Symbolic Equations and Inequalities

Sage can solve symbolic equations and inequalities. For example, we derive the quadratic formula as follows:

Sage

sage: a,b,c = var('a,b,c')
sage: qe = (a*x^2 + b*x + c == 0)
sage: qe
a*x^2 + b*x + c == 0
sage: print(solve(qe, x))
[
x == -1/2*(b + sqrt(b^2 - 4*a*c))/a,
x == -1/2*(b - sqrt(b^2 - 4*a*c))/a
]

Python

>>> from sage.all import *
>>> a,b,c = var('a,b,c')
>>> qe = (a*x**Integer(2) + b*x + c == Integer(0))
>>> qe
a*x^2 + b*x + c == 0
>>> print(solve(qe, x))
[
x == -1/2*(b + sqrt(b^2 - 4*a*c))/a,
x == -1/2*(b - sqrt(b^2 - 4*a*c))/a
]

The operator, left hand side, and right hand side

Operators:

Sage

sage: eqn = x^3 + 2/3 >= x - pi
sage: eqn.operator()
<built-in function ge>
sage: (x^3 + 2/3 < x - pi).operator()
<built-in function lt>
sage: (x^3 + 2/3 == x - pi).operator()
<built-in function eq>

Python

>>> from sage.all import *
>>> eqn = x**Integer(3) + Integer(2)/Integer(3) >= x - pi
>>> eqn.operator()
<built-in function ge>
>>> (x**Integer(3) + Integer(2)/Integer(3) < x - pi).operator()
<built-in function lt>
>>> (x**Integer(3) + Integer(2)/Integer(3) == x - pi).operator()
<built-in function eq>

Left hand side:

Sage

sage: eqn = x^3 + 2/3 >= x - pi
sage: eqn.lhs()
x^3 + 2/3
sage: eqn.left()
x^3 + 2/3
sage: eqn.left_hand_side()
x^3 + 2/3

Python

>>> from sage.all import *
>>> eqn = x**Integer(3) + Integer(2)/Integer(3) >= x - pi
>>> eqn.lhs()
x^3 + 2/3
>>> eqn.left()
x^3 + 2/3
>>> eqn.left_hand_side()
x^3 + 2/3

Right hand side:

Sage

sage: (x + sqrt(2) >= sqrt(3) + 5/2).right()
sqrt(3) + 5/2
sage: (x + sqrt(2) >= sqrt(3) + 5/2).rhs()
sqrt(3) + 5/2
sage: (x + sqrt(2) >= sqrt(3) + 5/2).right_hand_side()
sqrt(3) + 5/2

Python

>>> from sage.all import *
>>> (x + sqrt(Integer(2)) >= sqrt(Integer(3)) + Integer(5)/Integer(2)).right()
sqrt(3) + 5/2
>>> (x + sqrt(Integer(2)) >= sqrt(Integer(3)) + Integer(5)/Integer(2)).rhs()
sqrt(3) + 5/2
>>> (x + sqrt(Integer(2)) >= sqrt(Integer(3)) + Integer(5)/Integer(2)).right_hand_side()
sqrt(3) + 5/2

Arithmetic

Add two symbolic equations:

Sage

sage: var('a,b')
(a, b)
sage: m = 144 == -10 * a + b
sage: n = 136 == 10 * a + b
sage: m + n
280 == 2*b
sage: int(-144) + m
0 == -10*a + b - 144

Python

>>> from sage.all import *
>>> var('a,b')
(a, b)
>>> m = Integer(144) == -Integer(10) * a + b
>>> n = Integer(136) == Integer(10) * a + b
>>> m + n
280 == 2*b
>>> int(-Integer(144)) + m
0 == -10*a + b - 144

Subtract two symbolic equations:

Sage

sage: var('a,b')
(a, b)
sage: m = 144 == 20 * a + b
sage: n = 136 == 10 * a + b
sage: m - n
8 == 10*a
sage: int(144) - m
0 == -20*a - b + 144

Python

>>> from sage.all import *
>>> var('a,b')
(a, b)
>>> m = Integer(144) == Integer(20) * a + b
>>> n = Integer(136) == Integer(10) * a + b
>>> m - n
8 == 10*a
>>> int(Integer(144)) - m
0 == -20*a - b + 144

Multiply two symbolic equations:

Sage

sage: x = var('x')
sage: m = x == 5*x + 1
sage: n = sin(x) == sin(x+2*pi, hold=True)
sage: m * n
x*sin(x) == (5*x + 1)*sin(2*pi + x)
sage: m = 2*x == 3*x^2 - 5
sage: int(-1) * m
-2*x == -3*x^2 + 5

Python

>>> from sage.all import *
>>> x = var('x')
>>> m = x == Integer(5)*x + Integer(1)
>>> n = sin(x) == sin(x+Integer(2)*pi, hold=True)
>>> m * n
x*sin(x) == (5*x + 1)*sin(2*pi + x)
>>> m = Integer(2)*x == Integer(3)*x**Integer(2) - Integer(5)
>>> int(-Integer(1)) * m
-2*x == -3*x^2 + 5

Divide two symbolic equations:

Sage

sage: x = var('x')
sage: m = x == 5*x + 1
sage: n = sin(x) == sin(x+2*pi, hold=True)
sage: m/n
x/sin(x) == (5*x + 1)/sin(2*pi + x)
sage: m = x != 5*x + 1
sage: n = sin(x) != sin(x+2*pi, hold=True)
sage: m/n
x/sin(x) != (5*x + 1)/sin(2*pi + x)

Python

>>> from sage.all import *
>>> x = var('x')
>>> m = x == Integer(5)*x + Integer(1)
>>> n = sin(x) == sin(x+Integer(2)*pi, hold=True)
>>> m/n
x/sin(x) == (5*x + 1)/sin(2*pi + x)
>>> m = x != Integer(5)*x + Integer(1)
>>> n = sin(x) != sin(x+Integer(2)*pi, hold=True)
>>> m/n
x/sin(x) != (5*x + 1)/sin(2*pi + x)

Substitution

Substitution into relations:

Sage

sage: x, a = var('x, a')
sage: eq = (x^3 + a == sin(x/a)); eq
x^3 + a == sin(x/a)
sage: eq.substitute(x=5*x)
125*x^3 + a == sin(5*x/a)
sage: eq.substitute(a=1)
x^3 + 1 == sin(x)
sage: eq.substitute(a=x)
x^3 + x == sin(1)
sage: eq.substitute(a=x, x=1)
x + 1 == sin(1/x)
sage: eq.substitute({a:x, x:1})
x + 1 == sin(1/x)

Python

>>> from sage.all import *
>>> x, a = var('x, a')
>>> eq = (x**Integer(3) + a == sin(x/a)); eq
x^3 + a == sin(x/a)
>>> eq.substitute(x=Integer(5)*x)
125*x^3 + a == sin(5*x/a)
>>> eq.substitute(a=Integer(1))
x^3 + 1 == sin(x)
>>> eq.substitute(a=x)
x^3 + x == sin(1)
>>> eq.substitute(a=x, x=Integer(1))
x + 1 == sin(1/x)
>>> eq.substitute({a:x, x:Integer(1)})
x + 1 == sin(1/x)

You can even substitute multivariable and matrix expressions:

Sage

sage: x,y = var('x, y')
sage: M = Matrix([[x+1,y],[x^2,y^3]]); M
[x + 1     y]
[  x^2   y^3]
sage: M.substitute({x:0,y:1})
[1 1]
[0 1]

Python

>>> from sage.all import *
>>> x,y = var('x, y')
>>> M = Matrix([[x+Integer(1),y],[x**Integer(2),y**Integer(3)]]); M
[x + 1     y]
[  x^2   y^3]
>>> M.substitute({x:Integer(0),y:Integer(1)})
[1 1]
[0 1]

Solving

We can solve equations:

Sage

sage: x = var('x')
sage: S = solve(x^3 - 1 == 0, x)
sage: S
[x == 1/2*I*sqrt(3) - 1/2, x == -1/2*I*sqrt(3) - 1/2, x == 1]
sage: S[0]
x == 1/2*I*sqrt(3) - 1/2
sage: S[0].right()
1/2*I*sqrt(3) - 1/2
sage: S = solve(x^3 - 1 == 0, x, solution_dict=True)
sage: S
[{x: 1/2*I*sqrt(3) - 1/2}, {x: -1/2*I*sqrt(3) - 1/2}, {x: 1}]
sage: z = 5
sage: solve(z^2 == sqrt(3),z)
Traceback (most recent call last):
...
TypeError: 5 is not a valid variable.

Python

>>> from sage.all import *
>>> x = var('x')
>>> S = solve(x**Integer(3) - Integer(1) == Integer(0), x)
>>> S
[x == 1/2*I*sqrt(3) - 1/2, x == -1/2*I*sqrt(3) - 1/2, x == 1]
>>> S[Integer(0)]
x == 1/2*I*sqrt(3) - 1/2
>>> S[Integer(0)].right()
1/2*I*sqrt(3) - 1/2
>>> S = solve(x**Integer(3) - Integer(1) == Integer(0), x, solution_dict=True)
>>> S
[{x: 1/2*I*sqrt(3) - 1/2}, {x: -1/2*I*sqrt(3) - 1/2}, {x: 1}]
>>> z = Integer(5)
>>> solve(z**Integer(2) == sqrt(Integer(3)),z)
Traceback (most recent call last):
...
TypeError: 5 is not a valid variable.

We can also solve equations involving matrices. The following example defines a multivariable function f(x,y), then solves for where the partial derivatives with respect to x and y are zero. Then it substitutes one of the solutions into the Hessian matrix H for f:

Sage

sage: f(x,y) = x^2*y+y^2+y
sage: solutions = solve(list(f.diff()),[x,y],solution_dict=True)
sage: solutions == [{x: -I, y: 0}, {x: I, y: 0}, {x: 0, y: -1/2}]
True
sage: H = f.diff(2) # Hessian matrix
sage: H.subs(solutions[2])
[(x, y) |--> -1  (x, y) |--> 0]
[ (x, y) |--> 0  (x, y) |--> 2]
sage: H(x,y).subs(solutions[2])
[-1  0]
[ 0  2]

Python

>>> from sage.all import *
>>> __tmp__=var("x,y"); f = symbolic_expression(x**Integer(2)*y+y**Integer(2)+y).function(x,y)
>>> solutions = solve(list(f.diff()),[x,y],solution_dict=True)
>>> solutions == [{x: -I, y: Integer(0)}, {x: I, y: Integer(0)}, {x: Integer(0), y: -Integer(1)/Integer(2)}]
True
>>> H = f.diff(Integer(2)) # Hessian matrix
>>> H.subs(solutions[Integer(2)])
[(x, y) |--> -1  (x, y) |--> 0]
[ (x, y) |--> 0  (x, y) |--> 2]
>>> H(x,y).subs(solutions[Integer(2)])
[-1  0]
[ 0  2]

We illustrate finding multiplicities of solutions:

Sage

sage: f = (x-1)^5*(x^2+1)
sage: solve(f == 0, x)
[x == -I, x == I, x == 1]
sage: solve(f == 0, x, multiplicities=True)
([x == -I, x == I, x == 1], [1, 1, 5])

Python

>>> from sage.all import *
>>> f = (x-Integer(1))**Integer(5)*(x**Integer(2)+Integer(1))
>>> solve(f == Integer(0), x)
[x == -I, x == I, x == 1]
>>> solve(f == Integer(0), x, multiplicities=True)
([x == -I, x == I, x == 1], [1, 1, 5])

We can also solve many inequalities:

Sage

sage: solve(1/(x-1)<=8,x)
[[x < 1], [x >= (9/8)]]

Python

>>> from sage.all import *
>>> solve(Integer(1)/(x-Integer(1))<=Integer(8),x)
[[x < 1], [x >= (9/8)]]

We can numerically find roots of equations:

Sage

sage: (x == sin(x)).find_root(-2,2)
0.0
sage: (x^5 + 3*x + 2 == 0).find_root(-2,2,x)
-0.6328345202421523
sage: (cos(x) == sin(x)).find_root(10,20)
19.634954084936208

Python

>>> from sage.all import *
>>> (x == sin(x)).find_root(-Integer(2),Integer(2))
0.0
>>> (x**Integer(5) + Integer(3)*x + Integer(2) == Integer(0)).find_root(-Integer(2),Integer(2),x)
-0.6328345202421523
>>> (cos(x) == sin(x)).find_root(Integer(10),Integer(20))
19.634954084936208

We illustrate some valid error conditions:

Sage

sage: (cos(x) != sin(x)).find_root(10,20)
Traceback (most recent call last):
...
ValueError: Symbolic equation must be an equality.
sage: (SR(3)==SR(2)).find_root(-1,1)
Traceback (most recent call last):
...
RuntimeError: no zero in the interval, since constant expression is not 0.

Python

>>> from sage.all import *
>>> (cos(x) != sin(x)).find_root(Integer(10),Integer(20))
Traceback (most recent call last):
...
ValueError: Symbolic equation must be an equality.
>>> (SR(Integer(3))==SR(Integer(2))).find_root(-Integer(1),Integer(1))
Traceback (most recent call last):
...
RuntimeError: no zero in the interval, since constant expression is not 0.

There must be at most one variable:

Sage

sage: x, y = var('x,y')
sage: (x == y).find_root(-2,2)
Traceback (most recent call last):
...
NotImplementedError: root finding currently only implemented in 1 dimension.

Python

>>> from sage.all import *
>>> x, y = var('x,y')
>>> (x == y).find_root(-Integer(2),Integer(2))
Traceback (most recent call last):
...
NotImplementedError: root finding currently only implemented in 1 dimension.

Assumptions

Forgetting assumptions:

Sage

sage: var('x,y')
(x, y)
sage: forget() #Clear assumptions
sage: assume(x>0, y < 2)
sage: assumptions()
[x > 0, y < 2]
sage: (y < 2).forget()
sage: assumptions()
[x > 0]
sage: forget()
sage: assumptions()
[]

Python

>>> from sage.all import *
>>> var('x,y')
(x, y)
>>> forget() #Clear assumptions
>>> assume(x>Integer(0), y < Integer(2))
>>> assumptions()
[x > 0, y < 2]
>>> (y < Integer(2)).forget()
>>> assumptions()
[x > 0]
>>> forget()
>>> assumptions()
[]

Miscellaneous

Conversion to Maxima:

Sage

sage: x = var('x')
sage: eq = (x^(3/5) >= pi^2 + e^i)
sage: eq._maxima_init_()
'(_SAGE_VAR_x)^(3/5) >= ((%pi)^(2))+(exp(0+%i*1))'
sage: e1 = x^3 + x == sin(2*x)
sage: z = e1._maxima_()
sage: z.parent() is sage.calculus.calculus.maxima
True
sage: z = e1._maxima_(maxima)
sage: z.parent() is maxima
True
sage: z = maxima(e1)
sage: z.parent() is maxima
True

Python

>>> from sage.all import *
>>> x = var('x')
>>> eq = (x**(Integer(3)/Integer(5)) >= pi**Integer(2) + e**i)
>>> eq._maxima_init_()
'(_SAGE_VAR_x)^(3/5) >= ((%pi)^(2))+(exp(0+%i*1))'
>>> e1 = x**Integer(3) + x == sin(Integer(2)*x)
>>> z = e1._maxima_()
>>> z.parent() is sage.calculus.calculus.maxima
True
>>> z = e1._maxima_(maxima)
>>> z.parent() is maxima
True
>>> z = maxima(e1)
>>> z.parent() is maxima
True

Conversion to Maple:

Sage

sage: x = var('x')
sage: eq = (x == 2)
sage: eq._maple_init_()
'x = 2'

Python

>>> from sage.all import *
>>> x = var('x')
>>> eq = (x == Integer(2))
>>> eq._maple_init_()
'x = 2'

Comparison:

Sage

sage: x = var('x')
sage: (x>0) == (x>0)
True
sage: (x>0) == (x>1)
False
sage: (x>0) != (x>1)
True

Python

>>> from sage.all import *
>>> x = var('x')
>>> (x>Integer(0)) == (x>Integer(0))
True
>>> (x>Integer(0)) == (x>Integer(1))
False
>>> (x>Integer(0)) != (x>Integer(1))
True

Variables appearing in the relation:

Sage

sage: var('x,y,z,w')
(x, y, z, w)
sage: f =  (x+y+w) == (x^2 - y^2 - z^3);   f
w + x + y == -z^3 + x^2 - y^2
sage: f.variables()
(w, x, y, z)

Python

>>> from sage.all import *
>>> var('x,y,z,w')
(x, y, z, w)
>>> f =  (x+y+w) == (x**Integer(2) - y**Integer(2) - z**Integer(3));   f
w + x + y == -z^3 + x^2 - y^2
>>> f.variables()
(w, x, y, z)

LaTeX output:

Sage

sage: latex(x^(3/5) >= pi)
x^{\frac{3}{5}} \geq \pi

Python

>>> from sage.all import *
>>> latex(x**(Integer(3)/Integer(5)) >= pi)
x^{\frac{3}{5}} \geq \pi

When working with the symbolic complex number \(I\), notice that comparisons do not automatically simplify even in trivial situations:

Sage

sage: SR(I)^2 == -1
-1 == -1
sage: SR(I)^2 < 0
-1 < 0
sage: (SR(I)+1)^4 > 0
-4 > 0

Python

>>> from sage.all import *
>>> SR(I)**Integer(2) == -Integer(1)
-1 == -1
>>> SR(I)**Integer(2) < Integer(0)
-1 < 0
>>> (SR(I)+Integer(1))**Integer(4) > Integer(0)
-4 > 0

Nevertheless, if you force the comparison, you get the right answer (Issue #7160):

Sage

sage: bool(SR(I)^2 == -1)
True
sage: bool(SR(I)^2 < 0)
True
sage: bool((SR(I)+1)^4 > 0)
False

Python

>>> from sage.all import *
>>> bool(SR(I)**Integer(2) == -Integer(1))
True
>>> bool(SR(I)**Integer(2) < Integer(0))
True
>>> bool((SR(I)+Integer(1))**Integer(4) > Integer(0))
False

More Examples

Sage

sage: x,y,a = var('x,y,a')
sage: f = x^2 + y^2 == 1
sage: f.solve(x)
[x == -sqrt(-y^2 + 1), x == sqrt(-y^2 + 1)]

Python

>>> from sage.all import *
>>> x,y,a = var('x,y,a')
>>> f = x**Integer(2) + y**Integer(2) == Integer(1)
>>> f.solve(x)
[x == -sqrt(-y^2 + 1), x == sqrt(-y^2 + 1)]

Sage

sage: f = x^5 + a
sage: solve(f==0,x)
[x == 1/4*(-a)^(1/5)*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1), x == -1/4*(-a)^(1/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1), x == -1/4*(-a)^(1/5)*(sqrt(5) + I*sqrt(-2*sqrt(5) + 10) + 1), x == 1/4*(-a)^(1/5)*(sqrt(5) - I*sqrt(2*sqrt(5) + 10) - 1), x == (-a)^(1/5)]

Python

>>> from sage.all import *
>>> f = x**Integer(5) + a
>>> solve(f==Integer(0),x)
[x == 1/4*(-a)^(1/5)*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1), x == -1/4*(-a)^(1/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1), x == -1/4*(-a)^(1/5)*(sqrt(5) + I*sqrt(-2*sqrt(5) + 10) + 1), x == 1/4*(-a)^(1/5)*(sqrt(5) - I*sqrt(2*sqrt(5) + 10) - 1), x == (-a)^(1/5)]

You can also do arithmetic with inequalities, as illustrated below:

Sage

sage: var('x y')
(x, y)
sage: f = x + 3 == y - 2
sage: f
x + 3 == y - 2
sage: g = f - 3; g
x == y - 5
sage: h =  x^3 + sqrt(2) == x*y*sin(x)
sage: h
x^3 + sqrt(2) == x*y*sin(x)
sage: h - sqrt(2)
x^3 == x*y*sin(x) - sqrt(2)
sage: h + f
x^3 + x + sqrt(2) + 3 == x*y*sin(x) + y - 2
sage: f = x + 3 < y - 2
sage: g = 2 < x+10
sage: f - g
x + 1 < -x + y - 12
sage: f + g
x + 5 < x + y + 8
sage: f*(-1)
-x - 3 < -y + 2

Python

>>> from sage.all import *
>>> var('x y')
(x, y)
>>> f = x + Integer(3) == y - Integer(2)
>>> f
x + 3 == y - 2
>>> g = f - Integer(3); g
x == y - 5
>>> h =  x**Integer(3) + sqrt(Integer(2)) == x*y*sin(x)
>>> h
x^3 + sqrt(2) == x*y*sin(x)
>>> h - sqrt(Integer(2))
x^3 == x*y*sin(x) - sqrt(2)
>>> h + f
x^3 + x + sqrt(2) + 3 == x*y*sin(x) + y - 2
>>> f = x + Integer(3) < y - Integer(2)
>>> g = Integer(2) < x+Integer(10)
>>> f - g
x + 1 < -x + y - 12
>>> f + g
x + 5 < x + y + 8
>>> f*(-Integer(1))
-x - 3 < -y + 2

AUTHORS:

• Bobby Moretti: initial version (based on a trick that Robert Bradshaw suggested).

• William Stein: second version

• William Stein (2007-07-16): added arithmetic with symbolic equations

sage.symbolic.relation.solve(f, *args, **kwds)

Algebraically solve an equation or system of equations (over the complex numbers) for given variables. Inequalities and systems of inequalities are also supported.

INPUT:

• f – equation or system of equations (given by a list or tuple)

• *args – variables to solve for

• solution_dict – boolean (default: False); if True or nonzero, return a list of dictionaries containing the solutions. If there are no solutions, return an empty list (rather than a list containing an empty dictionary). Likewise, if there’s only a single solution, return a list containing one dictionary with that solution.

There are a few optional keywords if you are trying to solve a single equation. They may only be used in that context.

• multiplicities – boolean (default: False); if True, return corresponding multiplicities. This keyword is incompatible with to_poly_solve=True and does not make any sense when solving inequalities.

• explicit_solutions – boolean (default: False); require that all roots be explicit rather than implicit. Not used when solving inequalities.

• to_poly_solve – boolean (default: False) or string; use Maxima’s to_poly_solver package to search for more possible solutions, but possibly encounter approximate solutions. This keyword is incompatible with multiplicities=True and is not used when solving inequalities. Setting to_poly_solve to ‘force’ (string) omits Maxima’s solve command (useful when some solutions of trigonometric equations are lost).

• algorithm – string (default: 'maxima'); to use SymPy’s solvers set this to ‘sympy’. Note that SymPy is always used for diophantine equations. Another choice, if it is installed, is ‘giac’.

• domain – string (default: 'complex'); setting this to ‘real’ changes the way SymPy solves single equations; inequalities are always solved in the real domain.

EXAMPLES:

Sage

sage: x, y = var('x, y')
sage: solve([x+y==6, x-y==4], x, y)
[[x == 5, y == 1]]
sage: solve([x^2+y^2 == 1, y^2 == x^3 + x + 1], x, y)
[[x == -1/2*I*sqrt(3) - 1/2, y == -sqrt(-1/2*I*sqrt(3) + 3/2)],
 [x == -1/2*I*sqrt(3) - 1/2, y == sqrt(-1/2*I*sqrt(3) + 3/2)],
 [x == 1/2*I*sqrt(3) - 1/2, y == -sqrt(1/2*I*sqrt(3) + 3/2)],
 [x == 1/2*I*sqrt(3) - 1/2, y == sqrt(1/2*I*sqrt(3) + 3/2)],
 [x == 0, y == -1],
 [x == 0, y == 1]]
sage: solve([sqrt(x) + sqrt(y) == 5, x + y == 10], x, y)
[[x == -5/2*I*sqrt(5) + 5, y == 5/2*I*sqrt(5) + 5], [x == 5/2*I*sqrt(5) + 5, y == -5/2*I*sqrt(5) + 5]]
sage: solutions = solve([x^2+y^2 == 1, y^2 == x^3 + x + 1], x, y, solution_dict=True)
sage: for solution in solutions: print("{} , {}".format(solution[x].n(digits=3), solution[y].n(digits=3)))
-0.500 - 0.866*I , -1.27 + 0.341*I
-0.500 - 0.866*I , 1.27 - 0.341*I
-0.500 + 0.866*I , -1.27 - 0.341*I
-0.500 + 0.866*I , 1.27 + 0.341*I
0.000 , -1.00
0.000 , 1.00

Python

>>> from sage.all import *
>>> x, y = var('x, y')
>>> solve([x+y==Integer(6), x-y==Integer(4)], x, y)
[[x == 5, y == 1]]
>>> solve([x**Integer(2)+y**Integer(2) == Integer(1), y**Integer(2) == x**Integer(3) + x + Integer(1)], x, y)
[[x == -1/2*I*sqrt(3) - 1/2, y == -sqrt(-1/2*I*sqrt(3) + 3/2)],
 [x == -1/2*I*sqrt(3) - 1/2, y == sqrt(-1/2*I*sqrt(3) + 3/2)],
 [x == 1/2*I*sqrt(3) - 1/2, y == -sqrt(1/2*I*sqrt(3) + 3/2)],
 [x == 1/2*I*sqrt(3) - 1/2, y == sqrt(1/2*I*sqrt(3) + 3/2)],
 [x == 0, y == -1],
 [x == 0, y == 1]]
>>> solve([sqrt(x) + sqrt(y) == Integer(5), x + y == Integer(10)], x, y)
[[x == -5/2*I*sqrt(5) + 5, y == 5/2*I*sqrt(5) + 5], [x == 5/2*I*sqrt(5) + 5, y == -5/2*I*sqrt(5) + 5]]
>>> solutions = solve([x**Integer(2)+y**Integer(2) == Integer(1), y**Integer(2) == x**Integer(3) + x + Integer(1)], x, y, solution_dict=True)
>>> for solution in solutions: print("{} , {}".format(solution[x].n(digits=Integer(3)), solution[y].n(digits=Integer(3))))
-0.500 - 0.866*I , -1.27 + 0.341*I
-0.500 - 0.866*I , 1.27 - 0.341*I
-0.500 + 0.866*I , -1.27 - 0.341*I
-0.500 + 0.866*I , 1.27 + 0.341*I
0.000 , -1.00
0.000 , 1.00

Whenever possible, answers will be symbolic, but with systems of equations, at times approximations will be given by Maxima, due to the underlying algorithm:

Sage

sage: sols = solve([x^3==y,y^2==x], [x,y]); sols[-1], sols[0] # abs tol 1e-15
([x == 0, y == 0],
 [x == (0.3090169943749475 + 0.9510565162951535*I),
  y == (-0.8090169943749475 - 0.5877852522924731*I)])
sage: sols[0][0].rhs().pyobject().parent()
Complex Double Field

sage: solve([y^6==y],y)
[y == 1/4*sqrt(5) + 1/4*I*sqrt(2*sqrt(5) + 10) - 1/4,
 y == -1/4*sqrt(5) + 1/4*I*sqrt(-2*sqrt(5) + 10) - 1/4,
 y == -1/4*sqrt(5) - 1/4*I*sqrt(-2*sqrt(5) + 10) - 1/4,
 y == 1/4*sqrt(5) - 1/4*I*sqrt(2*sqrt(5) + 10) - 1/4,
 y == 1,
 y == 0]
sage: solve( [y^6 == y], y)==solve( y^6 == y, y)
True

Python

>>> from sage.all import *
>>> sols = solve([x**Integer(3)==y,y**Integer(2)==x], [x,y]); sols[-Integer(1)], sols[Integer(0)] # abs tol 1e-15
([x == 0, y == 0],
 [x == (0.3090169943749475 + 0.9510565162951535*I),
  y == (-0.8090169943749475 - 0.5877852522924731*I)])
>>> sols[Integer(0)][Integer(0)].rhs().pyobject().parent()
Complex Double Field

>>> solve([y**Integer(6)==y],y)
[y == 1/4*sqrt(5) + 1/4*I*sqrt(2*sqrt(5) + 10) - 1/4,
 y == -1/4*sqrt(5) + 1/4*I*sqrt(-2*sqrt(5) + 10) - 1/4,
 y == -1/4*sqrt(5) - 1/4*I*sqrt(-2*sqrt(5) + 10) - 1/4,
 y == 1/4*sqrt(5) - 1/4*I*sqrt(2*sqrt(5) + 10) - 1/4,
 y == 1,
 y == 0]
>>> solve( [y**Integer(6) == y], y)==solve( y**Integer(6) == y, y)
True

Here we demonstrate very basic use of the optional keywords:

Sage

sage: ((x^2-1)^2).solve(x)
[x == -1, x == 1]
sage: ((x^2-1)^2).solve(x,multiplicities=True)
([x == -1, x == 1], [2, 2])
sage: solve(sin(x)==x,x)
[x == sin(x)]
sage: solve(sin(x)==x,x,explicit_solutions=True)
[]
sage: solve(abs(1-abs(1-x)) == 10, x)
[abs(abs(x - 1) - 1) == 10]
sage: solve(abs(1-abs(1-x)) == 10, x, to_poly_solve=True)
[x == -10, x == 12]

sage: from sage.symbolic.expression import Expression
sage: Expression.solve(x^2==1,x)
[x == -1, x == 1]

Python

>>> from sage.all import *
>>> ((x**Integer(2)-Integer(1))**Integer(2)).solve(x)
[x == -1, x == 1]
>>> ((x**Integer(2)-Integer(1))**Integer(2)).solve(x,multiplicities=True)
([x == -1, x == 1], [2, 2])
>>> solve(sin(x)==x,x)
[x == sin(x)]
>>> solve(sin(x)==x,x,explicit_solutions=True)
[]
>>> solve(abs(Integer(1)-abs(Integer(1)-x)) == Integer(10), x)
[abs(abs(x - 1) - 1) == 10]
>>> solve(abs(Integer(1)-abs(Integer(1)-x)) == Integer(10), x, to_poly_solve=True)
[x == -10, x == 12]

>>> from sage.symbolic.expression import Expression
>>> Expression.solve(x**Integer(2)==Integer(1),x)
[x == -1, x == 1]

We must solve with respect to actual variables:

Sage

sage: z = 5
sage: solve([8*z + y == 3, -z +7*y == 0],y,z)
Traceback (most recent call last):
...
TypeError: 5 is not a valid variable.

Python

>>> from sage.all import *
>>> z = Integer(5)
>>> solve([Integer(8)*z + y == Integer(3), -z +Integer(7)*y == Integer(0)],y,z)
Traceback (most recent call last):
...
TypeError: 5 is not a valid variable.

If we ask for dictionaries containing the solutions, we get them:

Sage

sage: solve([x^2-1],x,solution_dict=True)
[{x: -1}, {x: 1}]
sage: solve([x^2-4*x+4],x,solution_dict=True)
[{x: 2}]
sage: res = solve([x^2 == y, y == 4],x,y,solution_dict=True)
sage: for soln in res: print("x: %s, y: %s" % (soln[x], soln[y]))
x: 2, y: 4
x: -2, y: 4

Python

>>> from sage.all import *
>>> solve([x**Integer(2)-Integer(1)],x,solution_dict=True)
[{x: -1}, {x: 1}]
>>> solve([x**Integer(2)-Integer(4)*x+Integer(4)],x,solution_dict=True)
[{x: 2}]
>>> res = solve([x**Integer(2) == y, y == Integer(4)],x,y,solution_dict=True)
>>> for soln in res: print("x: %s, y: %s" % (soln[x], soln[y]))
x: 2, y: 4
x: -2, y: 4

If there is a parameter in the answer, that will show up as a new variable. In the following example, r1 is an arbitrary constant (because of the r):

Sage

sage: forget()
sage: x, y = var('x,y')
sage: solve([x+y == 3, 2*x+2*y == 6],x,y)
[[x == -r1 + 3, y == r1]]

sage: var('b, c')
(b, c)
sage: solve((b-1)*(c-1), [b,c])
[[b == 1, c == r...], [b == r..., c == 1]]

Python

>>> from sage.all import *
>>> forget()
>>> x, y = var('x,y')
>>> solve([x+y == Integer(3), Integer(2)*x+Integer(2)*y == Integer(6)],x,y)
[[x == -r1 + 3, y == r1]]

>>> var('b, c')
(b, c)
>>> solve((b-Integer(1))*(c-Integer(1)), [b,c])
[[b == 1, c == r...], [b == r..., c == 1]]

Especially with trigonometric functions, the dummy variable may be implicitly an integer (hence the z):

Sage

sage: solve( sin(x)==cos(x), x, to_poly_solve=True)
[x == 1/4*pi + pi*z...]
sage: solve([cos(x)*sin(x) == 1/2, x+y == 0],x,y)
[[x == 1/4*pi + pi*z..., y == -1/4*pi - pi*z...]]

Python

>>> from sage.all import *
>>> solve( sin(x)==cos(x), x, to_poly_solve=True)
[x == 1/4*pi + pi*z...]
>>> solve([cos(x)*sin(x) == Integer(1)/Integer(2), x+y == Integer(0)],x,y)
[[x == 1/4*pi + pi*z..., y == -1/4*pi - pi*z...]]

Expressions which are not equations are assumed to be set equal to zero, as with \(x\) in the following example:

Sage

sage: solve([x, y == 2],x,y)
[[x == 0, y == 2]]

Python

>>> from sage.all import *
>>> solve([x, y == Integer(2)],x,y)
[[x == 0, y == 2]]

If True appears in the list of equations it is ignored, and if False appears in the list then no solutions are returned. E.g., note that the first 3==3 evaluates to True, not to a symbolic equation.

Sage

sage: solve([3==3, 1.00000000000000*x^3 == 0], x)
[x == 0]
sage: solve([1.00000000000000*x^3 == 0], x)
[x == 0]

Python

>>> from sage.all import *
>>> solve([Integer(3)==Integer(3), RealNumber('1.00000000000000')*x**Integer(3) == Integer(0)], x)
[x == 0]
>>> solve([RealNumber('1.00000000000000')*x**Integer(3) == Integer(0)], x)
[x == 0]

Here, the first equation evaluates to False, so there are no solutions:

Sage

sage: solve([1==3, 1.00000000000000*x^3 == 0], x)
[]

Python

>>> from sage.all import *
>>> solve([Integer(1)==Integer(3), RealNumber('1.00000000000000')*x**Integer(3) == Integer(0)], x)
[]

Completely symbolic solutions are supported:

Sage

sage: var('s,j,b,m,g')
(s, j, b, m, g)
sage: sys = [ m*(1-s) - b*s*j, b*s*j-g*j ]
sage: solve(sys,s,j)
[[s == 1, j == 0], [s == g/b, j == (b - g)*m/(b*g)]]
sage: solve(sys,(s,j))
[[s == 1, j == 0], [s == g/b, j == (b - g)*m/(b*g)]]
sage: solve(sys,[s,j])
[[s == 1, j == 0], [s == g/b, j == (b - g)*m/(b*g)]]

sage: z = var('z')
sage: solve((x-z)^2==2, x)
[x == z - sqrt(2), x == z + sqrt(2)]

Python

>>> from sage.all import *
>>> var('s,j,b,m,g')
(s, j, b, m, g)
>>> sys = [ m*(Integer(1)-s) - b*s*j, b*s*j-g*j ]
>>> solve(sys,s,j)
[[s == 1, j == 0], [s == g/b, j == (b - g)*m/(b*g)]]
>>> solve(sys,(s,j))
[[s == 1, j == 0], [s == g/b, j == (b - g)*m/(b*g)]]
>>> solve(sys,[s,j])
[[s == 1, j == 0], [s == g/b, j == (b - g)*m/(b*g)]]

>>> z = var('z')
>>> solve((x-z)**Integer(2)==Integer(2), x)
[x == z - sqrt(2), x == z + sqrt(2)]

Inequalities can be also solved:

Sage

sage: solve(x^2>8,x)
[[x < -2*sqrt(2)], [x > 2*sqrt(2)]]
sage: x,y = var('x,y'); (ln(x)-ln(y)>0).solve(x)
[[log(x) - log(y) > 0]]
sage: x,y = var('x,y'); (ln(x)>ln(y)).solve(x)  # random
[[0 < y, y < x, 0 < x]]
[[y < x, 0 < y]]

Python

>>> from sage.all import *
>>> solve(x**Integer(2)>Integer(8),x)
[[x < -2*sqrt(2)], [x > 2*sqrt(2)]]
>>> x,y = var('x,y'); (ln(x)-ln(y)>Integer(0)).solve(x)
[[log(x) - log(y) > 0]]
>>> x,y = var('x,y'); (ln(x)>ln(y)).solve(x)  # random
[[0 < y, y < x, 0 < x]]
[[y < x, 0 < y]]

A simple example to show the use of the keyword multiplicities:

Sage

sage: ((x^2-1)^2).solve(x)
[x == -1, x == 1]
sage: ((x^2-1)^2).solve(x,multiplicities=True)
([x == -1, x == 1], [2, 2])
sage: ((x^2-1)^2).solve(x,multiplicities=True,to_poly_solve=True)
Traceback (most recent call last):
...
NotImplementedError: to_poly_solve does not return multiplicities

Python

>>> from sage.all import *
>>> ((x**Integer(2)-Integer(1))**Integer(2)).solve(x)
[x == -1, x == 1]
>>> ((x**Integer(2)-Integer(1))**Integer(2)).solve(x,multiplicities=True)
([x == -1, x == 1], [2, 2])
>>> ((x**Integer(2)-Integer(1))**Integer(2)).solve(x,multiplicities=True,to_poly_solve=True)
Traceback (most recent call last):
...
NotImplementedError: to_poly_solve does not return multiplicities

Here is how the explicit_solutions keyword functions:

Sage

sage: solve(sin(x)==x,x)
[x == sin(x)]
sage: solve(sin(x)==x,x,explicit_solutions=True)
[]
sage: solve(x*sin(x)==x^2,x)
[x == 0, x == sin(x)]
sage: solve(x*sin(x)==x^2,x,explicit_solutions=True)
[x == 0]

Python

>>> from sage.all import *
>>> solve(sin(x)==x,x)
[x == sin(x)]
>>> solve(sin(x)==x,x,explicit_solutions=True)
[]
>>> solve(x*sin(x)==x**Integer(2),x)
[x == 0, x == sin(x)]
>>> solve(x*sin(x)==x**Integer(2),x,explicit_solutions=True)
[x == 0]

The following examples show the use of the keyword to_poly_solve:

Sage

sage: solve(abs(1-abs(1-x)) == 10, x)
[abs(abs(x - 1) - 1) == 10]
sage: solve(abs(1-abs(1-x)) == 10, x, to_poly_solve=True)
[x == -10, x == 12]

sage: var('Q')
Q
sage: solve(Q*sqrt(Q^2 + 2) - 1, Q)
[Q == 1/sqrt(Q^2 + 2)]

Python

>>> from sage.all import *
>>> solve(abs(Integer(1)-abs(Integer(1)-x)) == Integer(10), x)
[abs(abs(x - 1) - 1) == 10]
>>> solve(abs(Integer(1)-abs(Integer(1)-x)) == Integer(10), x, to_poly_solve=True)
[x == -10, x == 12]

>>> var('Q')
Q
>>> solve(Q*sqrt(Q**Integer(2) + Integer(2)) - Integer(1), Q)
[Q == 1/sqrt(Q^2 + 2)]

The following example is a regression in Maxima 5.39.0. It used to be possible to get one more solution here, namely 1/sqrt(sqrt(2) + 1), see https://sourceforge.net/p/maxima/bugs/3276/:

Sage

sage: solve(Q*sqrt(Q^2 + 2) - 1, Q, to_poly_solve=True)
[Q == -sqrt(-sqrt(2) - 1), Q == sqrt(sqrt(2) + 1)*(sqrt(2) - 1)]

Python

>>> from sage.all import *
>>> solve(Q*sqrt(Q**Integer(2) + Integer(2)) - Integer(1), Q, to_poly_solve=True)
[Q == -sqrt(-sqrt(2) - 1), Q == sqrt(sqrt(2) + 1)*(sqrt(2) - 1)]

An effort is made to only return solutions that satisfy the current assumptions:

Sage

sage: solve(x^2==4, x)
[x == -2, x == 2]
sage: assume(x<0)
sage: solve(x^2==4, x)
[x == -2]
sage: solve((x^2-4)^2 == 0, x, multiplicities=True)
([x == -2], [2])
sage: solve(x^2==2, x)
[x == -sqrt(2)]
sage: z = var('z')
sage: solve(x^2==2-z, x)
[x == -sqrt(-z + 2)]
sage: assume(x, 'rational')
sage: solve(x^2 == 2, x)
[]

Python

>>> from sage.all import *
>>> solve(x**Integer(2)==Integer(4), x)
[x == -2, x == 2]
>>> assume(x<Integer(0))
>>> solve(x**Integer(2)==Integer(4), x)
[x == -2]
>>> solve((x**Integer(2)-Integer(4))**Integer(2) == Integer(0), x, multiplicities=True)
([x == -2], [2])
>>> solve(x**Integer(2)==Integer(2), x)
[x == -sqrt(2)]
>>> z = var('z')
>>> solve(x**Integer(2)==Integer(2)-z, x)
[x == -sqrt(-z + 2)]
>>> assume(x, 'rational')
>>> solve(x**Integer(2) == Integer(2), x)
[]

In some cases it may be worthwhile to directly use to_poly_solve if one suspects some answers are being missed:

Sage

sage: forget()
sage: solve(cos(x)==0, x)
[x == 1/2*pi]
sage: solve(cos(x)==0, x, to_poly_solve=True)
[x == 1/2*pi]
sage: solve(cos(x)==0, x, to_poly_solve='force')
[x == 1/2*pi + pi*z...]

Python

>>> from sage.all import *
>>> forget()
>>> solve(cos(x)==Integer(0), x)
[x == 1/2*pi]
>>> solve(cos(x)==Integer(0), x, to_poly_solve=True)
[x == 1/2*pi]
>>> solve(cos(x)==Integer(0), x, to_poly_solve='force')
[x == 1/2*pi + pi*z...]

The same may also apply if a returned unsolved expression has a denominator, but the original one did not:

Sage

sage: solve(cos(x) * sin(x) == 1/2, x, to_poly_solve=True)
[sin(x) == 1/2/cos(x)]
sage: solve(cos(x) * sin(x) == 1/2, x, to_poly_solve=True, explicit_solutions=True)
[x == 1/4*pi + pi*z...]
sage: solve(cos(x) * sin(x) == 1/2, x, to_poly_solve='force')
[x == 1/4*pi + pi*z...]

Python

>>> from sage.all import *
>>> solve(cos(x) * sin(x) == Integer(1)/Integer(2), x, to_poly_solve=True)
[sin(x) == 1/2/cos(x)]
>>> solve(cos(x) * sin(x) == Integer(1)/Integer(2), x, to_poly_solve=True, explicit_solutions=True)
[x == 1/4*pi + pi*z...]
>>> solve(cos(x) * sin(x) == Integer(1)/Integer(2), x, to_poly_solve='force')
[x == 1/4*pi + pi*z...]

We use use_grobner in Maxima if no solution is obtained from Maxima’s to_poly_solve:

Sage

sage: x,y = var('x y')
sage: c1(x,y) = (x-5)^2+y^2-16
sage: c2(x,y) = (y-3)^2+x^2-9
sage: solve([c1(x,y),c2(x,y)],[x,y])
[[x == -9/68*sqrt(55) + 135/68, y == -15/68*sqrt(55) + 123/68],
 [x == 9/68*sqrt(55) + 135/68, y == 15/68*sqrt(55) + 123/68]]

Python

>>> from sage.all import *
>>> x,y = var('x y')
>>> __tmp__=var("x,y"); c1 = symbolic_expression((x-Integer(5))**Integer(2)+y**Integer(2)-Integer(16)).function(x,y)
>>> __tmp__=var("x,y"); c2 = symbolic_expression((y-Integer(3))**Integer(2)+x**Integer(2)-Integer(9)).function(x,y)
>>> solve([c1(x,y),c2(x,y)],[x,y])
[[x == -9/68*sqrt(55) + 135/68, y == -15/68*sqrt(55) + 123/68],
 [x == 9/68*sqrt(55) + 135/68, y == 15/68*sqrt(55) + 123/68]]

We use SymPy for Diophantine equations, see Expression.solve_diophantine:

Sage

sage: assume(x, 'integer')
sage: assume(z, 'integer')
sage: solve((x-z)^2==2, x)
[]

sage: forget()

Python

>>> from sage.all import *
>>> assume(x, 'integer')
>>> assume(z, 'integer')
>>> solve((x-z)**Integer(2)==Integer(2), x)
[]

>>> forget()

The following shows some more of SymPy’s capabilities that cannot be handled by Maxima:

Sage

sage: _ = var('t')
sage: r = solve([x^2 - y^2/exp(x), y-1], x, y, algorithm='sympy')
sage: (r[0][x], r[0][y])
(2*lambert_w(-1/2), 1)
sage: solve(-2*x**3 + 4*x**2 - 2*x + 6 > 0, x, algorithm='sympy')
[x < 1/3*(1/2)^(1/3)*(9*sqrt(77) + 79)^(1/3) + 2/3*(1/2)^(2/3)/(9*sqrt(77) + 79)^(1/3) + 2/3]
sage: solve(sqrt(2*x^2 - 7) - (3 - x),x,algorithm='sympy')
[x == -8, x == 2]
sage: solve(sqrt(2*x + 9) - sqrt(x + 1) - sqrt(x + 4),x,algorithm='sympy')
[x == 0]
sage: r = solve([x + y + z + t, -z - t], x, y, z, t, algorithm='sympy')
sage: (r[0][x], r[0][z])
(-y, -t)
sage: r = solve([x^2+y+z, y+x^2+z, x+y+z^2], x, y,z, algorithm='sympy')
sage: (r[0][x], r[0][y])
(z, -(z + 1)*z)
sage: (r[1][x], r[1][y])
(-z + 1, -z^2 + z - 1)
sage: solve(abs(x + 3) - 2*abs(x - 3),x,algorithm='sympy',domain='real')
[x == 1, x == 9]

Python

>>> from sage.all import *
>>> _ = var('t')
>>> r = solve([x**Integer(2) - y**Integer(2)/exp(x), y-Integer(1)], x, y, algorithm='sympy')
>>> (r[Integer(0)][x], r[Integer(0)][y])
(2*lambert_w(-1/2), 1)
>>> solve(-Integer(2)*x**Integer(3) + Integer(4)*x**Integer(2) - Integer(2)*x + Integer(6) > Integer(0), x, algorithm='sympy')
[x < 1/3*(1/2)^(1/3)*(9*sqrt(77) + 79)^(1/3) + 2/3*(1/2)^(2/3)/(9*sqrt(77) + 79)^(1/3) + 2/3]
>>> solve(sqrt(Integer(2)*x**Integer(2) - Integer(7)) - (Integer(3) - x),x,algorithm='sympy')
[x == -8, x == 2]
>>> solve(sqrt(Integer(2)*x + Integer(9)) - sqrt(x + Integer(1)) - sqrt(x + Integer(4)),x,algorithm='sympy')
[x == 0]
>>> r = solve([x + y + z + t, -z - t], x, y, z, t, algorithm='sympy')
>>> (r[Integer(0)][x], r[Integer(0)][z])
(-y, -t)
>>> r = solve([x**Integer(2)+y+z, y+x**Integer(2)+z, x+y+z**Integer(2)], x, y,z, algorithm='sympy')
>>> (r[Integer(0)][x], r[Integer(0)][y])
(z, -(z + 1)*z)
>>> (r[Integer(1)][x], r[Integer(1)][y])
(-z + 1, -z^2 + z - 1)
>>> solve(abs(x + Integer(3)) - Integer(2)*abs(x - Integer(3)),x,algorithm='sympy',domain='real')
[x == 1, x == 9]

We cannot translate all results from SymPy but we can at least print them:

Sage

sage: solve(sinh(x) - 2*cosh(x),x,algorithm='sympy')
[ImageSet(Lambda(_n, I*(2*_n*pi + pi/2) + log(sqrt(3))), Integers),
 ImageSet(Lambda(_n, I*(2*_n*pi - pi/2) + log(sqrt(3))), Integers)]
sage: solve(2*sin(x) - 2*sin(2*x), x,algorithm='sympy')
[ImageSet(Lambda(_n, 2*_n*pi), Integers),
 ImageSet(Lambda(_n, 2*_n*pi + pi), Integers),
 ImageSet(Lambda(_n, 2*_n*pi + 5*pi/3), Integers),
 ImageSet(Lambda(_n, 2*_n*pi + pi/3), Integers)]

sage: solve(x^5 + 3*x^3 + 7, x, algorithm='sympy')[0] # known bug
complex_root_of(x^5 + 3*x^3 + 7, 0)

Python

>>> from sage.all import *
>>> solve(sinh(x) - Integer(2)*cosh(x),x,algorithm='sympy')
[ImageSet(Lambda(_n, I*(2*_n*pi + pi/2) + log(sqrt(3))), Integers),
 ImageSet(Lambda(_n, I*(2*_n*pi - pi/2) + log(sqrt(3))), Integers)]
>>> solve(Integer(2)*sin(x) - Integer(2)*sin(Integer(2)*x), x,algorithm='sympy')
[ImageSet(Lambda(_n, 2*_n*pi), Integers),
 ImageSet(Lambda(_n, 2*_n*pi + pi), Integers),
 ImageSet(Lambda(_n, 2*_n*pi + 5*pi/3), Integers),
 ImageSet(Lambda(_n, 2*_n*pi + pi/3), Integers)]

>>> solve(x**Integer(5) + Integer(3)*x**Integer(3) + Integer(7), x, algorithm='sympy')[Integer(0)] # known bug
complex_root_of(x^5 + 3*x^3 + 7, 0)

A basic interface to Giac is provided:

Sage

sage: # needs sage.libs.giac
sage: solve([(2/3)^x-2], [x], algorithm='giac')
...[[-log(2)/(log(3) - log(2))]]

sage: # needs sage.libs.giac
sage: f = (sin(x) - 8*cos(x)*sin(x))*(sin(x)^2 + cos(x)) - (2*cos(x)*sin(x) - sin(x))*(-2*sin(x)^2 + 2*cos(x)^2 - cos(x))
sage: solve(f, x, algorithm='giac')
...[-2*arctan(sqrt(2)), 0, 2*arctan(sqrt(2)), pi]

sage: # needs sage.libs.giac
sage: x, y = SR.var('x,y')
sage: solve([x+y-4,x*y-3],[x,y],algorithm='giac')
[[1, 3], [3, 1]]

Python

>>> from sage.all import *
>>> # needs sage.libs.giac
>>> solve([(Integer(2)/Integer(3))**x-Integer(2)], [x], algorithm='giac')
...[[-log(2)/(log(3) - log(2))]]

>>> # needs sage.libs.giac
>>> f = (sin(x) - Integer(8)*cos(x)*sin(x))*(sin(x)**Integer(2) + cos(x)) - (Integer(2)*cos(x)*sin(x) - sin(x))*(-Integer(2)*sin(x)**Integer(2) + Integer(2)*cos(x)**Integer(2) - cos(x))
>>> solve(f, x, algorithm='giac')
...[-2*arctan(sqrt(2)), 0, 2*arctan(sqrt(2)), pi]

>>> # needs sage.libs.giac
>>> x, y = SR.var('x,y')
>>> solve([x+y-Integer(4),x*y-Integer(3)],[x,y],algorithm='giac')
[[1, 3], [3, 1]]

sage.symbolic.relation.solve_ineq(ineq, vars=None)

Solve inequalities and systems of inequalities using Maxima. Switches between rational inequalities (sage.symbolic.relation.solve_ineq_rational) and Fourier elimination (sage.symbolic.relation.solve_ineq_fouried). See the documentation of these functions for more details.

INPUT:

• ineq – one inequality or a list of inequalities

  Case1: If ineq is one equality, then it should be rational expression in one variable. This input is passed to sage.symbolic.relation.solve_ineq_univar function.

  Case2: If ineq is a list involving one or more inequalities, than the input is passed to sage.symbolic.relation.solve_ineq_fourier function. This function can be used for system of linear inequalities and for some types of nonlinear inequalities. See http://maxima.cvs.sourceforge.net/viewvc/maxima/maxima/share/contrib/fourier_elim/rtest_fourier_elim.mac for a big gallery of problems covered by this algorithm.

• vars – (optional) parameter with list of variables. This list is used only if Fourier elimination is used. If omitted or if rational inequality is solved, then variables are determined automatically.

OUTPUT:

• list – output is list of solutions as a list of simple inequalities output [A,B,C] means (A or B or C) each A, B, C is again a list and if A=[a,b], then A means (a and b).

EXAMPLES:

Sage

sage: from sage.symbolic.relation import solve_ineq

Python

>>> from sage.all import *
>>> from sage.symbolic.relation import solve_ineq

Inequalities in one variable. The variable is detected automatically:

Sage

sage: solve_ineq(x^2-1>3)
[[x < -2], [x > 2]]

sage: solve_ineq(1/(x-1)<=8)
[[x < 1], [x >= (9/8)]]

Python

>>> from sage.all import *
>>> solve_ineq(x**Integer(2)-Integer(1)>Integer(3))
[[x < -2], [x > 2]]

>>> solve_ineq(Integer(1)/(x-Integer(1))<=Integer(8))
[[x < 1], [x >= (9/8)]]

System of inequalities with automatically detected inequalities:

Sage

sage: y = var('y')
sage: solve_ineq([x-y<0,x+y-3<0],[y,x])
[[x < y, y < -x + 3, x < (3/2)]]
sage: solve_ineq([x-y<0,x+y-3<0],[x,y])
[[x < min(-y + 3, y)]]

Python

>>> from sage.all import *
>>> y = var('y')
>>> solve_ineq([x-y<Integer(0),x+y-Integer(3)<Integer(0)],[y,x])
[[x < y, y < -x + 3, x < (3/2)]]
>>> solve_ineq([x-y<Integer(0),x+y-Integer(3)<Integer(0)],[x,y])
[[x < min(-y + 3, y)]]

Note that although Sage will detect the variables automatically, the order it puts them in may depend on the system, so the following command is only guaranteed to give you one of the above answers:

Sage

sage: solve_ineq([x-y<0,x+y-3<0])  # random
[[x < y, y < -x + 3, x < (3/2)]]

Python

>>> from sage.all import *
>>> solve_ineq([x-y<Integer(0),x+y-Integer(3)<Integer(0)])  # random
[[x < y, y < -x + 3, x < (3/2)]]

ALGORITHM:

Calls solve_ineq_fourier if inequalities are list and solve_ineq_univar of the inequality is symbolic expression. See the description of these commands for more details related to the set of inequalities which can be solved. The list is empty if there is no solution.

AUTHORS:

• Robert Marik (01-2010)

sage.symbolic.relation.solve_ineq_fourier(ineq, vars=None)

Solve system of inequalities using Maxima and Fourier elimination.

Can be used for system of linear inequalities and for some types of nonlinear inequalities. For examples, see the example section below and http://maxima.cvs.sourceforge.net/viewvc/maxima/maxima/share/contrib/fourier_elim/rtest_fourier_elim.mac

INPUT:

• ineq – list with system of inequalities

• vars – optionally list with variables for Fourier elimination

OUTPUT:

• list – output is list of solutions as a list of simple inequalities output [A,B,C] means (A or B or C) each A, B, C is again a list and if A=[a,b], then A means (a and b). The list is empty if there is no solution.

EXAMPLES:

Sage

sage: from sage.symbolic.relation import solve_ineq_fourier
sage: y = var('y')
sage: solve_ineq_fourier([x+y<9,x-y>4],[x,y])
[[y + 4 < x, x < -y + 9, y < (5/2)]]
sage: solve_ineq_fourier([x+y<9,x-y>4],[y,x])[0][0](x=42)
y < -33

sage: solve_ineq_fourier([x^2>=0])
[[x < +Infinity]]

sage: solve_ineq_fourier([log(x)>log(y)],[x,y])
[[y < x, 0 < y]]
sage: solve_ineq_fourier([log(x)>log(y)],[y,x])
[[0 < y, y < x, 0 < x]]

Python

>>> from sage.all import *
>>> from sage.symbolic.relation import solve_ineq_fourier
>>> y = var('y')
>>> solve_ineq_fourier([x+y<Integer(9),x-y>Integer(4)],[x,y])
[[y + 4 < x, x < -y + 9, y < (5/2)]]
>>> solve_ineq_fourier([x+y<Integer(9),x-y>Integer(4)],[y,x])[Integer(0)][Integer(0)](x=Integer(42))
y < -33

>>> solve_ineq_fourier([x**Integer(2)>=Integer(0)])
[[x < +Infinity]]

>>> solve_ineq_fourier([log(x)>log(y)],[x,y])
[[y < x, 0 < y]]
>>> solve_ineq_fourier([log(x)>log(y)],[y,x])
[[0 < y, y < x, 0 < x]]

Note that different systems will find default variables in different orders, so the following is not tested:

Sage

sage: solve_ineq_fourier([log(x)>log(y)])  # random (one of the following appears)
[[0 < y, y < x, 0 < x]]
[[y < x, 0 < y]]

Python

>>> from sage.all import *
>>> solve_ineq_fourier([log(x)>log(y)])  # random (one of the following appears)
[[0 < y, y < x, 0 < x]]
[[y < x, 0 < y]]

ALGORITHM:

Calls Maxima command fourier_elim

AUTHORS:

• Robert Marik (01-2010)

sage.symbolic.relation.solve_ineq_univar(ineq)

Function solves rational inequality in one variable.

INPUT:

• ineq – inequality in one variable

OUTPUT:

• list – output is list of solutions as a list of simple inequalities output [A,B,C] means (A or B or C) each A, B, C is again a list and if A=[a,b], then A means (a and b). The list is empty if there is no solution.

EXAMPLES:

Sage

sage: from sage.symbolic.relation import solve_ineq_univar
sage: solve_ineq_univar(x-1/x>0)
[[x > -1, x < 0], [x > 1]]

sage: solve_ineq_univar(x^2-1/x>0)
[[x < 0], [x > 1]]

sage: solve_ineq_univar((x^3-1)*x<=0)
[[x >= 0, x <= 1]]

Python

>>> from sage.all import *
>>> from sage.symbolic.relation import solve_ineq_univar
>>> solve_ineq_univar(x-Integer(1)/x>Integer(0))
[[x > -1, x < 0], [x > 1]]

>>> solve_ineq_univar(x**Integer(2)-Integer(1)/x>Integer(0))
[[x < 0], [x > 1]]

>>> solve_ineq_univar((x**Integer(3)-Integer(1))*x<=Integer(0))
[[x >= 0, x <= 1]]

ALGORITHM:

Calls Maxima command solve_rat_ineq

AUTHORS:

• Robert Marik (01-2010)

sage.symbolic.relation.solve_mod(eqns, modulus, solution_dict=False)

Return all solutions to an equation or list of equations modulo the given integer modulus. Each equation must involve only polynomials in 1 or many variables.

By default the solutions are returned as \(n\)-tuples, where \(n\) is the number of variables appearing anywhere in the given equations. The variables are in alphabetical order.

INPUT:

• eqns – equation or list of equations

• modulus – integer

• solution_dict – boolean (default: False); if True or nonzero, return a list of dictionaries containing the solutions. If there are no solutions, return an empty list (rather than a list containing an empty dictionary). Likewise, if there’s only a single solution, return a list containing one dictionary with that solution.

EXAMPLES:

Sage

sage: var('x,y')
(x, y)
sage: solve_mod([x^2 + 2 == x, x^2 + y == y^2], 14)
[(4, 2), (4, 6), (4, 9), (4, 13)]
sage: solve_mod([x^2 == 1, 4*x  == 11], 15)
[(14,)]

Python

>>> from sage.all import *
>>> var('x,y')
(x, y)
>>> solve_mod([x**Integer(2) + Integer(2) == x, x**Integer(2) + y == y**Integer(2)], Integer(14))
[(4, 2), (4, 6), (4, 9), (4, 13)]
>>> solve_mod([x**Integer(2) == Integer(1), Integer(4)*x  == Integer(11)], Integer(15))
[(14,)]

Fermat’s equation modulo 3 with exponent 5:

Sage

sage: var('x,y,z')
(x, y, z)
sage: solve_mod([x^5 + y^5 == z^5], 3)
[(0, 0, 0), (0, 1, 1), (0, 2, 2), (1, 0, 1), (1, 1, 2), (1, 2, 0), (2, 0, 2), (2, 1, 0), (2, 2, 1)]

Python

>>> from sage.all import *
>>> var('x,y,z')
(x, y, z)
>>> solve_mod([x**Integer(5) + y**Integer(5) == z**Integer(5)], Integer(3))
[(0, 0, 0), (0, 1, 1), (0, 2, 2), (1, 0, 1), (1, 1, 2), (1, 2, 0), (2, 0, 2), (2, 1, 0), (2, 2, 1)]

We can solve with respect to a bigger modulus if it consists only of small prime factors:

Sage

sage: [d] = solve_mod([5*x + y == 3, 2*x - 3*y == 9], 3*5*7*11*19*23*29, solution_dict = True)
sage: d[x]
12915279
sage: d[y]
8610183

Python

>>> from sage.all import *
>>> [d] = solve_mod([Integer(5)*x + y == Integer(3), Integer(2)*x - Integer(3)*y == Integer(9)], Integer(3)*Integer(5)*Integer(7)*Integer(11)*Integer(19)*Integer(23)*Integer(29), solution_dict = True)
>>> d[x]
12915279
>>> d[y]
8610183

For cases where there are relatively few solutions and the prime factors are small, this can be efficient even if the modulus itself is large:

Sage

sage: sorted(solve_mod([x^2 == 41], 10^20))
[(4538602480526452429,), (11445932736758703821,), (38554067263241296179,),
(45461397519473547571,), (54538602480526452429,), (61445932736758703821,),
(88554067263241296179,), (95461397519473547571,)]

Python

>>> from sage.all import *
>>> sorted(solve_mod([x**Integer(2) == Integer(41)], Integer(10)**Integer(20)))
[(4538602480526452429,), (11445932736758703821,), (38554067263241296179,),
(45461397519473547571,), (54538602480526452429,), (61445932736758703821,),
(88554067263241296179,), (95461397519473547571,)]

We solve a simple equation modulo 2:

Sage

sage: x,y = var('x,y')
sage: solve_mod([x == y], 2)
[(0, 0), (1, 1)]

Python

>>> from sage.all import *
>>> x,y = var('x,y')
>>> solve_mod([x == y], Integer(2))
[(0, 0), (1, 1)]

Warning

The current implementation splits the modulus into prime powers, then naively enumerates all possible solutions (starting modulo primes and then working up through prime powers), and finally combines the solution using the Chinese Remainder Theorem. The interface is good, but the algorithm is very inefficient if the modulus has some larger prime factors! Sage does have the ability to do something much faster in certain cases at least by using Groebner basis, linear algebra techniques, etc. But for a lot of toy problems this function as is might be useful. At least it establishes an interface.

sage.symbolic.relation.string_to_list_of_solutions(s)

Used internally by the symbolic solve command to convert the output of Maxima’s solve command to a list of solutions in Sage’s symbolic package.

EXAMPLES:

We derive the (monic) quadratic formula:

Sage

sage: var('x,a,b')
(x, a, b)
sage: solve(x^2 + a*x + b == 0, x)
[x == -1/2*a - 1/2*sqrt(a^2 - 4*b), x == -1/2*a + 1/2*sqrt(a^2 - 4*b)]

Python

>>> from sage.all import *
>>> var('x,a,b')
(x, a, b)
>>> solve(x**Integer(2) + a*x + b == Integer(0), x)
[x == -1/2*a - 1/2*sqrt(a^2 - 4*b), x == -1/2*a + 1/2*sqrt(a^2 - 4*b)]

Behind the scenes when the above is evaluated the function string_to_list_of_solutions() is called with input the string \(s\) below:

Sage

sage: s = '[x=-(sqrt(a^2-4*b)+a)/2,x=(sqrt(a^2-4*b)-a)/2]'
sage: sage.symbolic.relation.string_to_list_of_solutions(s)
 [x == -1/2*a - 1/2*sqrt(a^2 - 4*b), x == -1/2*a + 1/2*sqrt(a^2 - 4*b)]

Python

>>> from sage.all import *
>>> s = '[x=-(sqrt(a^2-4*b)+a)/2,x=(sqrt(a^2-4*b)-a)/2]'
>>> sage.symbolic.relation.string_to_list_of_solutions(s)
 [x == -1/2*a - 1/2*sqrt(a^2 - 4*b), x == -1/2*a + 1/2*sqrt(a^2 - 4*b)]

sage.symbolic.relation.test_relation_maxima(relation)

Return True if this (in)equality is definitely true. Return False if it is false or the algorithm for testing (in)equality is inconclusive.

EXAMPLES:

Sage

sage: from sage.symbolic.relation import test_relation_maxima
sage: k = var('k')
sage: pol = 1/(k-1) - 1/k -1/k/(k-1)
sage: test_relation_maxima(pol == 0)
True
sage: f = sin(x)^2 + cos(x)^2 - 1
sage: test_relation_maxima(f == 0)
True
sage: test_relation_maxima( x == x )
True
sage: test_relation_maxima( x != x )
False
sage: test_relation_maxima( x > x )
False
sage: test_relation_maxima( x^2 > x )
False
sage: test_relation_maxima( x + 2 > x )
True
sage: test_relation_maxima( x - 2 > x )
False

Python

>>> from sage.all import *
>>> from sage.symbolic.relation import test_relation_maxima
>>> k = var('k')
>>> pol = Integer(1)/(k-Integer(1)) - Integer(1)/k -Integer(1)/k/(k-Integer(1))
>>> test_relation_maxima(pol == Integer(0))
True
>>> f = sin(x)**Integer(2) + cos(x)**Integer(2) - Integer(1)
>>> test_relation_maxima(f == Integer(0))
True
>>> test_relation_maxima( x == x )
True
>>> test_relation_maxima( x != x )
False
>>> test_relation_maxima( x > x )
False
>>> test_relation_maxima( x**Integer(2) > x )
False
>>> test_relation_maxima( x + Integer(2) > x )
True
>>> test_relation_maxima( x - Integer(2) > x )
False

Here are some examples involving assumptions:

Sage

sage: x, y, z = var('x, y, z')
sage: assume(x>=y,y>=z,z>=x)
sage: test_relation_maxima(x==z)
True
sage: test_relation_maxima(z<x)
False
sage: test_relation_maxima(z>y)
False
sage: test_relation_maxima(y==z)
True
sage: forget()
sage: assume(x>=1,x<=1)
sage: test_relation_maxima(x==1)
True
sage: test_relation_maxima(x>1)
False
sage: test_relation_maxima(x>=1)
True
sage: test_relation_maxima(x!=1)
False
sage: forget()
sage: assume(x>0)
sage: test_relation_maxima(x==0)
False
sage: test_relation_maxima(x>-1)
True
sage: test_relation_maxima(x!=0)
True
sage: test_relation_maxima(x!=1)
False
sage: forget()

Python

>>> from sage.all import *
>>> x, y, z = var('x, y, z')
>>> assume(x>=y,y>=z,z>=x)
>>> test_relation_maxima(x==z)
True
>>> test_relation_maxima(z<x)
False
>>> test_relation_maxima(z>y)
False
>>> test_relation_maxima(y==z)
True
>>> forget()
>>> assume(x>=Integer(1),x<=Integer(1))
>>> test_relation_maxima(x==Integer(1))
True
>>> test_relation_maxima(x>Integer(1))
False
>>> test_relation_maxima(x>=Integer(1))
True
>>> test_relation_maxima(x!=Integer(1))
False
>>> forget()
>>> assume(x>Integer(0))
>>> test_relation_maxima(x==Integer(0))
False
>>> test_relation_maxima(x>-Integer(1))
True
>>> test_relation_maxima(x!=Integer(0))
True
>>> test_relation_maxima(x!=Integer(1))
False
>>> forget()