Mixing
Symmetrize a polynomial forgetting the commutativity property of multiplication
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up vote
1
down vote
favorite
I need a script to authomatically symmetrize a given polynomial.
For example, if the input is
xy
the output should be
(xy+yx)/2
The same principle should work also for higher order polynomial. For example, if the input is
xyz
the output should be
(xyz+xzy+yxz+yzx+zxy+zyx)/6
The input is, in general, a polynomial. If the input is
xy+wz
the output should be
(xy+yx+wz+zw)/2
It may happen that some terms come with some powers. If the input is
x^2 z
the output should be either
(xxz + xxz +xzx+ xzx + zxx + zxx)/6
or
(2x^2z+2xzx+2zx^2)/6
Both these ouputs are good.
Thanks in advance for the help.
polynomials algebraic-manipulation algebra
asked 2 hours ago
AndreaPaco
1324
add a comment |Â
up vote
1
down vote
favorite
I need a script to authomatically symmetrize a given polynomial.
For example, if the input is
xy
the output should be
(xy+yx)/2
The same principle should work also for higher order polynomial. For example, if the input is
xyz
the output should be
(xyz+xzy+yxz+yzx+zxy+zyx)/6
The input is, in general, a polynomial. If the input is
xy+wz
the output should be
(xy+yx+wz+zw)/2
It may happen that some terms come with some powers. If the input is
x^2 z
the output should be either
(xxz + xxz +xzx+ xzx + zxx + zxx)/6
or
(2x^2z+2xzx+2zx^2)/6
Both these ouputs are good.
Thanks in advance for the help.
polynomials algebraic-manipulation algebra
asked 2 hours ago
AndreaPaco
1324
Are you always dealing with monomials, or should a solution expect things likex y + w z?
– J. M. is computer-less♦
2 hours ago
The input could be a polynomial as well. For example, if the input is $xy+wz$, the output should be $(xy+yx+wz+zw)/2$, but I know a priori what are the variables that come into play. Moreover, of course, it may appen to have things like $x^2y$ as input. In this case, the output should be either $(xxz+xzx+xxz+xzx+zxx+zxx)/6$ or $(2x^2z + 2xzx+ 2zx^2)/6$. Both outputs are ok for me.
– AndreaPaco
2 hours ago
@J.M.iscomputer-less, I've edited the question in order to meke my question more precise.
– AndreaPaco
1 hour ago
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I need a script to authomatically symmetrize a given polynomial.
For example, if the input is
xy
the output should be
(xy+yx)/2
The same principle should work also for higher order polynomial. For example, if the input is
xyz
the output should be
(xyz+xzy+yxz+yzx+zxy+zyx)/6
The input is, in general, a polynomial. If the input is
xy+wz
the output should be
(xy+yx+wz+zw)/2
It may happen that some terms come with some powers. If the input is
x^2 z
the output should be either
(xxz + xxz +xzx+ xzx + zxx + zxx)/6
or
(2x^2z+2xzx+2zx^2)/6
Both these ouputs are good.
Thanks in advance for the help.
polynomials algebraic-manipulation algebra
asked 2 hours ago
AndreaPaco
1324
I need a script to authomatically symmetrize a given polynomial.
For example, if the input is
xy
the output should be
(xy+yx)/2
The same principle should work also for higher order polynomial. For example, if the input is
xyz
the output should be
(xyz+xzy+yxz+yzx+zxy+zyx)/6
The input is, in general, a polynomial. If the input is
xy+wz
the output should be
(xy+yx+wz+zw)/2
It may happen that some terms come with some powers. If the input is
x^2 z
the output should be either
(xxz + xxz +xzx+ xzx + zxx + zxx)/6
or
(2x^2z+2xzx+2zx^2)/6
Both these ouputs are good.
Thanks in advance for the help.
polynomials algebraic-manipulation algebra
polynomials algebraic-manipulation algebra
asked 2 hours ago
AndreaPaco
1324
asked 2 hours ago
AndreaPaco
1324
edited 1 hour ago
asked 2 hours ago
AndreaPaco
1324
asked 2 hours ago
AndreaPaco
1324
asked 2 hours ago
AndreaPaco
1324
1324
Are you always dealing with monomials, or should a solution expect things likex y + w z?
– J. M. is computer-less♦
2 hours ago
The input could be a polynomial as well. For example, if the input is $xy+wz$, the output should be $(xy+yx+wz+zw)/2$, but I know a priori what are the variables that come into play. Moreover, of course, it may appen to have things like $x^2y$ as input. In this case, the output should be either $(xxz+xzx+xxz+xzx+zxx+zxx)/6$ or $(2x^2z + 2xzx+ 2zx^2)/6$. Both outputs are ok for me.
– AndreaPaco
2 hours ago
@J.M.iscomputer-less, I've edited the question in order to meke my question more precise.
– AndreaPaco
1 hour ago
add a comment |Â
Are you always dealing with monomials, or should a solution expect things likex y + w z?
– J. M. is computer-less♦
2 hours ago
The input could be a polynomial as well. For example, if the input is $xy+wz$, the output should be $(xy+yx+wz+zw)/2$, but I know a priori what are the variables that come into play. Moreover, of course, it may appen to have things like $x^2y$ as input. In this case, the output should be either $(xxz+xzx+xxz+xzx+zxx+zxx)/6$ or $(2x^2z + 2xzx+ 2zx^2)/6$. Both outputs are ok for me.
– AndreaPaco
2 hours ago
@J.M.iscomputer-less, I've edited the question in order to meke my question more precise.
– AndreaPaco
1 hour ago
Are you always dealing with monomials, or should a solution expect things like
x y + w z?– J. M. is computer-less♦
2 hours ago
Are you always dealing with monomials, or should a solution expect things like
x y + w z?– J. M. is computer-less♦
2 hours ago
The input could be a polynomial as well. For example, if the input is $xy+wz$, the output should be $(xy+yx+wz+zw)/2$, but I know a priori what are the variables that come into play. Moreover, of course, it may appen to have things like $x^2y$ as input. In this case, the output should be either $(xxz+xzx+xxz+xzx+zxx+zxx)/6$ or $(2x^2z + 2xzx+ 2zx^2)/6$. Both outputs are ok for me.
– AndreaPaco
2 hours ago
The input could be a polynomial as well. For example, if the input is $xy+wz$, the output should be $(xy+yx+wz+zw)/2$, but I know a priori what are the variables that come into play. Moreover, of course, it may appen to have things like $x^2y$ as input. In this case, the output should be either $(xxz+xzx+xxz+xzx+zxx+zxx)/6$ or $(2x^2z + 2xzx+ 2zx^2)/6$. Both outputs are ok for me.
– AndreaPaco
2 hours ago
@J.M.iscomputer-less, I've edited the question in order to meke my question more precise.
– AndreaPaco
1 hour ago
@J.M.iscomputer-less, I've edited the question in order to meke my question more precise.
– AndreaPaco
1 hour ago
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
What about this?
f[NonCommutativeMultiply[x__]] := Mean[Permutations[NonCommutativeMultiply[x]]]
f[x ** y ** z]
1/6 (x ** y ** z + x ** z ** y + y ** x ** z + y ** z ** x +
z ** x ** y + z ** y ** x)
Adding
f[x_Times] := f /@ x
f[x_Plus] := f /@ x
f[x_?NumericQ] := x
allows us to treat also polynomials:
f[3 x ** x ** y + y ** y ** y + 54 x ** z ** y]
x ** x ** y + x ** y ** x + y ** x ** x + y ** y ** y +
9 (x ** y ** z + x ** z ** y + y ** x ** z + y ** z ** x +
z ** x ** y + z ** y ** x)
answered 1 hour ago
Henrik Schumacher
41.8k260125
Thanks for your help. Can it handle a generic polynomial $P(x,y,z)$ such as $P(x,y,z)=3x^2y+y^3+54xzy$?
– AndreaPaco
1 hour ago
Thanks a lot for the update. You gave me a huge help. Just a curiosity more. In the input the non-commutative multiplication $**$ is already made explicit. Is there a way to authomatically turn a traditional multiplication (such as $x*y$) into a non-commutative one, (such as $x**y$)?
– AndreaPaco
1 hour ago
1
Yes, you can always tuse replacement rules, e.g.x x y //. Power[a_, k_Integer?Positive] :> NonCommutativeMultiply @@ ConstantArray[a, k], Times -> NonCommutativeMultiply(however this won't work out with the coefficients).
– Henrik Schumacher
1 hour ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
What about this?
f[NonCommutativeMultiply[x__]] := Mean[Permutations[NonCommutativeMultiply[x]]]
f[x ** y ** z]
1/6 (x ** y ** z + x ** z ** y + y ** x ** z + y ** z ** x +
z ** x ** y + z ** y ** x)
Adding
f[x_Times] := f /@ x
f[x_Plus] := f /@ x
f[x_?NumericQ] := x
allows us to treat also polynomials:
f[3 x ** x ** y + y ** y ** y + 54 x ** z ** y]
x ** x ** y + x ** y ** x + y ** x ** x + y ** y ** y +
9 (x ** y ** z + x ** z ** y + y ** x ** z + y ** z ** x +
z ** x ** y + z ** y ** x)
answered 1 hour ago
Henrik Schumacher
41.8k260125
Thanks for your help. Can it handle a generic polynomial $P(x,y,z)$ such as $P(x,y,z)=3x^2y+y^3+54xzy$?
– AndreaPaco
1 hour ago
Thanks a lot for the update. You gave me a huge help. Just a curiosity more. In the input the non-commutative multiplication $**$ is already made explicit. Is there a way to authomatically turn a traditional multiplication (such as $x*y$) into a non-commutative one, (such as $x**y$)?
– AndreaPaco
1 hour ago
1
Yes, you can always tuse replacement rules, e.g.x x y //. Power[a_, k_Integer?Positive] :> NonCommutativeMultiply @@ ConstantArray[a, k], Times -> NonCommutativeMultiply(however this won't work out with the coefficients).
– Henrik Schumacher
1 hour ago
add a comment |Â
up vote
2
down vote
accepted
What about this?
f[NonCommutativeMultiply[x__]] := Mean[Permutations[NonCommutativeMultiply[x]]]
f[x ** y ** z]
1/6 (x ** y ** z + x ** z ** y + y ** x ** z + y ** z ** x +
z ** x ** y + z ** y ** x)
Adding
f[x_Times] := f /@ x
f[x_Plus] := f /@ x
f[x_?NumericQ] := x
allows us to treat also polynomials:
f[3 x ** x ** y + y ** y ** y + 54 x ** z ** y]
x ** x ** y + x ** y ** x + y ** x ** x + y ** y ** y +
9 (x ** y ** z + x ** z ** y + y ** x ** z + y ** z ** x +
z ** x ** y + z ** y ** x)
answered 1 hour ago
Henrik Schumacher
41.8k260125
Thanks for your help. Can it handle a generic polynomial $P(x,y,z)$ such as $P(x,y,z)=3x^2y+y^3+54xzy$?
– AndreaPaco
1 hour ago
Thanks a lot for the update. You gave me a huge help. Just a curiosity more. In the input the non-commutative multiplication $**$ is already made explicit. Is there a way to authomatically turn a traditional multiplication (such as $x*y$) into a non-commutative one, (such as $x**y$)?
– AndreaPaco
1 hour ago
1
Yes, you can always tuse replacement rules, e.g.x x y //. Power[a_, k_Integer?Positive] :> NonCommutativeMultiply @@ ConstantArray[a, k], Times -> NonCommutativeMultiply(however this won't work out with the coefficients).
– Henrik Schumacher
1 hour ago
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
What about this?
f[NonCommutativeMultiply[x__]] := Mean[Permutations[NonCommutativeMultiply[x]]]
f[x ** y ** z]
1/6 (x ** y ** z + x ** z ** y + y ** x ** z + y ** z ** x +
z ** x ** y + z ** y ** x)
Adding
f[x_Times] := f /@ x
f[x_Plus] := f /@ x
f[x_?NumericQ] := x
allows us to treat also polynomials:
f[3 x ** x ** y + y ** y ** y + 54 x ** z ** y]
x ** x ** y + x ** y ** x + y ** x ** x + y ** y ** y +
9 (x ** y ** z + x ** z ** y + y ** x ** z + y ** z ** x +
z ** x ** y + z ** y ** x)
answered 1 hour ago
Henrik Schumacher
41.8k260125
What about this?
f[NonCommutativeMultiply[x__]] := Mean[Permutations[NonCommutativeMultiply[x]]]
f[x ** y ** z]
1/6 (x ** y ** z + x ** z ** y + y ** x ** z + y ** z ** x +
z ** x ** y + z ** y ** x)
Adding
f[x_Times] := f /@ x
f[x_Plus] := f /@ x
f[x_?NumericQ] := x
allows us to treat also polynomials:
f[3 x ** x ** y + y ** y ** y + 54 x ** z ** y]
x ** x ** y + x ** y ** x + y ** x ** x + y ** y ** y +
9 (x ** y ** z + x ** z ** y + y ** x ** z + y ** z ** x +
z ** x ** y + z ** y ** x)
answered 1 hour ago
Henrik Schumacher
41.8k260125
edited 1 hour ago
answered 1 hour ago
Henrik Schumacher
41.8k260125
answered 1 hour ago
Henrik Schumacher
41.8k260125
answered 1 hour ago
Henrik Schumacher
41.8k260125
41.8k260125
Thanks for your help. Can it handle a generic polynomial $P(x,y,z)$ such as $P(x,y,z)=3x^2y+y^3+54xzy$?
– AndreaPaco
1 hour ago
Thanks a lot for the update. You gave me a huge help. Just a curiosity more. In the input the non-commutative multiplication $**$ is already made explicit. Is there a way to authomatically turn a traditional multiplication (such as $x*y$) into a non-commutative one, (such as $x**y$)?
– AndreaPaco
1 hour ago
1
Yes, you can always tuse replacement rules, e.g.x x y //. Power[a_, k_Integer?Positive] :> NonCommutativeMultiply @@ ConstantArray[a, k], Times -> NonCommutativeMultiply(however this won't work out with the coefficients).
– Henrik Schumacher
1 hour ago
add a comment |Â
Thanks for your help. Can it handle a generic polynomial $P(x,y,z)$ such as $P(x,y,z)=3x^2y+y^3+54xzy$?
– AndreaPaco
1 hour ago
Thanks a lot for the update. You gave me a huge help. Just a curiosity more. In the input the non-commutative multiplication $**$ is already made explicit. Is there a way to authomatically turn a traditional multiplication (such as $x*y$) into a non-commutative one, (such as $x**y$)?
– AndreaPaco
1 hour ago
1
Yes, you can always tuse replacement rules, e.g.x x y //. Power[a_, k_Integer?Positive] :> NonCommutativeMultiply @@ ConstantArray[a, k], Times -> NonCommutativeMultiply(however this won't work out with the coefficients).
– Henrik Schumacher
1 hour ago
Thanks for your help. Can it handle a generic polynomial $P(x,y,z)$ such as $P(x,y,z)=3x^2y+y^3+54xzy$?
– AndreaPaco
1 hour ago
Thanks for your help. Can it handle a generic polynomial $P(x,y,z)$ such as $P(x,y,z)=3x^2y+y^3+54xzy$?
– AndreaPaco
1 hour ago
Thanks a lot for the update. You gave me a huge help. Just a curiosity more. In the input the non-commutative multiplication $**$ is already made explicit. Is there a way to authomatically turn a traditional multiplication (such as $x*y$) into a non-commutative one, (such as $x**y$)?
– AndreaPaco
1 hour ago
Thanks a lot for the update. You gave me a huge help. Just a curiosity more. In the input the non-commutative multiplication $**$ is already made explicit. Is there a way to authomatically turn a traditional multiplication (such as $x*y$) into a non-commutative one, (such as $x**y$)?
– AndreaPaco
1 hour ago
1
1
Yes, you can always tuse replacement rules, e.g.
x x y //. Power[a_, k_Integer?Positive] :> NonCommutativeMultiply @@ ConstantArray[a, k], Times -> NonCommutativeMultiply (however this won't work out with the coefficients).– Henrik Schumacher
1 hour ago
Yes, you can always tuse replacement rules, e.g.
x x y //. Power[a_, k_Integer?Positive] :> NonCommutativeMultiply @@ ConstantArray[a, k], Times -> NonCommutativeMultiply (however this won't work out with the coefficients).– Henrik Schumacher
1 hour ago
add a comment |Â
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Are you always dealing with monomials, or should a solution expect things like
x y + w z?– J. M. is computer-less♦
2 hours ago
The input could be a polynomial as well. For example, if the input is $xy+wz$, the output should be $(xy+yx+wz+zw)/2$, but I know a priori what are the variables that come into play. Moreover, of course, it may appen to have things like $x^2y$ as input. In this case, the output should be either $(xxz+xzx+xxz+xzx+zxx+zxx)/6$ or $(2x^2z + 2xzx+ 2zx^2)/6$. Both outputs are ok for me.
– AndreaPaco
2 hours ago
@J.M.iscomputer-less, I've edited the question in order to meke my question more precise.
– AndreaPaco
1 hour ago