Sage can solve symbolic equations and inequalities. For example, we derive the quadratic formula as follows:
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(-4*a*c + b^2))/a,
x == -1/2*(b - sqrt(-4*a*c + b^2))/a
]
Operators:
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>
Left hand side:
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
Right hand side:
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
Add two symbolic equations:
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
Subtract two symbolic equations:
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
Multiply two symbolic equations:
sage: x = var('x')
sage: m = x == 5*x + 1
sage: n = sin(x) == sin(x+2*pi)
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
Divide two symbolic equations:
sage: x = var('x')
sage: m = x == 5*x + 1
sage: n = sin(x) == sin(x+2*pi)
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)
sage: m/n
x/sin(x) != (5*x + 1)/sin(2*pi + x)
Substitution into relations:
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)
We can solve equations:
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)
...
TypeError: 5 is not a valid variable.
We illustrate finding multiplicities of solutions:
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])
We can also solve many inequalities:
sage: solve(1/(x-1)<=8,x)
[[x < 1], [x >= (9/8)]]
We can numerically find roots of equations:
sage: (x == sin(x)).find_root(-2,2)
0.0
sage: (x^5 + 3*x + 2 == 0).find_root(-2,2,x)
-0.63283452024215225
sage: (cos(x) == sin(x)).find_root(10,20)
19.634954084936208
We illustrate some valid error conditions:
sage: (cos(x) != sin(x)).find_root(10,20)
...
ValueError: Symbolic equation must be an equality.
sage: (SR(3)==SR(2)).find_root(-1,1)
...
RuntimeError: no zero in the interval, since constant expression is not 0.
There must be at most one variable:
sage: x, y = var('x,y')
sage: (x == y).find_root(-2,2)
...
NotImplementedError: root finding currently only implemented in 1 dimension.
Forgetting assumptions:
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()
[]
Conversion to Maxima:
sage: x = var('x')
sage: eq = (x^(3/5) >= pi^2 + e^i)
sage: eq._maxima_init_()
'(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
Conversion to Maple:
sage: x = var('x')
sage: eq = (x == 2)
sage: eq._maple_init_()
'x = 2'
Comparison:
sage: x = var('x')
sage: (x>0) == (x>0)
True
sage: (x>0) == (x>1)
False
sage: (x>0) != (x>1)
True
Variables appearing in the relation:
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 == x^2 - y^2 - z^3
sage: f.variables()
(w, x, y, z)
LaTeX output:
sage: latex(x^(3/5) >= pi)
x^{\left(\frac{3}{5}\right)} \geq \pi
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)]
sage: f = x^5 + a
sage: solve(f==0,x)
[x == (-a)^(1/5)*e^(2/5*I*pi), x == (-a)^(1/5)*e^(4/5*I*pi), x == (-a)^(1/5)*e^(-4/5*I*pi), x == (-a)^(1/5)*e^(-2/5*I*pi), x == (-a)^(1/5)]
You can also do arithmetic with inequalities, as illustrated below:
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
sqrt(2) + x^3 == x*y*sin(x)
sage: h - sqrt(2)
x^3 == x*y*sin(x) - sqrt(2)
sage: h + f
x + sqrt(2) + x^3 + 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
TESTS:
We test serializing symbolic equations:
sage: eqn = x^3 + 2/3 >= x
sage: loads(dumps(eqn))
x^3 + 2/3 >= x
sage: loads(dumps(eqn)) == eqn
True
AUTHORS:
Algebraically solve an equation or system of equations (over the complex numbers) for given variables. Inequalities and systems of inequalities are also supported.
INPUT:
EXAMPLES:
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 == -1/2*sqrt(-I*sqrt(3) + 3)*sqrt(2)],
[x == -1/2*I*sqrt(3) - 1/2, y == 1/2*sqrt(-I*sqrt(3) + 3)*sqrt(2)],
[x == 1/2*I*sqrt(3) - 1/2, y == -1/2*sqrt(I*sqrt(3) + 3)*sqrt(2)],
[x == 1/2*I*sqrt(3) - 1/2, y == 1/2*sqrt(I*sqrt(3) + 3)*sqrt(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 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
Whenever possible, answers will be symbolic, but with systems of equations, at times approximations will be given, due to the underlying algorithm in Maxima:
sage: sols = solve([x^3==y,y^2==x],[x,y]); sols[-1], sols[0]
([x == 0, y == 0], [x == (0.309016994375 + 0.951056516295*I), y == (-0.809016994375 - 0.587785252292*I)])
sage: sols[0][0].rhs().pyobject().parent()
Complex Double Field
If f is only one equation or expression, we use the solve method for symbolic expressions, which defaults to exact answers only:
sage: solve([y^6==y],y)
[y == e^(2/5*I*pi), y == e^(4/5*I*pi), y == e^(-4/5*I*pi), y == e^(-2/5*I*pi), y == 1, y == 0]
sage: solve( [y^6 == y], y)==solve( y^6 == y, y)
True
Note
For more details about solving a single equations, see the documentation for its solve.
sage: from sage.symbolic.expression import Expression
sage: Expression.solve(x^2==1,x)
[x == -1, x == 1]
We must solve with respect to actual variables:
sage: z = 5
sage: solve([8*z + y == 3, -z +7*y == 0],y,z)
...
TypeError: 5 is not a valid variable.
If we ask for a dictionary for the solutions, we get it:
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
If there is a parameter in the answer, that will show up as a new variable. In the following example, r1 is a real free variable (because of the r):
sage: solve([x+y == 3, 2*x+2*y == 6],x,y)
[[x == -r1 + 3, y == r1]]
Especially with trigonometric functions, the dummy variable may be implicitly an integer (hence the z):
sage: solve([cos(x)*sin(x) == 1/2, x+y == 0],x,y)
[[x == 1/4*pi + pi*z38, y == -1/4*pi - pi*z38]]
Expressions which are not equations are assumed to be set equal
to zero, as with 
in the following example:
sage: solve([x, y == 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: solve([3==3, 1.00000000000000*x^3 == 0], x)
[x == 0]
sage: solve([1.00000000000000*x^3 == 0], x)
[x == 0]
Here, the first equation evaluates to False, so there are no solutions:
sage: solve([1==3, 1.00000000000000*x^3 == 0], x)
[]
Completely symbolic solutions are supported:
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)]]
Inequalities can be also solved:
sage: solve(x^2>8,x)
[[x < -2*sqrt(2)], [x > 2*sqrt(2)]]
TESTS:
sage: solve([sin(x)==x,y^2==x],x,y)
[sin(x) == x, y^2 == x]
Use use_grobner if no solution is obtained from to_poly_solve:
sage: x,y=var('x y'); c1(x,y)=(x-5)^2+y^2-16; 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(5)*sqrt(11) + 123/68], [x == 9/68*sqrt(55) + 135/68, y == 15/68*sqrt(5)*sqrt(11) + 123/68]]
Solves 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 varible. 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:
EXAMPLES:
sage: from sage.symbolic.relation import solve_ineq
Inequalities in one variable. The variable is detected automatically:
sage: solve_ineq(x^2-1>3)
[[x < -2], [x > 2]]
sage: solve_ineq(1/(x-1)<=8)
[[x < 1], [x >= (9/8)]]
System of inequalities with automatically detected inequalities:
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)]]
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: solve_ineq([x-y<0,x+y-3<0]) # not tested - 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:
Solves sytem 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 section EXAMPLES below and http://maxima.cvs.sourceforge.net/viewvc/maxima/maxima/share/contrib/fourier_elim/rtest_fourier_elim.mac
INPUT:
OUTPUT:
EXAMPLES:
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])
[[y < min(x - 4, -x + 9)]]
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]]
Note that different systems will find default variables in different orders, so the following is not tested:
sage: solve_ineq_fourier([log(x)>log(y)]) # not tested - one of the following appears
[[0 < y, y < x, 0 < x]]
[[y < x, 0 < y]]
ALGORITHM:
Calls Maxima command fourier_elim
AUTHORS:
Function solves rational inequality in one variable.
INPUT:
OUTPUT:
EXAMPLES:
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]]
ALGORITHM:
Calls Maxima command solve_rat_ineq
AUTHORS:
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 
-tuples, where
is the number of variables appearing anywhere in the given
equations. The variables are in alphabetical order.
INPUT:
EXAMPLES:
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,)]
Fermat’s equation modulo 3 with exponent 5:
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)]
We can solve with respect to a bigger modulus if it consists only of small prime factors:
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
We solve an simple equation modulo 2:
sage: x,y = var('x,y')
sage: solve_mod([x == y], 2)
[(0, 0), (1, 1)]
Warning
The current implementation splits the modulus into prime powers, then naively enumerates all possible solutions and finally combines the solution using the Chinese Remainder Theorem. The interface is good, but the algorithm is horrible 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.
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.
The solutions are returned as -tuples, where
is the number of variables appearing anywhere in the given
equations. The variables are in alphabetical order.
INPUT:
EXAMPLES:
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,)]
Fermat’s equation modulo 3 with exponent 5:
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)]
We solve an simple equation modulo 2:
sage: x,y = var('x,y')
sage: solve_mod([x == y], 2)
[(0, 0), (1, 1)]
Warning
Currently this naively enumerates all possible solutions. The interface is good, but the algorithm is horrible if the modulus is at all large! Sage does have the ability to do something much faster in certain cases at least by using the Chinese Remainder Theorem, Groebner basis, linear algebra techniques, etc. But for a lot of toy problems this function as is might be useful. At the very least, it establishes an interface.
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: 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)]
Behind the scenes when the above is evaluated the function
string_to_list_of_solutions() is called with input the
string 
below:
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)]
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: 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
Here are some examples involving assumptions:
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: test_relation_maxima(x<>1) # alternate syntax for not equal
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()