def test_linsolve():
    x, y, z, u, v, w = symbols("x, y, z, u, v, w")
    x1, x2, x3, x4 = symbols('x1, x2, x3, x4')

    # Test for different input forms

    M = Matrix([[1, 2, 1, 1, 7], [1, 2, 2, -1, 12], [2, 4, 0, 6, 4]])
    system1 = A, b = M[:, :-1], M[:, -1]
    Eqns = [x1 + 2*x2 + x3 + x4 - 7, x1 + 2*x2 + 2*x3 - x4 - 12,
            2*x1 + 4*x2 + 6*x4 - 4]

    sol = FiniteSet((-2*x2 - 3*x4 + 2, x2, 2*x4 + 5, x4))
    assert linsolve(M, (x1, x2, x3, x4)) == sol
    assert linsolve(Eqns, (x1, x2, x3, x4)) == sol
    assert linsolve(system1, (x1, x2, x3, x4)) == sol

    # raise ValueError if no symbols are given
    raises(ValueError, lambda: linsolve(system1))

    # raise ValueError if, A & b is not given as tuple
    raises(ValueError, lambda: linsolve(A, b, x1, x2, x3, x4))

    # raise ValueError for garbage value
    raises(ValueError, lambda: linsolve(Eqns[0], x1, x2, x3, x4))

    # Fully symbolic test
    a, b, c, d, e, f = symbols('a, b, c, d, e, f')
    A = Matrix([[a, b], [c, d]])
    B = Matrix([[e], [f]])
    system2 = (A, B)
    sol = FiniteSet((-b*(f - c*e/a)/(a*(d - b*c/a)) + e/a,
                    (f - c*e/a)/(d - b*c/a)))
    assert linsolve(system2, [x, y]) == sol

    # Test for Dummy Symbols issue #9667
    x1 = Dummy('x1')
    x2 = Dummy('x2')
    x3 = Dummy('x3')
    x4 = Dummy('x4')

    assert linsolve(system1, x1, x2, x3, x4) == FiniteSet((-2*x2 - 3*x4 + 2, x2, 2*x4 + 5, x4))

    # No solution
    A = Matrix([[1, 2, 3], [2, 4, 6], [3, 6, 9]])
    b = Matrix([0, 0, 1])
    assert linsolve((A, b), (x, y, z)) == EmptySet()

    # Issue #10121 - Assignment of free variables
    a, b, c, d, e = symbols('a, b, c, d, e')
    Augmatrix = Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0]])
    assert linsolve(Augmatrix, a, b, c, d, e) == FiniteSet((a, 0, c, 0, e))

def _transform_explike_DE(DE, g, x, order, syms):
    """Converts DE with free parameters into DE with constant coefficients."""
    from sympy.solvers.solveset import linsolve

    eq = []
    highest_coeff = DE.coeff(Derivative(g(x), x, order))
    for i in range(order):
        coeff = DE.coeff(Derivative(g(x), x, i))
        coeff = (coeff / highest_coeff).expand().collect(x)
        for t in Add.make_args(coeff):
            eq.append(t)
    temp = []
    for e in eq:
        if e.has(x):
            break
        elif e.has(Symbol):
            temp.append(e)
    else:
        eq = temp
    if eq:
        sol = dict(zip(syms, (i for s in linsolve(eq, list(syms)) for i in s)))
        if sol:
            DE = DE.subs(sol)
            DE = DE.factor().as_coeff_mul(Derivative)[1][0]
            DE = DE.collect(Derivative(g(x)))
    return DE

def test_linsolve():
    x, y, z, u, v, w = symbols("x, y, z, u, v, w")
    x1, x2, x3, x4 = symbols("x1, x2, x3, x4")

    # Test for different input forms

    M = Matrix([[1, 2, 1, 1, 7], [1, 2, 2, -1, 12], [2, 4, 0, 6, 4]])
    system = A, b = M[:, :-1], M[:, -1]
    Eqns = [x1 + 2 * x2 + x3 + x4 - 7, x1 + 2 * x2 + 2 * x3 - x4 - 12, 2 * x1 + 4 * x2 + 6 * x4 - 4]

    sol = FiniteSet((-2 * x2 - 3 * x4 + 2, x2, 2 * x4 + 5, x4))
    assert linsolve(M, (x1, x2, x3, x4)) == sol
    assert linsolve(Eqns, (x1, x2, x3, x4)) == sol
    assert linsolve(system, (x1, x2, x3, x4)) == sol

    # raise ValueError if no symbols are given
    raises(ValueError, lambda: linsolve(system))

    # raise ValueError if, A & b is not given as tuple
    raises(ValueError, lambda: linsolve(A, b, x1, x2, x3, x4))

    # raise ValueError for garbage value
    raises(ValueError, lambda: linsolve(Eqns[0], x1, x2, x3, x4))

    # Fully symbolic test
    a, b, c, d, e, f = symbols("a, b, c, d, e, f")
    A = Matrix([[a, b], [c, d]])
    B = Matrix([[e], [f]])
    system = (A, B)
    sol = FiniteSet((-b * (f - c * e / a) / (a * (d - b * c / a)) + e / a, (f - c * e / a) / (d - b * c / a)))
    assert linsolve(system, [x, y]) == sol

 def _contains(self, other):
     from sympy.matrices import Matrix
     from sympy.solvers.solveset import solveset, linsolve
     from sympy.utilities.iterables import iterable, cartes
     L = self.lamda
     if self._is_multivariate():
         if not iterable(L.expr):
             if iterable(other):
                 return S.false
             return other.as_numer_denom() in self.func(
                 Lambda(L.variables, L.expr.as_numer_denom()), self.base_set)
         if len(L.expr) != len(self.lamda.variables):
             raise NotImplementedError(filldedent('''
 Dimensions of input and output of Lambda are different.'''))
         eqs = [expr - val for val, expr in zip(other, L.expr)]
         variables = L.variables
         free = set(variables)
         if all(i.is_number for i in list(Matrix(eqs).jacobian(variables))):
             solns = list(linsolve([e - val for e, val in
             zip(L.expr, other)], variables))
         else:
             syms = [e.free_symbols & free for e in eqs]
             solns = {}
             for i, (e, s, v) in enumerate(zip(eqs, syms, other)):
                 if not s:
                     if e != v:
                         return S.false
                     solns[vars[i]] = [v]
                     continue
                 elif len(s) == 1:
                     sy = s.pop()
                     sol = solveset(e, sy)
                     if sol is S.EmptySet:
                         return S.false
                     elif isinstance(sol, FiniteSet):
                         solns[sy] = list(sol)
                     else:
                         raise NotImplementedError
                 else:
                     raise NotImplementedError
             solns = cartes(*[solns[s] for s in variables])
     else:
         # assume scalar -> scalar mapping
         solnsSet = solveset(L.expr - other, L.variables[0])
         if solnsSet.is_FiniteSet:
             solns = list(solnsSet)
         else:
             raise NotImplementedError(filldedent('''
             Determining whether an ImageSet contains %s has not
             been implemented.''' % func_name(other)))
     for soln in solns:
         try:
             if soln in self.base_set:
                 return S.true
         except TypeError:
             return self.base_set.contains(soln.evalf())
     return S.false

def main():
    equations = get_equations(layout)
    symbols = set()
    for eqn in equations:
        symbols = symbols.union(eqn.atoms(Symbol))
    symbols = list(symbols)
    solved_equations = linsolve(equations, *symbols)
    if not solved_equations:
        raise OverconstrainedError("A solution could not be found.")
    solve = list(solved_equations)[0]
    print(list(zip(symbols, solve)))

    def _contains(self, other):
        from sympy.solvers.solveset import solveset, linsolve
        L = self.lamda
        if self._is_multivariate():
            solns = list(linsolve([expr - val for val, expr in zip(other, L.expr)],
                         L.variables).args[0])
        else:
            solns = list(solveset(L.expr - other, L.variables[0]))

        for soln in solns:
            try:
                if soln in self.base_set:
                    return S.true
            except TypeError:
                return self.base_set.contains(soln.evalf())
        return S.false

def simpleDE(f, x, g, order=4):
    r"""Generates simple DE.

    DE is of the form

    .. math::
        f^k(x) + \sum\limits_{j=0}^{k-1} A_j f^j(x) = 0

    where :math:`A_j` should be rational function in x.

    Generates DE's upto order 4 (default). DE's can also have free parameters.

    By increasing order, higher order DE's can be found.

    Yields a tuple of (DE, order).
    """
    from sympy.solvers.solveset import linsolve

    a = symbols('a:%d' % (order))

    def _makeDE(k):
        eq = f.diff(x, k) + Add(*[a[i]*f.diff(x, i) for i in range(0, k)])
        DE = g(x).diff(x, k) + Add(*[a[i]*g(x).diff(x, i) for i in range(0, k)])
        return eq, DE

    eq, DE = _makeDE(order)

    found = False
    for k in range(1, order + 1):
        eq, DE = _makeDE(k)
        eq = eq.expand()
        terms = eq.as_ordered_terms()
        ind = rational_independent(terms, x)
        if found or len(ind) == k:
            sol = dict(zip(a, (i for s in linsolve(ind, a[:k]) for i in s)))
            if sol:
                found = True
                DE = DE.subs(sol)
            DE = DE.as_numer_denom()[0]
            DE = DE.factor().as_coeff_mul(Derivative)[1][0]
            yield DE.collect(Derivative(g(x))), k

def _transform_DE_RE(DE, g, k, order, syms):
    """Converts DE with free parameters into RE of hypergeometric type."""
    from sympy.solvers.solveset import linsolve

    RE = hyper_re(DE, g, k)

    eq = []
    for i in range(1, order):
        coeff = RE.coeff(g(k + i))
        eq.append(coeff)
    sol = dict(zip(syms, (i for s in linsolve(eq, list(syms)) for i in s)))
    if sol:
        m = Wild('m')
        RE = RE.subs(sol)
        RE = RE.factor().as_numer_denom()[0].collect(g(k + m))
        RE = RE.as_coeff_mul(g)[1][0]
        for i in range(order):  # smallest order should be g(k)
            if RE.coeff(g(k + i)) and i:
                RE = RE.subs(k, k - i)
                break
    return RE

 def _contains(self, other):
     from sympy.solvers.solveset import solveset, linsolve
     L = self.lamda
     if self._is_multivariate():
         solns = list(linsolve([expr - val for val, expr in
         zip(other, L.expr)], L.variables).args[0])
     else:
         solnsSet = solveset(L.expr-other, L.variables[0])
         if solnsSet.is_FiniteSet:
             solns = list(solveset(L.expr - other, L.variables[0]))
         else:
             raise NotImplementedError(filldedent('''
             Determining whether an ImageSet contains %s has not
             been implemented.''' % func_name(other)))
     for soln in solns:
         try:
             if soln in self.base_set:
                 return S.true
         except TypeError:
             return self.base_set.contains(soln.evalf())
     return S.false

def test_issue_9953():
    assert linsolve([ ], x) == S.EmptySet

#.........这里部分代码省略.........

        >>> l1 = Line3D(Point3D(4,19,12), Point3D(5,25,17))
        >>> l2 = Line3D(Point3D(-3, -15, -19), direction_ratio=[2,8,8])
        >>> l1.intersection(l2)
        [Point3D(1, 1, -3)]

        >>> p6, p7 = Point3D(0, 5, 2), Point3D(2, 6, 3)
        >>> s1 = Segment3D(p6, p7)
        >>> l1.intersection(s1)
        []

        """
        if isinstance(o, Point3D):
            if o in self:
                return [o]
            else:
                return []

        elif isinstance(o, LinearEntity3D):
            if self == o:
                return [self]
            elif self.is_parallel(o):
                if isinstance(self, Line3D):
                    if o.p1 in self:
                        return [o]
                    return []
                elif isinstance(self, Ray3D):
                    if isinstance(o, Ray3D):
                        # case 1, rays in the same direction
                        if self.xdirection == o.xdirection and \
                                self.ydirection == o.ydirection and \
                                self.zdirection == o.zdirection:
                            return [self] if (self.source in o) else [o]
                        # case 2, rays in the opposite directions
                        else:
                            if o.source in self:
                                if self.source == o.source:
                                    return [self.source]
                                return [Segment3D(o.source, self.source)]
                            return []
                    elif isinstance(o, Segment3D):
                        if o.p1 in self:
                            if o.p2 in self:
                                return [o]
                            return [Segment3D(o.p1, self.source)]
                        elif o.p2 in self:
                            return [Segment3D(o.p2, self.source)]
                        return []
                elif isinstance(self, Segment3D):
                    if isinstance(o, Segment3D):
                        # A reminder that the points of Segments are ordered
                        # in such a way that the following works. See
                        # Segment3D.__new__ for details on the ordering.
                        if self.p1 not in o:
                            if self.p2 not in o:
                                # Neither of the endpoints are in o so either
                                # o is contained in this segment or it isn't
                                if o in self:
                                    return [o]
                                return []
                            else:
                                # p1 not in o but p2 is. Either there is a
                                # segment as an intersection, or they only
                                # intersect at an endpoint
                                if self.p2 == o.p1:
                                    return [o.p1]
                                return [Segment3D(o.p1, self.p2)]
                        elif self.p2 not in o:
                            # p2 not in o but p1 is. Either there is a
                            # segment as an intersection, or they only
                            # intersect at an endpoint
                            if self.p1 == o.p2:
                                return [o.p2]
                            return [Segment3D(o.p2, self.p1)]

                        # Both points of self in o so the whole segment
                        # is in o
                        return [self]

                else:  # unrecognized LinearEntity
                    raise NotImplementedError

            else:
                # If the lines are not parallel then solve their arbitrary points
                # to obtain the point of intersection
                t = t1, t2 = Dummy(), Dummy()
                a = self.arbitrary_point(t1)
                b = o.arbitrary_point(t2)
                dx = a.x - b.x
                c = linsolve([dx, a.y - b.y], t).args[0]
                d = linsolve([dx, a.z - b.z], t).args[0]
                if len(c.free_symbols) == 1 and len(d.free_symbols) == 1:
                    return []
                e = a.subs(t1, c[0])
                if e in self and e in o:
                    return [e]
                else:
                    return []

        return o.intersection(self)

    coeff = '1'
    Q,rest = str_term.split('*',1)
  else:
    coeff,Q,rest = str_term.split('*',2)
  if Q not in our_Qs:
    our_Qs[Q] = [coeff + "*" + rest]
  else:
    our_Qs[Q].append(coeff + "*" + rest)

for key,value in our_Qs.items():
  print("%s: %s" % (key,value))

variables = symbols(" ".join(list(our_Qs)))                # assumes our_Qs is an ordered dictionary, not a standard one.
print(variables)

result = linsolve(system_of_eqns,variables)
print("result:",result)

Q_values = OrderedDict()
for k,Q in enumerate(our_Qs):
  value = list(result)[0][k]
  print("%s: %s" % (Q,value))
  Q_values[Q] = value

sys.exit(0)


# bah this is ugly!!!
R_det = R.det()
#det_table = {}
det_table = OrderedDict()

def test_linsolve():
    x, y, z, u, v, w = symbols("x, y, z, u, v, w")
    x1, x2, x3, x4 = symbols("x1, x2, x3, x4")

    # Test for different input forms

    M = Matrix([[1, 2, 1, 1, 7], [1, 2, 2, -1, 12], [2, 4, 0, 6, 4]])
    system1 = A, b = M[:, :-1], M[:, -1]
    Eqns = [x1 + 2 * x2 + x3 + x4 - 7, x1 + 2 * x2 + 2 * x3 - x4 - 12, 2 * x1 + 4 * x2 + 6 * x4 - 4]

    sol = FiniteSet((-2 * x2 - 3 * x4 + 2, x2, 2 * x4 + 5, x4))
    assert linsolve(M, (x1, x2, x3, x4)) == sol
    assert linsolve(Eqns, (x1, x2, x3, x4)) == sol
    assert linsolve(system1, (x1, x2, x3, x4)) == sol

    # raise ValueError if no symbols are given
    raises(ValueError, lambda: linsolve(system1))

    # raise ValueError if, A & b is not given as tuple
    raises(ValueError, lambda: linsolve(A, b, x1, x2, x3, x4))

    # raise ValueError for garbage value
    raises(ValueError, lambda: linsolve(Eqns[0], x1, x2, x3, x4))

    # Fully symbolic test
    a, b, c, d, e, f = symbols("a, b, c, d, e, f")
    A = Matrix([[a, b], [c, d]])
    B = Matrix([[e], [f]])
    system2 = (A, B)
    sol = FiniteSet(((-b * f + d * e) / (a * d - b * c), (a * f - c * e) / (a * d - b * c)))
    assert linsolve(system2, [x, y]) == sol

    # Test for Dummy Symbols issue #9667
    x1 = Dummy("x1")
    x2 = Dummy("x2")
    x3 = Dummy("x3")
    x4 = Dummy("x4")

    assert linsolve(system1, x1, x2, x3, x4) == FiniteSet((-2 * x2 - 3 * x4 + 2, x2, 2 * x4 + 5, x4))

    # No solution
    A = Matrix([[1, 2, 3], [2, 4, 6], [3, 6, 9]])
    b = Matrix([0, 0, 1])
    assert linsolve((A, b), (x, y, z)) == EmptySet()

    # Issue #10056
    A, B, J1, J2 = symbols("A B J1 J2")
    Augmatrix = Matrix(
        [[2 * I * J1, 2 * I * J2, -2 / J1], [-2 * I * J2, -2 * I * J1, 2 / J2], [0, 2, 2 * I / (J1 * J2)], [2, 0, 0]]
    )

    assert linsolve(Augmatrix, A, B) == FiniteSet((0, I / (J1 * J2)))

    # Issue #10121 - Assignment of free variables
    a, b, c, d, e = symbols("a, b, c, d, e")
    Augmatrix = Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0]])
    assert linsolve(Augmatrix, a, b, c, d, e) == FiniteSet((a, 0, c, 0, e))

def interpolating_spline(d, x, X, Y):
    """Return spline of degree ``d``, passing through the given ``X``
    and ``Y`` values.

    This function returns a piecewise function such that each part is
    a polynomial of degree not greater than ``d``. The value of ``d``
    must be 1 or greater and the values of ``X`` must be strictly
    increasing.

    Examples
    ========

    >>> from sympy import interpolating_spline
    >>> from sympy.abc import x
    >>> interpolating_spline(1, x, [1, 2, 4, 7], [3, 6, 5, 7])
    Piecewise((3*x, (x >= 1) & (x <= 2)),
            (7 - x/2, (x >= 2) & (x <= 4)),
            (2*x/3 + 7/3, (x >= 4) & (x <= 7)))
    >>> interpolating_spline(3, x, [-2, 0, 1, 3, 4], [4, 2, 1, 1, 3])
    Piecewise((-x**3/36 - x**2/36 - 17*x/18 + 2, (x >= -2) & (x <= 1)),
            (5*x**3/36 - 13*x**2/36 - 11*x/18 + 7/3, (x >= 1) & (x <= 4)))

    See Also
    ========

    bsplines_basis_set, sympy.polys.specialpolys.interpolating_poly
    """
    from sympy import symbols, Number, Dummy, Rational
    from sympy.solvers.solveset import linsolve
    from sympy.matrices.dense import Matrix

    # Input sanitization
    d = sympify(d)
    if not(d.is_Integer and d.is_positive):
        raise ValueError(
            "Spline degree must be a positive integer, not %s." % d)
    if len(X) != len(Y):
        raise ValueError(
            "Number of X and Y coordinates must be the same.")
    if len(X) < d + 1:
        raise ValueError(
            "Degree must be less than the number of control points.")
    if not all(a < b for a, b in zip(X, X[1:])):
        raise ValueError(
            "The x-coordinates must be strictly increasing.")

    # Evaluating knots value
    if d.is_odd:
        j = (d + 1) // 2
        interior_knots = X[j:-j]
    else:
        j = d // 2
        interior_knots = [Rational(a + b, 2) for a, b in
            zip(X[j:-j - 1], X[j + 1:-j])]

    knots = [X[0]] * (d + 1) + list(interior_knots) + [X[-1]] * (d + 1)

    basis = bspline_basis_set(d, knots, x)

    A = [[b.subs(x, v) for b in basis] for v in X]

    coeff = linsolve((Matrix(A), Matrix(Y)), symbols('c0:{}'.format(
        len(X)), cls=Dummy))
    coeff = list(coeff)[0]
    intervals = set([c for b in basis for (e, c) in b.args
        if c != True])

    # Sorting the intervals
    #  ival contains the end-points of each interval
    ival = [e.atoms(Number) for e in intervals]
    ival = [list(sorted(e))[0] for e in ival]
    com = zip(ival, intervals)
    com = sorted(com, key=lambda x: x[0])
    intervals = [y for x, y in com]

    basis_dicts = [dict((c, e) for (e, c) in b.args) for b in basis]
    spline = []
    for i in intervals:
        piece = sum([c*d.get(i, S.Zero) for (c, d) in
            zip(coeff, basis_dicts)], S.Zero)
        spline.append((piece, i))
    return(Piecewise(*spline))