinstance(f, (sin, csc)): F = asin if isinstance(f, sin) else acsc return (lambda a: n*pi + (-1)**n*F(a),) if isinstance(f, (cos, sec)): F = acos if isinstance(f, cos) else asec return ( lambda a: 2*n*pi + F(a), lambda a: 2*n*pi - F(a),) if isinstance(f, (tan, cot)): return (lambda a: n*pi + f.inverse()(a),) n = Dummy('n', integer=True) invs = S.EmptySet for L in inv(f): invs += Union(*[imageset(Lambda(n, L(g)), S.Integers) for g in g_ys]) return _invert_real(f.args[0], invs, symbol) return (f, g_ys) def _invert_complex(f, g_ys, symbol): """Helper function for _invert.""" if f == symbol: return (f, g_ys) n = Dummy('n') if f.is_Add: # f = g + h g, h = f.as_independent(symbol) if g is not S.Zero: return _invert_complex(h, imageset(Lambda(n, n - g), g_ys), symbol) if f.is_Mul: # f = g*h g, h = f.as_independent(symbol) if g is not S.One: if g in set([S.NegativeInfinity, S.ComplexInfinity, S.Infinity]): return (h, S.EmptySet) return _invert_complex(h, imageset(Lambda(n, n/g), g_ys), symbol) if hasattr(f, 'inverse') and \ not isinstance(f, TrigonometricFunction) and \ not isinstance(f, exp): if len(f.args) > 1: raise ValueError("Only functions with one argument are supported.") return _invert_complex(f.args[0], imageset(Lambda(n, f.inverse()(n)), g_ys), symbol) if isinstance(f, exp): if isinstance(g_ys, FiniteSet): exp_invs = Union(*[imageset(Lambda(n, I*(2*n*pi + arg(g_y)) + log(Abs(g_y))), S.Integers) for g_y in g_ys if g_y != 0]) return _invert_complex(f.args[0], exp_invs, symbol) return (f, g_ys)
[docs]def domain_check(f, symbol, p): """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 """ f, p = sympify(f), sympify(p) if p.is_infinite: return False return _domain_check(f, symbol, p)
def _domain_check(f, symbol, p): # helper for domain check if f.is_Atom and f.is_finite: return True elif f.subs(symbol, p).is_infinite: return False else: return all([_domain_check(g, symbol, p) for g in f.args]) def _is_finite_with_finite_vars(f, domain=S.Complexes): """ Return True if the given expression is finite. For symbols that don't assign a value for `complex` and/or `real`, the domain will be used to assign a value; symbols that don't assign a value for `finite` will be made finite. All other assumptions are left unmodified. """ def assumptions(s): A = s.assumptions0 A.setdefault('finite', A.get('finite', True)) if domain.is_subset(S.Reals): # if this gets set it will make complex=True, too A.setdefault('real', True) else: # don't change 'real' because being complex implies # nothing about being real A.setdefault('complex', True) return A reps = {s: Dummy(**assumptions(s)) for s in f.free_symbols} return f.xreplace(reps).is_finite def _is_function_class_equation(func_class, f, symbol): """ Tests whether the equation is an equation of the given function class. The given equation belongs to the given function class if it is comprised of functions of the function class which are multiplied by or added to expressions independent of the symbol. In addition, the arguments of all such functions must be linear in the symbol as well. Examples ======== >>> from sympy.solvers.solveset import _is_function_class_equation >>> from sympy import tan, sin, tanh, sinh, exp >>> from sympy.abc import x >>> from sympy.functions.elementary.trigonometric import (TrigonometricFunction, ... HyperbolicFunction) >>> _is_function_class_equation(TrigonometricFunction, exp(x) + tan(x), x) False >>> _is_function_class_equation(TrigonometricFunction, tan(x) + sin(x), x) True >>> _is_function_class_equation(TrigonometricFunction, tan(x**2), x) False >>> _is_function_class_equation(TrigonometricFunction, tan(x + 2), x) True >>> _is_function_class_equation(HyperbolicFunction, tanh(x) + sinh(x), x) True """ if f.is_Mul or f.is_Add: return all(_is_function_class_equation(func_class, arg, symbol) for arg in f.args) if f.is_Pow: if not f.exp.has(symbol): return _is_function_class_equation(func_class, f.base, symbol) else: return False if not f.has(symbol): return True if isinstance(f, func_class): try: g = Poly(f.args[0], symbol) return g.degree() <= 1 except PolynomialError: return False else: return False def _solve_as_rational(f, symbol, domain): """ solve rational functions""" f = together(f, deep=True) g, h = fraction(f) if not h.has(symbol): try: return _solve_as_poly(g, symbol, domain) except NotImplementedError: # The polynomial formed from g could end up having # coefficients in a ring over which finding roots # isn't implemented yet, e.g. ZZ[a] for some symbol a return ConditionSet(f, symbol, domain) else: valid_solns = _solveset(g, symbol, domain) invalid_solns = _solveset(h, symbol, domain) return valid_solns - invalid_solns def _solve_trig(f, symbol, domain): """ Helper to solve trigonometric equations """ f = trigsimp(f) f_original = f f = f.rewrite(exp) f = together(f) g, h = fraction(f) y = Dummy('y') g, h = g.expand(), h.expand() g, h = g.subs(exp(I*symbol), y), h.subs(exp(I*symbol), y) if g.has(symbol) or h.has(symbol): return ConditionSet(symbol, Eq(f, 0), S.Reals) solns = solveset_complex(g, y) - solveset_complex(h, y) if isinstance(solns, FiniteSet): result = Union(*[invert_complex(exp(I*symbol), s, symbol)[1] for s in solns]) return Intersection(result, domain) elif solns is S.EmptySet: return S.EmptySet else: return ConditionSet(symbol, Eq(f_original, 0), S.Reals) def _solve_as_poly(f, symbol, domain=S.Complexes): """ Solve the equation using polynomial techniques if it already is a polynomial equation or, with a change of variables, can be made so. """ result = None if f.is_polynomial(symbol): solns = roots(f, symbol, cubics=True, quartics=True, quintics=True, domain='EX') num_roots = sum(solns.values()) if degree(f, symbol) <= num_roots: result = FiniteSet(*solns.keys()) else: poly = Poly(f, symbol) solns = poly.all_roots() if poly.degree() <= len(solns): result = FiniteSet(*solns) else: result = ConditionSet(symbol, Eq(f, 0), domain) else: poly = Poly(f) if poly is None: result = ConditionSet(symbol, Eq(f, 0), domain) gens = [g for g in poly.gens if g.has(symbol)] if len(gens) == 1: poly = Poly(poly, gens[0]) gen = poly.gen deg = poly.degree() poly = Poly(poly.as_expr(), poly.gen, composite=True) poly_solns = FiniteSet(*roots(poly, cubics=True, quartics=True, quintics=True).keys()) if len(poly_solns) < deg: result = ConditionSet(symbol, Eq(f, 0), domain) if gen != symbol: y = Dummy('y') inverter = invert_real if domain.is_subset(S.Reals) else invert_complex lhs, rhs_s = inverter(gen, y, symbol) if lhs == symbol: result = Union(*[rhs_s.subs(y, s) for s in poly_solns]) else: result = ConditionSet(symbol, Eq(f, 0), domain) else: result = ConditionSet(symbol, Eq(f, 0), domain) if result is not None: if isinstance(result, FiniteSet): # this is to simplify solutions like -sqrt(-I) to sqrt(2)/2 # - sqrt(2)*I/2. We are not expanding for solution with free # variables because that makes the solution more complicated. For # example expand_complex(a) returns re(a) + I*im(a) if all([s.free_symbols == set() and not isinstance(s, RootOf) for s in result]): s = Dummy('s') result = imageset(Lambda(s, expand_complex(s)), result) if isinstance(result, FiniteSet): result = result.intersection(domain) return result else: return ConditionSet(symbol, Eq(f, 0), domain) def _has_rational_power(expr, symbol): """ Returns (bool, den) where bool is True if the term has a non-integer rational power and den is the denominator of the expression's exponent. Examples ======== >>> from sympy.solvers.solveset import _has_rational_power >>> from sympy import sqrt >>> from sympy.abc import x >>> _has_rational_power(sqrt(x), x) (True, 2) >>> _has_rational_power(x**2, x) (False, 1) """ a, p, q = Wild('a'), Wild('p'), Wild('q') pattern_match = expr.match(a*p**q) or {} if pattern_match.get(a, S.Zero) is S.Zero: return (False, S.One) elif p not in pattern_match.keys(): return (False, S.One) elif isinstance(pattern_match[q], Rational) \ and pattern_match[p].has(symbol): if not pattern_match[q].q == S.One: return (True, pattern_match[q].q) if not isinstance(pattern_match[a], Pow) \ or isinstance(pattern_match[a], Mul): return (False, S.One) else: return _has_rational_power(pattern_match[a], symbol) def _solve_radical(f, symbol, solveset_solver): """ Helper function to solve equations with radicals """ eq, cov = unrad(f) if not cov: result = solveset_solver(eq, symbol) - \ Union(*[solveset_solver(g, symbol) for g in denoms(f, symbol)]) else: y, yeq = cov if not solveset_solver(y - I, y): yreal = Dummy('yreal', real=True) yeq = yeq.xreplace({y: yreal}) eq = eq.xreplace({y: yreal}) y = yreal g_y_s = solveset_solver(yeq, symbol) f_y_sols = solveset_solver(eq, y) result = Union(*[imageset(Lambda(y, g_y), f_y_sols) for g_y in g_y_s]) if isinstance(result, Complement) or isinstance(result,ConditionSet): solution_set = result else: f_set = [] # solutions for FiniteSet c_set = [] # solutions for ConditionSet for s in result: if checksol(f, symbol, s): f_set.append(s) else: c_set.append(s) solution_set = FiniteSet(*f_set) + ConditionSet(symbol, Eq(f, 0), FiniteSet(*c_set)) return solution_set def _solve_abs(f, symbol, domain): """ Helper function to solve equation involving absolute value function """ if not domain.is_subset(S.Reals): raise ValueError(filldedent(''' Absolute values cannot be inverted in the complex domain.''')) p, q, r = Wild('p'), Wild('q'), Wild('r') pattern_match = f.match(p*Abs(q) + r) or {} if not pattern_match.get(p, S.Zero).is_zero: f_p, f_q, f_r = pattern_match[p], pattern_match[q], pattern_match[r] domain = continuous_domain(f_q, symbol, domain) q_pos_cond = solve_univariate_inequality(f_q >= 0, symbol, relational=False, domain=domain, continuous=True) q_neg_cond = q_pos_cond.complement(domain) sols_q_pos = solveset_real(f_p*f_q + f_r, symbol).intersect(q_pos_cond) sols_q_neg = solveset_real(f_p*(-f_q) + f_r, symbol).intersect(q_neg_cond) return Union(sols_q_pos, sols_q_neg) else: return ConditionSet(symbol, Eq(f, 0), domain) def solve_decomposition(f, symbol, domain): """ Function to solve equations via the principle of "Decomposition and Rewriting". Examples ======== >>> from sympy import exp, sin, Symbol, pprint, S >>> from sympy.solvers.solveset import solve_decomposition as sd >>> x = Symbol('x') >>> f1 = exp(2*x) - 3*exp(x) + 2 >>> sd(f1, x, S.Reals) {0, log(2)} >>> f2 = sin(x)**2 + 2*sin(x) + 1 >>> pprint(sd(f2, x, S.Reals), use_unicode=False) 3*pi {2*n*pi + ---- | n in S.Integers} 2 >>> f3 = sin(x + 2) >>> pprint(sd(f3, x, S.Reals), use_unicode=False) {2*n*pi - 2 | n in S.Integers} U {pi*(2*n + 1) - 2 | n in S.Integers} """ from sympy.solvers.decompogen import decompogen from sympy.calculus.util import function_range # decompose the given function g_s = decompogen(f, symbol) # `y_s` represents the set of values for which the function `g` is to be # solved. # `solutions` represent the solutions of the equations `g = y_s` or # `g = 0` depending on the type of `y_s`. # As we are interested in solving the equation: f = 0 y_s = FiniteSet(0) for g in g_s: frange = function_range(g, symbol, domain) y_s = Intersection(frange, y_s) result = S.EmptySet if isinstance(y_s, FiniteSet): for y in y_s: solutions = solveset(Eq(g, y), symbol, domain) if not isinstance(solutions, ConditionSet): result += solutions else: if isinstance(y_s, ImageSet): iter_iset = (y_s,) elif isinstance(y_s, Union): iter_iset = y_s.args for iset in iter_iset: new_solutions = solveset(Eq(iset.lamda.expr, g), symbol, domain) dummy_var = tuple(iset.lamda.expr.free_symbols)[0] base_set = iset.base_set if isinstance(new_solutions, FiniteSet): new_exprs = new_solutions elif isinstance(new_solutions, Intersection): if isinstance(new_solutions.args[1], FiniteSet): new_exprs = new_solutions.args[1] for new_expr in new_exprs: result += ImageSet(Lambda(dummy_var, new_expr), base_set) if result is S.EmptySet: return ConditionSet(symbol, Eq(f, 0), domain) y_s = result return y_s def _solveset(f, symbol, domain, _check=False): """Helper for solveset to return a result from an expression that has already been sympify'ed and is known to contain the given symbol.""" # _check controls whether the answer is checked or not from sympy.simplify.simplify import signsimp orig_f = f tf = f = together(f) if f.is_Mul: coeff, f = f.as_independent(symbol, as_Add=False) if coeff in set([S.ComplexInfinity, S.NegativeInfinity, S.Infinity]): f = tf if f.is_Add: a, h = f.as_independent(symbol) m, h = h.as_independent(symbol, as_Add=False) if m not in set([S.ComplexInfinity, S.Zero, S.Infinity, S.NegativeInfinity]): f = a/m + h # XXX condition `m != 0` should be added to soln f = piecewise_fold(f) # assign the solvers to use solver = lambda f, x, domain=domain: _solveset(f, x, domain) if domain.is_subset(S.Reals): inverter_func = invert_real else: inverter_func = invert_complex inverter = lambda f, rhs, symbol: inverter_func(f, rhs, symbol, domain) result = EmptySet() if f.expand().is_zero: return domain elif not f.has(symbol): return EmptySet() elif f.is_Mul and all(_is_finite_with_finite_vars(m, domain) for m in f.args): # if f(x) and g(x) are both finite we can say that the solution of # f(x)*g(x) == 0 is same as Union(f(x) == 0, g(x) == 0) is not true in # general. g(x) can grow to infinitely large for the values where # f(x) == 0. To be sure that we are not silently allowing any # wrong solutions we are using this technique only if both f and g are # finite for a finite input. result = Union(*[solver(m, symbol) for m in f.args]) elif _is_function_class_equation(TrigonometricFunction, f, symbol) or \ _is_function_class_equation(HyperbolicFunction, f, symbol): result = _solve_trig(f, symbol, domain) elif f.is_Piecewise: dom = domain result = EmptySet() expr_set_pairs = f.as_expr_set_pairs() for (expr, in_set) in expr_set_pairs: if in_set.is_Relational: in_set = in_set.as_set() if in_set.is_Interval: dom -= in_set solns = solver(expr, symbol, in_set) result += solns else: lhs, rhs_s = inverter(f, 0, symbol) if lhs == symbol: # do some very minimal simplification since # repeated inversion may have left the result # in a state that other solvers (e.g. poly) # would have simplified; this is done here # rather than in the inverter since here it # is only done once whereas there it would # be repeated for each step of the inversion if isinstance(rhs_s, FiniteSet): rhs_s = FiniteSet(*[Mul(* signsimp(i).as_content_primitive()) for i in rhs_s]) result = rhs_s elif isinstance(rhs_s, FiniteSet): for equation in [lhs - rhs for rhs in rhs_s]: if equation == f: if any(_has_rational_power(g, symbol)[0] for g in equation.args) or _has_rational_power( equation, symbol)[0]: result += _solve_radical(equation, symbol, solver) elif equation.has(Abs): result += _solve_abs(f, symbol, domain) else: result += _solve_as_rational(equation, symbol, domain) else: result += solver(equation, symbol) elif rhs_s is not S.EmptySet: result = ConditionSet(symbol, Eq(f, 0), domain) if isinstance(result, ConditionSet): num, den = f.as_numer_denom() if den.has(symbol): _result = _solveset(num, symbol, domain) if not isinstance(_result, ConditionSet): singularities = _solveset(den, symbol, domain) result = _result - singularities if _check: if isinstance(result, ConditionSet): # it wasn't solved or has enumerated all conditions # -- leave it alone return result # whittle away all but the symbol-containing core # to use this for testing fx = orig_f.as_independent(symbol, as_Add=True)[1] fx = fx.as_independent(symbol, as_Add=False)[1] if isinstance(result, FiniteSet): # check the result for invalid solutions result = FiniteSet(*[s for s in result if isinstance(s, RootOf) or domain_check(fx, symbol, s)]) return result
[docs]def solveset(f, symbol=None, domain=S.Complexes): r"""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. 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). See Also ======== solveset_real: solver for real domain solveset_complex: solver for complex domain 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) """ f = sympify(f) if f is S.true: return domain if f is S.false: return S.EmptySet if not isinstance(f, (Expr, Number)): raise ValueError("%s is not a valid SymPy expression" % (f)) free_symbols = f.free_symbols if not free_symbols: b = Eq(f, 0) if b is S.true: return domain elif b is S.false: return S.EmptySet else: raise NotImplementedError(filldedent(''' relationship between value and 0 is unknown: %s''' % b)) if symbol is None: if len(free_symbols) == 1: symbol = free_symbols.pop() else: raise ValueError(filldedent(''' The independent variable must be specified for a multivariate equation.''')) elif not getattr(symbol, 'is_Symbol', False): raise ValueError('A Symbol must be given, not type %s: %s' % (type(symbol), symbol)) if isinstance(f, Eq): from sympy.core import Add f = Add(f.lhs, - f.rhs, evaluate=False) elif f.is_Relational: if not domain.is_subset(S.Reals): raise NotImplementedError(filldedent(''' Inequalities in the complex domain are not supported. Try the real domain by setting domain=S.Reals''')) try: result = solve_univariate_inequality( f, symbol, domain=domain, relational=False) except NotImplementedError: result = ConditionSet(symbol, f, domain) return result return _solveset(f, symbol, domain, _check=True)
[docs]def solveset_real(f, symbol): return solveset(f, symbol, S.Reals)
[docs]def solveset_complex(f, symbol): return solveset(f, symbol, S.Complexes)
def solvify(f, symbol, domain): """Solves an equation using solveset and returns the solution in accordance with the `solve` output API. Returns ======= We classify the output based on the type of solution returned by `solveset`. Solution | Output ---------------------------------------- FiniteSet | list ImageSet, | list (if `f` is periodic) Union | EmptySet | empty list Others | None Raises ====== NotImplementedError A ConditionSet is the input. Examples ======== >>> from sympy.solvers.solveset import solvify, solveset >>> from sympy.abc import x >>> from sympy import S, tan, sin, exp >>> solvify(x**2 - 9, x, S.Reals) [-3, 3] >>> solvify(sin(x) - 1, x, S.Reals) [pi/2] >>> solvify(tan(x), x, S.Reals) [0] >>> solvify(exp(x) - 1, x, S.Complexes) >>> solvify(exp(x) - 1, x, S.Reals) [0] """ solution_set = solveset(f, symbol, domain) result = None if solution_set is S.EmptySet: result = [] elif isinstance(solution_set, ConditionSet): raise NotImplementedError('solveset is unable to solve this equation.') elif isinstance(solution_set, FiniteSet): result = list(solution_set) else: period = periodicity(f, symbol) if period is not None: solutions = S.EmptySet if isinstance(solution_set, ImageSet): iter_solutions = (solution_set,) elif isinstance(solution_set, Union): if all(isinstance(i, ImageSet) for i in solution_set.args): iter_solutions = solution_set.args for solution in iter_solutions: solutions += solution.intersect(Interval(0, period, False, True)) if isinstance(solutions, FiniteSet): result = list(solutions) else: solution = solution_set.intersect(domain) if isinstance(solution, FiniteSet): result += solution return result ############################################################################### ################################ LINSOLVE ##################################### ###############################################################################
[docs]def linear_eq_to_matrix(equations, *symbols): r""" 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: .. math:: 4x + 2y + 3z = 1 .. math:: 3x + y + z = -6 .. math:: 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]]) """ if not symbols: raise ValueError('Symbols must be given, for which coefficients \ are to be found.') if hasattr(symbols[0], '__iter__'): symbols = symbols[0] M = Matrix([symbols]) # initialise Matrix with symbols + 1 columns M = M.col_insert(len(symbols), Matrix([1])) row_no = 1 for equation in equations: f = sympify(equation) if isinstance(f, Equality): f = f.lhs - f.rhs # Extract coeff of symbols coeff_list = [] for symbol in symbols: coeff_list.append(f.coeff(symbol)) # append constant term (term free from symbols) coeff_list.append(-f.as_coeff_add(*symbols)[0]) # insert equations coeff's into rows M = M.row_insert(row_no, Matrix([coeff_list])) row_no += 1 # delete the initialised (Ist) trivial row M.row_del(0) A, b = M[:, :-1], M[:, -1:] return A, b
[docs]def linsolve(system, *symbols): r""" 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: .. math:: 3x + 2y - z = 1 .. math:: 2x - 2y + 4z = -2 .. math:: 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() """ if not system: return S.EmptySet if not symbols: raise ValueError('Symbols must be given, for which solution of the ' 'system is to be found.') if hasattr(symbols[0], '__iter__'): symbols = symbols[0] try: sym = symbols[0].is_Symbol or symbols[0].is_Function except AttributeError: sym = False if not sym: raise ValueError('Symbols or iterable of symbols must be given as ' 'second argument, not type %s: %s' % (type(symbols[0]), symbols[0])) # 1). Augmented Matrix input Form if isinstance(system, Matrix): A, b = system[:, :-1], system[:, -1:] elif hasattr(system, '__iter__'): # 2). A & b as input Form if len(system) == 2 and system[0].is_Matrix: A, b = system[0], system[1] # 3). List of equations Form if not system[0].is_Matrix: A, b = linear_eq_to_matrix(system, symbols) else: raise ValueError("Invalid arguments") # Solve using Gauss-Jordan elimination try: sol, params, free_syms = A.gauss_jordan_solve(b, freevar=True) except ValueError: # No solution return EmptySet() # Replace free parameters with free symbols solution = [] if params: for s in sol: for k, v in enumerate(params): s = s.xreplace({v: symbols[free_syms[k]]}) solution.append(simplify(s)) else: for s in sol: solution.append(simplify(s)) # Return solutions solution = FiniteSet(tuple(solution)) return solution
############################################################################## # ------------------------------nonlinsolve ---------------------------------# ############################################################################## def _return_conditionset(eqs, symbols): # return conditionset condition_set = ConditionSet( FiniteSet(*symbols), FiniteSet(*eqs), S.Complexes) return condition_set def substitution(system, symbols, result=[{}], known_symbols=[], exclude=[], all_symbols=None): r""" Solves the `system` using substitution method. It is used in `nonlinsolve`. This will be called from `nonlinsolve` when any equation(s) is non polynomial equation. Parameters ========== system : list of equations The target system of equations symbols : list of symbols to be solved. The variable(s) for which the system is solved known_symbols : list of solved symbols Values are known for these variable(s) result : An empty list or list of dict If No symbol values is known then empty list otherwise symbol as keys and corresponding value in dict. exclude : Set of expression. Mostly denominator expression(s) of the equations of the system. Final solution should not satisfy these expressions. all_symbols : known_symbols + symbols(unsolved). Returns ======= A FiniteSet of ordered tuple of values of `all_symbols` for which the `system` has solution. Order of values in the tuple is same as symbols present in the parameter `all_symbols`. If parameter `all_symbols` is None then 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. 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 >>> x, y = symbols('x, y', real=True) >>> from sympy.solvers.solveset import substitution >>> substitution([x + y], [x], [{y: 1}], [y], set([]), [x, y]) {(-1, 1)} * when you want soln should not satisfy eq `x + 1 = 0` >>> substitution([x + y], [x], [{y: 1}], [y], set([x + 1]), [y, x]) EmptySet() >>> substitution([x + y], [x], [{y: 1}], [y], set([x - 1]), [y, x]) {(1, -1)} >>> substitution([x + y - 1, y - x**2 + 5], [x, y]) {(-3, 4), (2, -1)} * Returns both real and complex solution >>> x, y, z = symbols('x, y, z') >>> from sympy import exp, sin >>> substitution([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)} >>> eqs = [z**2 + exp(2*x) - sin(y), -3 + exp(-y)] >>> substitution(eqs, [y, z]) {(-log(3), -sqrt(-exp(2*x) - sin(log(3)))), (-log(3), sqrt(-exp(2*x) - sin(log(3)))), (ImageSet(Lambda(_n, 2*_n*I*pi + Mod(-log(3), 2*I*pi)), S.Integers), ImageSet(Lambda(_n, -sqrt(-exp(2*x) + sin(2*_n*I*pi + Mod(-log(3), 2*I*pi)))), S.Integers)), (ImageSet(Lambda(_n, 2*_n*I*pi + Mod(-log(3), 2*I*pi)), S.Integers), ImageSet(Lambda(_n, sqrt(-exp(2*x) + sin(2*_n*I*pi + Mod(-log(3), 2*I*pi)))), S.Integers))} """ from sympy import Complement from sympy.core.compatibility import is_sequence if not system: return S.EmptySet if not symbols: msg = ('Symbols must be given, for which solution of the ' 'system is to be found.') raise ValueError(filldedent(msg)) if not is_sequence(symbols): msg = ('symbols should be given as a sequence, e.g. a list.' 'Not type %s: %s') raise TypeError(filldedent(msg % (type(symbols), symbols))) try: sym = symbols[0].is_Symbol except AttributeError: sym = False if not sym: msg = ('Iterable of symbols must be given as ' 'second argument, not type %s: %s') raise ValueError(filldedent(msg % (type(symbols[0]), symbols[0]))) # By default `all_symbols` will be same as `symbols` if all_symbols is None: all_symbols = symbols old_result = result # storing complements and intersection for particular symbol complements = {} intersections = {} # when total_solveset_call is equals to total_conditionset # means solvest fail to solve all the eq. total_conditionset = -1 total_solveset_call = -1 def _unsolved_syms(eq, sort=False): """Returns the unsolved symbol present in the equation `eq`. """ free = eq.free_symbols unsolved = (free - set(known_symbols)) & set(all_symbols) if sort: unsolved = list(unsolved) unsolved.sort(key=default_sort_key) return unsolved # end of _unsolved_syms() # sort such that equation with the fewest potential symbols is first. # means eq with less number of variable first in the list. eqs_in_better_order = list( ordered(system, lambda _: len(_unsolved_syms(_)))) def add_intersection_complement(result, sym_set, **flags): # If solveset have returned some intersection/complement # for any symbol. It will be added in final solution. final_result = [] for res in result: res_copy = res for key_res, value_res in res.items(): # Intersection/complement is in Interval or Set. intersection_true = flags.get('Intersection', True) complements_true = flags.get('Complement', True) for key_sym, value_sym in sym_set.items(): if key_sym == key_res: if intersection_true: # testcase is not added for this line(intersection) new_value = \ Intersection(FiniteSet(value_res), value_sym) if new_value is not S.EmptySet: res_copy[key_res] = new_value if complements_true: new_value = \ Complement(FiniteSet(value_res), value_sym) if new_value is not S.EmptySet: res_copy[key_res] = new_value final_result.append(res_copy) return final_result # end of def add_intersection_complement() def _extract_main_soln(sol, soln_imageset): """separate the Complements, Intersections, ImageSet lambda expr and it's base_set. """ # if there is union, then need to check # Complement, Intersection, Imageset. # Order should not be changed. if isinstance(sol, Complement): # extract solution and complement complements[sym] = sol.args[1] sol = sol.args[0] # complement will be added at the end # using `add_intersection_complement` method if isinstance(sol, Intersection): # Interval/Set will be at 0th index always if sol.args[0] != Interval(-oo, oo): # sometimes solveset returns soln # with intersection `S.Reals`, to confirm that # soln is in `domain=S.Reals` or not. We don't consider # that intersecton. intersections[sym] = sol.args[0] sol = sol.args[1] # after intersection and complement Imageset should # be checked. if isinstance(sol, ImageSet): soln_imagest = sol expr2 = sol.lamda.expr sol = FiniteSet(expr2) soln_imageset[expr2] = soln_imagest # if there is union of Imageset or other in soln. # no testcase is written for this if block if isinstance(sol, Union): sol_args = sol.args sol = S.EmptySet # We need in sequence so append finteset elements # and then imageset or other. for sol_arg2 in sol_args: if isinstance(sol_arg2, FiniteSet): sol += sol_arg2 else: # ImageSet, Intersection, complement then # append them directly sol += FiniteSet(sol_arg2) if not isinstance(sol, FiniteSet): sol = FiniteSet(sol) return sol, soln_imageset # end of def _extract_main_soln() # helper function for _append_new_soln def _check_exclude(rnew, imgset_yes): rnew_ = rnew if imgset_yes: # replace all dummy variables (Imageset lambda variables) # with zero before `checksol`. Considering fundamental soln # for `checksol`. rnew_copy = rnew.copy() dummy_n = imgset_yes[0] for key_res, value_res in rnew_copy.items(): rnew_copy[key_res] = value_res.subs(dummy_n, 0) rnew_ = rnew_copy # satisfy_exclude == true if it satisfies the expr of `exclude` list. try: # something like : `Mod(-log(3), 2*I*pi)` can't be # simplified right now, so `checksol` returns `TypeError`. # when this issue is fixed this try block should be # removed. Mod(-log(3), 2*I*pi) == -log(3) satisfy_exclude = any( checksol(d, rnew_) for d in exclude) except TypeError: satisfy_exclude = None return satisfy_exclude # end of def _check_exclude() # helper function for _append_new_soln def _restore_imgset(rnew, original_imageset, newresult): restore_sym = set(rnew.keys()) & \ set(original_imageset.keys()) for key_sym in restore_sym: img = original_imageset[key_sym] rnew[key_sym] = img if rnew not in newresult: newresult.append(rnew) # end of def _restore_imgset() def _append_eq(eq, result, res, delete_soln, n=None): u = Dummy('u') if n: eq = eq.subs(n, 0) satisfy = checksol(u, u, eq, minimal=True) if satisfy is False: delete_soln = True res = {} else: result.append(res) return result, res, delete_soln def _append_new_soln(rnew, sym, sol, imgset_yes, soln_imageset, original_imageset, newresult, eq=None): """If `rnew` (A dict <symbol: soln>) contains valid soln append it to `newresult` list. `imgset_yes` is (base, dummy_var) if there was imageset in previously calculated result(otherwise empty tuple). `original_imageset` is dict of imageset expr and imageset from this result. `soln_imageset` dict of imageset expr and imageset of new soln. """ satisfy_exclude = _check_exclude(rnew, imgset_yes) delete_soln = False # soln should not satisfy expr present in `exclude` list. if not satisfy_exclude: local_n = None # if it is imageset if imgset_yes: local_n = imgset_yes[0] base = imgset_yes[1] if sym and sol: # when `sym` and `sol` is `None` means no new # soln. In that case we will append rnew directly after # substituting original imagesets in rnew values if present # (second last line of this function using _restore_imgset) dummy_list = list(sol.atoms(Dummy)) # use one dummy `n` which is in # previous imageset local_n_list = [ local_n for i in range( 0, len(dummy_list))] dummy_zip = zip(dummy_list, local_n_list) lam = Lambda(local_n, sol.subs(dummy_zip)) rnew[sym] = ImageSet(lam, base) if eq is not None: newresult, rnew, delete_soln = _append_eq( eq, newresult, rnew, delete_soln, local_n) elif eq is not None: newresult, rnew, delete_soln = _append_eq( eq, newresult, rnew, delete_soln) elif soln_imageset: rnew[sym] = soln_imageset[sol] # restore original imageset _restore_imgset(rnew, original_imageset, newresult) else: newresult.append(rnew) elif satisfy_exclude: delete_soln = True rnew = {} _restore_imgset(rnew, original_imageset, newresult) return newresult, delete_soln # end of def _append_new_soln() def _new_order_result(result, eq): # separate first, second priority. `res` that makes `eq` value equals # to zero, should be used first then other result(second priority). # If it is not done then we may miss some soln. first_priority = [] second_priority = [] for res in result: if not any(isinstance(val, ImageSet) for val in res.values()): if eq.subs(res) == 0: first_priority.append(res) else: second_priority.append(res) if first_priority or second_priority: return first_priority + second_priority return result def _solve_using_known_values(result, solver): """Solves the system using already known solution (result contains the dict <symbol: value>). solver is `solveset_complex` or `solveset_real`. """ # stores imageset <expr: imageset(Lambda(n, expr), base)>. soln_imageset = {} total_solvest_call = 0 total_conditionst = 0 # sort such that equation with the fewest potential symbols is first. # means eq with less variable first for index, eq in enumerate(eqs_in_better_order): newresult = [] original_imageset = {} # if imageset expr is used to solve other symbol imgset_yes = False result = _new_order_result(result, eq) for res in result: got_symbol = set() # symbols solved in one iteration if soln_imageset: # find the imageset and use its expr. for key_res, value_res in res.items(): if isinstance(value_res, ImageSet): res[key_res] = value_res.lamda.expr original_imageset[key_res] = value_res dummy_n = value_res.lamda.expr.atoms(Dummy).pop() base = value_res.base_set imgset_yes = (dummy_n, base) # update eq with everything that is known so far eq2 = eq.subs(res) unsolved_syms = _unsolved_syms(eq2, sort=True) if not unsolved_syms: if res: newresult, delete_res = _append_new_soln( res, None, None, imgset_yes, soln_imageset, original_imageset, newresult, eq2) if delete_res: # `delete_res` is true, means substituting `res` in # eq2 doesn't return `zero` or deleting the `res` # (a soln) since it staisfies expr of `exclude` # list. result.remove(res) continue # skip as it's independent of desired symbols depen = eq2.as_independent(unsolved_syms)[0] if depen.has(Abs) and solver == solveset_complex: # Absolute values cannot be inverted in the # complex domain continue soln_imageset = {} for sym in unsolved_syms: not_solvable = False try: soln = solver(eq2, sym) total_solvest_call += 1 soln_new = S.EmptySet if isinstance(soln, Complement): # separate solution and complement complements[sym] = soln.args[1] soln = soln.args[0] # complement will be added at the end if isinstance(soln, Intersection): # Interval will be at 0th index always if soln.args[0] != Interval(-oo, oo): # sometimes solveset returns soln # with intersection S.Reals, to confirm that # soln is in domain=S.Reals intersections[sym] = soln.args[0] soln_new += soln.args[1] soln = soln_new if soln_new else soln if index > 0 and solver == solveset_real: # one symbol's real soln , another symbol may have # corresponding complex soln. if not isinstance(soln, (ImageSet, ConditionSet)): soln += solveset_complex(eq2, sym) except NotImplementedError: # If sovleset is not able to solve equation `eq2`. Next # time we may get soln using next equation `eq2` continue if isinstance(soln, ConditionSet): soln = S.EmptySet # don't do `continue` we may get soln # in terms of other symbol(s) not_solvable = True total_conditionst += 1 if soln is not S.EmptySet: soln, soln_imageset = _extract_main_soln( soln, soln_imageset) for sol in soln: # sol is not a `Union` since we checked it # before this loop sol, soln_imageset = _extract_main_soln( sol, soln_imageset) sol = set(sol).pop() free = sol.free_symbols if got_symbol and any([ ss in free for ss in got_symbol ]): # sol depends on previously solved symbols # then continue continue rnew = res.copy() # put each solution in res and append the new result # in the new result list (solution for symbol `s`) # along with old results. for k, v in res.items(): if isinstance(v, Expr): # if any unsolved symbol is present # Then subs known value rnew[k] = v.subs(sym, sol) # and add this new solution if soln_imageset: # replace all lambda variables with 0. imgst = soln_imageset[sol] rnew[sym] = imgst.lamda( *[0 for i in range(0, len( imgst.lamda.variables))]) else: rnew[sym] = sol newresult, delete_res = _append_new_soln( rnew, sym, sol, imgset_yes, soln_imageset, original_imageset, newresult) if delete_res: # deleting the `res` (a soln) since it staisfies # eq of `exclude` list result.remove(res) # solution got for sym if not not_solvable: got_symbol.add(sym) # next time use this new soln if newresult: result = newresult return result, total_solvest_call, total_conditionst # end def _solve_using_know_values() new_result_real, solve_call1, cnd_call1 = _solve_using_known_values( old_result, solveset_real) new_result_complex, solve_call2, cnd_call2 = _solve_using_known_values( old_result, solveset_complex) # when `total_solveset_call` is equals to `total_conditionset` # means solvest fails to solve all the eq. # return conditionset in this case total_conditionset += (cnd_call1 + cnd_call2) total_solveset_call += (solve_call1 + solve_call2) if total_conditionset == total_solveset_call and total_solveset_call != -1: return _return_conditionset(eqs_in_better_order, all_symbols) # overall result result = new_result_real + new_result_complex result_all_variables = [] result_infinite = [] for res in result: if not res: # means {None : None} continue # If length < len(all_symbols) means infinite soln. # Some or all the soln is dependent on 1 symbol. # eg. {x: y+2} then final soln {x: y+2, y: y} if len(res) < len(all_symbols): solved_symbols = res.keys() unsolved = list(filter( lambda x: x not in solved_symbols, all_symbols)) for unsolved_sym in unsolved: res[unsolved_sym] = unsolved_sym result_infinite.append(res) if res not in result_all_variables: result_all_variables.append(res) if result_infinite: # we have general soln # eg : [{x: -1, y : 1}, {x : -y , y: y}] then # return [{x : -y, y : y}] result_all_variables = result_infinite if intersections and complements: # no testcase is added for this block result_all_variables = add_intersection_complement( result_all_variables, intersections, Intersection=True, Complement=True) elif intersections: result_all_variables = add_intersection_complement( result_all_variables, intersections, Intersection=True) elif complements: result_all_variables = add_intersection_complement( result_all_variables, complements, Complement=True) # convert to ordered tuple result = S.EmptySet for r in result_all_variables: temp = [r[symb] for symb in all_symbols] result += FiniteSet(tuple(temp)) return result # end of def substitution() def _solveset_work(system, symbols): soln = solveset(system[0], symbols[0]) if isinstance(soln, FiniteSet): _soln = FiniteSet(*[tuple((s,)) for s in soln]) return _soln else: return FiniteSet(tuple(FiniteSet(soln))) def _handle_positive_dimensional(polys, symbols, denominators): from sympy.polys.polytools import groebner # substitution method where new system is groebner basis of the system _symbols = list(symbols) _symbols.sort(key=default_sort_key) basis = groebner(polys, _symbols, polys=True) new_system = [] for poly_eq in basis: new_system.append(poly_eq.as_expr()) result = [{}] result = substitution( new_system, symbols, result, [], denominators) return result # end of def _handle_positive_dimensional() def _handle_zero_dimensional(polys, symbols, system): # solve 0 dimensional poly system using `solve_poly_system` result = solve_poly_system(polys, *symbols) # May be some extra soln is added because # we used `unrad` in `_separate_poly_nonpoly`, so # need to check and remove if it is not a soln. result_update = S.EmptySet for res in result: dict_sym_value = dict(list(zip(symbols, res))) if all(checksol(eq, dict_sym_value) for eq in system): result_update += FiniteSet(res) return result_update # end of def _handle_zero_dimensional() def _separate_poly_nonpoly(system, symbols): polys = [] polys_expr = [] nonpolys = [] denominators = set() poly = None for eq in system: # Store denom expression if it contains symbol denominators.update(_simple_dens(eq, symbols)) # try to remove sqrt and rational power without_radicals = unrad(simplify(eq)) if without_radicals: eq_unrad, cov = without_radicals if not cov: eq = eq_unrad if isinstance(eq, Expr): eq = eq.as_numer_denom()[0] poly = eq.as_poly(*symbols, extension=True) elif simplify(eq).is_number: continue if poly is not None: polys.append(poly) polys_expr.append(poly.as_expr()) else: nonpolys.append(eq) return polys, polys_expr, nonpolys, denominators # end of def _separate_poly_nonpoly()
[docs]def nonlinsolve(system, *symbols): r""" 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: .. math:: x*y - 1 = 0 .. math:: 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. """ from sympy.polys.polytools import is_zero_dimensional if not system: return S.EmptySet if not symbols: msg = ('Symbols must be given, for which solution of the ' 'system is to be found.') raise ValueError(filldedent(msg)) if hasattr(symbols[0], '__iter__'): symbols = symbols[0] try: sym = symbols[0].is_Symbol except AttributeError: sym = False except IndexError: msg = ('Symbols must be given, for which solution of the ' 'system is to be found.') raise IndexError(filldedent(msg)) if not sym: msg = ('Symbols or iterable of symbols must be given as ' 'second argument, not type %s: %s') raise ValueError(filldedent(msg % (type(symbols[0]), symbols[0]))) if len(system) == 1 and len(symbols) == 1: return _solveset_work(system, symbols) # main code of def nonlinsolve() starts from here polys, polys_expr, nonpolys, denominators = _separate_poly_nonpoly( system, symbols) if len(symbols) == len(polys): # If all the equations in the system is poly if is_zero_dimensional(polys, symbols): # finite number of soln (Zero dimensional system) try: return _handle_zero_dimensional(polys, symbols, system) except NotImplementedError: # Right now it doesn't fail for any polynomial system of # equation. If `solve_poly_system` fails then `substitution` # method will handle it. result = substitution( polys_expr, symbols, exclude=denominators) return result # positive dimensional system return _handle_positive_dimensional(polys, symbols, denominators) else: # If alll the equations are not polynomial. # Use `substitution` method for the system result = substitution( polys_expr + nonpolys, symbols, exclude=denominators) return result