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Integrate a polynomial over a polyhedron
Overview
The question is about integrating a polynomial over a region defined by linear inequalities.
Such a region is called a polytope or a polyhedron.
In Sage one can construct such regions using the
Polyhedron class.
This class provides a method integrate which allows,
when the latte_int optional package is installed,
to integrate a polynomial function over a polyhedron.
In a way, the trick is: rather than start from the function
integrate and somehow specify a region, we fist define
a polyhedron and then use the method integrate to compute
the integral of a polynomial over that polyhedron.
Steps
The region is defined by five inequalities:
x + 2 y + 3 z < 2
-1 < x
x < y
y < z
z < 1
and the function to integrate is
x^2 + 2 y z
The five inequalities can be rewritten as
2 + (-1) * x + (-2) * y + (-3) * z > 0
1 + (+1) * x > 0
0 + (-1) * x + (+1) * y > 0
0 + (-1) * y + (+1) * z > 0
1 + (-1) * z > 0
Encode these inequalities in Sage as
sage: ieqs = [[2, -1, -2, -3],
....: [1, 1, 0, 0],
....: [0, -1, 1, 0],
....: [0, 0, -1, 1],
....: [1, 0, 0, -1], ]
Create a polyhedron from those inequalities:
sage: P = Polyhedron(ieqs=ieqs)
Define polynomial variables and a polynomial:
sage: x, y, z = polygens(QQ, names='x, y, z')
sage: f = x^2 + 2*y*z
Integrate that polynomial over the polyhedron we defined:
sage: P.integrate(f)
53833/151875
This last command requires the latte_int optional package
for Sage to be installed.
The latte_int optional package
To install that package, if you installed Sage from source or from binaries downloaded from the Sage website, run the following command in a terminal:
$ sage -i latte_int
Documentation
Related
- Ask Sage question 50839: (defining polyhedron via inequalities)
- Sage Trac ticket 22824: Add see also section to integrate over a polytope
| | 2 | No.2 Revision |
Integrate a polynomial over a polyhedron
Overview
The question is about integrating a polynomial over a region defined by linear inequalities.
Such a region is called a polytope or a polyhedron.
In Sage one can construct such regions using the
Polyhedron class.
This class provides a method integrate which allows,
when the latte_int optional package is installed,
to integrate a polynomial function over a polyhedron.
In a way, the trick is: rather than start from the function
integrate and somehow specify a region, we fist define
a polyhedron and then use the method integrate to compute
the integral of a polynomial over that polyhedron.
Steps
The region is defined by five inequalities:
x + 2 y + 3 z < 2
-1 < x
x < y
y < z
z < 1
and the function to integrate is
x^2 + 2 y z
The five inequalities can be rewritten as
2 + (-1) * x + (-2) * y + (-3) * z > 0
1 + (+1) * x > 0
0 + (-1) * x + (+1) * y > 0
0 + (-1) * y + (+1) * z > 0
1 + (-1) * z > 0
Encode these inequalities in Sage as
sage: ieqs = [[2, -1, -2, -3],
....: [1, 1, 0, 0],
....: [0, -1, 1, 0],
....: [0, 0, -1, 1],
....: [1, 0, 0, -1], ]
Create a polyhedron from those inequalities:
sage: P = Polyhedron(ieqs=ieqs)
Define polynomial variables and a polynomial:
sage: x, y, z = polygens(QQ, names='x, y, z')
sage: f = x^2 + 2*y*z
Integrate that polynomial over the polyhedron we defined:
sage: P.integrate(f)
53833/151875
This last command requires the latte_int optional package
for Sage to be installed.
The latte_int optional package
To install that package, if you installed Sage from source or from binaries downloaded from the Sage website, run the following command in a terminal:
$ sage -i latte_int
Documentation
Related
Ask Sage question 50839: (defining polyhedron via inequalities)- Sage Trac ticket 22824: Add see also section to integrate over a polytope
| | 3 | No.3 Revision |
Integrate a polynomial over a polyhedron
Overview
The question is about integrating a polynomial over a region defined by linear inequalities.
Such a region is called a polytope or a polyhedron.
In Sage one can construct such regions using the
Polyhedron class.
This class provides a method integrate which allows,
when the latte_int optional package is installed,
to integrate a polynomial function over a polyhedron.
In a way, the trick is: rather than start from the function
integrate and somehow specify a region, we fist define
a polyhedron and then use the method integrate to compute
the integral of a polynomial over that polyhedron.
Steps
The region is defined by five inequalities:
x + 2 y + 3 z < 2
-1 < x
x < y
y < z
z < 1
and the function to integrate is
x^2 + 2 y z
The five inequalities can be rewritten as
2 + (-1) * x + (-2) * y + (-3) * z > 0
1 + (+1) * x > 0
0 + (-1) * x + (+1) * y > 0
0 + (-1) * y + (+1) * z > 0
1 + (-1) * z > 0
Encode these inequalities in Sage as
sage: ieqs = [[2, -1, -2, -3],
....: [1, 1, 0, 0],
....: [0, -1, 1, 0],
....: [0, 0, -1, 1],
....: [1, 0, 0, -1], ]
Create a polyhedron from those inequalities:
sage: P = Polyhedron(ieqs=ieqs)
Define polynomial variables and a polynomial:
sage: x, y, z = polygens(QQ, names='x, y, z')
sage: f = x^2 + 2*y*z
Integrate that polynomial over the polyhedron we defined:
sage: P.integrate(f)
53833/151875
This last command requires the latte_int optional package
for Sage to be installed.
The latte_int optional package
To install that package, if you installed Sage from source or from binaries downloaded from the Sage website, run the following command in a terminal:
$ sage -i latte_int
