+.. _reference:
+
Module Reference
================
+
+.. _reference_symbols:
+
Symbols
-------
True
+.. _reference_linexprs:
+
Linear Expressions
------------------
For example, if ``x`` is a :class:`Symbol`, then ``x + 1`` is an instance of :class:`LinExpr`.
.. class:: LinExpr(coefficients=None, constant=0)
- LinExpr(string)
+ LinExpr(string)
Return a linear expression from a dictionary or a sequence, that maps symbols to their coefficients, and a constant term.
The coefficients and the constant term must be rational numbers.
- For example, the linear expression ``x + 2y + 1`` can be constructed using one of the following instructions:
+ For example, the linear expression ``x + 2*y + 1`` can be constructed using one of the following instructions:
>>> x, y = symbols('x y')
>>> LinExpr({x: 1, y: 2}, 1)
Alternatively, linear expressions can be constructed from a string:
- >>> LinExpr('x + 2*y + 1')
+ >>> LinExpr('x + 2y + 1')
:class:`LinExpr` instances are hashable, and should be treated as immutable.
.. method:: __eq__(expr)
Test whether two linear expressions are equal.
+ Unlike methods :meth:`LinExpr.__lt__`, :meth:`LinExpr.__le__`, :meth:`LinExpr.__ge__`, :meth:`LinExpr.__gt__`, the result is a boolean value, not a polyhedron.
+ To express that two linear expressions are equal or not equal, use functions :func:`Eq` and :func:`Ne` instead.
As explained below, it is possible to create polyhedra from linear expressions using comparison methods.
>>> x, y = symbols('x y')
>>> x < y
- Le(x - y + 1, 0)
-
+ x + 1 <= y
.. method:: scaleint()
.. classmethod:: fromsympy(expr)
Create a linear expression from a :mod:`sympy` expression.
- Raise :exc:`ValueError` is the :mod:`sympy` expression is not linear.
+ Raise :exc:`TypeError` is the :mod:`sympy` expression is not linear.
.. method:: tosympy()
They are implemented by the :class:`Rational` class, that inherits from both :class:`LinExpr` and :class:`fractions.Fraction` classes.
.. class:: Rational(numerator, denominator=1)
- Rational(string)
+ Rational(string)
The first version requires that the *numerator* and *denominator* are instances of :class:`numbers.Rational` and returns a new :class:`Rational` instance with the value ``numerator/denominator``.
If the denominator is ``0``, it raises a :exc:`ZeroDivisionError`.
See the documentation of :class:`fractions.Fraction` for more information and examples.
+
+.. _reference_polyhedra:
+
Polyhedra
---------
A *convex polyhedron* (or simply "polyhedron") is the space defined by a system of linear equalities and inequalities.
This space can be unbounded.
+A *Z-polyhedron* (simply called "polyhedron" in LinPy) is the set of integer points in a convex polyhedron.
.. class:: Polyhedron(equalities, inequalities)
- Polyhedron(string)
- Polyhedron(geometric object)
+ Polyhedron(string)
+ Polyhedron(geometric object)
Return a polyhedron from two sequences of linear expressions: *equalities* is a list of expressions equal to ``0``, and *inequalities* is a list of expressions greater or equal to ``0``.
For example, the polyhedron ``0 <= x <= 2, 0 <= y <= 2`` can be constructed with:
>>> x, y = symbols('x y')
- >>> square = Polyhedron([], [x, 2 - x, y, 2 - y])
+ >>> square1 = Polyhedron([], [x, 2 - x, y, 2 - y])
+ >>> square1
+ And(0 <= x, x <= 2, 0 <= y, y <= 2)
It may be easier to use comparison operators :meth:`LinExpr.__lt__`, :meth:`LinExpr.__le__`, :meth:`LinExpr.__ge__`, :meth:`LinExpr.__gt__`, or functions :func:`Lt`, :func:`Le`, :func:`Eq`, :func:`Ge` and :func:`Gt`, using one of the following instructions:
>>> x, y = symbols('x y')
- >>> square = (0 <= x) & (x <= 2) & (0 <= y) & (y <= 2)
- >>> square = Le(0, x, 2) & Le(0, y, 2)
+ >>> square1 = (0 <= x) & (x <= 2) & (0 <= y) & (y <= 2)
+ >>> square1 = Le(0, x, 2) & Le(0, y, 2)
It is also possible to build a polyhedron from a string.
- >>> square = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
+ >>> square1 = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
Finally, a polyhedron can be constructed from a :class:`GeometricObject` instance, calling the :meth:`GeometricObject.aspolyedron` method.
This way, it is possible to compute the polyhedral hull of a :class:`Domain` instance, i.e., the convex hull of two polyhedra:
- >>> square = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
- >>> square2 = Polyhedron('2 <= x <= 4, 2 <= y <= 4')
- >>> Polyhedron(square | square2)
+ >>> square1 = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
+ >>> square2 = Polyhedron('1 <= x <= 3, 1 <= y <= 3')
+ >>> Polyhedron(square1 | square2)
+ And(0 <= x, 0 <= y, x <= y + 2, y <= x + 2, x <= 3, y <= 3)
A polyhedron is a :class:`Domain` instance, and, therefore, inherits the functionalities of this class.
It is also a :class:`GeometricObject` instance.
The tuple of constraints, i.e., equalities and inequalities.
This is semantically equivalent to: ``equalities + inequalities``.
+ .. method:: convex_union(polyhedron[, ...])
+
+ Return the convex union of two or more polyhedra.
+
+ .. method:: asinequalities()
+
+ Express the polyhedron using inequalities, given as a list of expressions greater or equal to 0.
+
.. method:: widen(polyhedron)
Compute the *standard widening* of two polyhedra, à la Halbwachs.
+ In its current implementation, this method is slow and should not be used on large polyhedra.
+
.. data:: Empty
The universe polyhedron, whose set of constraints is always satisfiable, i.e. is empty.
+
+.. _reference_domains:
+
Domains
-------
A *domain* is a union of polyhedra.
-Unlike polyhedra, domains allow exact computation of union and complementary operations.
+Unlike polyhedra, domains allow exact computation of union, subtraction and complementary operations.
.. class:: Domain(*polyhedra)
- Domain(string)
- Domain(geometric object)
+ Domain(string)
+ Domain(geometric object)
Return a domain from a sequence of polyhedra.
- >>> square = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
- >>> square2 = Polyhedron('2 <= x <= 4, 2 <= y <= 4')
- >>> dom = Domain([square, square2])
+ >>> square1 = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
+ >>> square2 = Polyhedron('1 <= x <= 3, 1 <= y <= 3')
+ >>> dom = Domain(square1, square2)
+ >>> dom
+ Or(And(x <= 2, 0 <= x, y <= 2, 0 <= y), And(x <= 3, 1 <= x, y <= 3, 1 <= y))
- It is also possible to build domains from polyhedra using arithmetic operators :meth:`Domain.__and__`, :meth:`Domain.__or__` or functions :func:`And` and :func:`Or`, using one of the following instructions:
+ It is also possible to build domains from polyhedra using arithmetic operators :meth:`Domain.__or__`, :meth:`Domain.__invert__` or functions :func:`Or` and :func:`Not`, using one of the following instructions:
- >>> square = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
- >>> square2 = Polyhedron('2 <= x <= 4, 2 <= y <= 4')
- >>> dom = square | square2
- >>> dom = Or(square, square2)
+ >>> dom = square1 | square2
+ >>> dom = Or(square1, square2)
Alternatively, a domain can be built from a string:
- >>> dom = Domain('0 <= x <= 2, 0 <= y <= 2; 2 <= x <= 4, 2 <= y <= 4')
+ >>> dom = Domain('0 <= x <= 2, 0 <= y <= 2; 1 <= x <= 3, 1 <= y <= 3')
Finally, a domain can be built from a :class:`GeometricObject` instance, calling the :meth:`GeometricObject.asdomain` method.
Convert the domain to a sympy expression.
+.. _reference_operators:
+
Comparison and Logic Operators
------------------------------
.. function:: Ne(expr1, expr2[, expr3, ...])
Create the domain such that ``expr1 != expr2 != expr3 ...``.
- The result is a :class:`Domain`, not a :class:`Polyhedron`.
+ The result is a :class:`Domain` object, not a :class:`Polyhedron`.
.. function:: Ge(expr1, expr2[, expr3, ...])
The following functions combine :class:`Polyhedron` or :class:`Domain` instances using logic operators:
-.. function:: Or(domain1, domain2[, ...])
-
- Create the union domain of the domains given in arguments.
-
.. function:: And(domain1, domain2[, ...])
Create the intersection domain of the domains given in arguments.
+.. function:: Or(domain1, domain2[, ...])
+
+ Create the union domain of the domains given in arguments.
+
.. function:: Not(domain)
Create the complementary domain of the domain given in argument.
+.. _reference_geometry:
+
Geometric Objects
-----------------
.. method:: cross(vector)
Compute the cross product of two 3D vectors.
- If either one of the vectors is not tridimensional, a :exc:`ValueError` exception is raised.
+ If either one of the vectors is not three-dimensional, a :exc:`ValueError` exception is raised.
.. method:: dot(vector)
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