+.. _reference:
+
Module Reference
================
+
+.. _reference_symbols:
+
Symbols
-------
Return a symbol with the name string given in argument.
Alternatively, the function :func:`symbols` allows to create several symbols at once.
- Symbols are instances of class :class:`LinExpr` and, as such, inherit its functionalities.
+ Symbols are instances of class :class:`LinExpr` and inherit its functionalities.
>>> x = Symbol('x')
>>> x
>>> x, y = symbols(['x', 'y'])
-Sometimes, you need to have a unique symbol, for example as a temporary one in some calculation, which is going to be substituted for something else at the end anyway.
+Sometimes you need to have a unique symbol. For example, you might need a temporary one in some calculation, which is going to be substituted for something else at the end anyway.
This is achieved using ``Dummy('x')``.
.. class:: Dummy(name=None)
- A variation of :class:`Symbol` which are all unique, identified by an internal count index.
+ A variation of :class:`Symbol` in which all symbols are unique and identified by an internal count index.
If a name is not supplied then a string value of the count index will be used.
This is useful when a unique, temporary variable is needed and the name of the variable used in the expression is not important.
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 must be rational numbers.
+ 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)
>>> LinExpr([(x, 1), (y, 2)], 1)
- although it may be easier to use overloaded operators:
+ However, it may be easier to use overloaded operators:
>>> x, y = symbols('x y')
>>> x + 2*y + 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.
A linear expression with no symbol, only a constant term, is automatically subclassed as a :class:`Rational` instance.
.. method:: coefficient(symbol)
- __getitem__(symbol)
+ __getitem__(symbol)
Return the coefficient value of the given symbol, or ``0`` if the symbol does not appear in the expression.
.. 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.
.. method:: __lt__(expr)
- __le__(expr)
- __ge__(expr)
- __gt__(expr)
+ __le__(expr)
+ __ge__(expr)
+ __gt__(expr)
Create a new :class:`Polyhedron` instance whose unique constraint is the comparison between two linear expressions.
As an alternative, functions :func:`Lt`, :func:`Le`, :func:`Ge` and :func:`Gt` can be used.
>>> x, y = symbols('x y')
>>> x < y
- Le(x - y + 1, 0)
-
+ x + 1 <= y
.. method:: scaleint()
Return the expression multiplied by its lowest common denominator to make all values integer.
.. method:: subs(symbol, expression)
- subs(pairs)
+ subs(pairs)
Substitute the given symbol by an expression and return the resulting expression.
- Raise :exc:`TypeError` is the resulting expression is not linear.
+ Raise :exc:`TypeError` if the resulting expression is not linear.
>>> x, y = symbols('x y')
>>> e = x + 2*y + 1
.. 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 *numerator* and *denominator* are instances of :class:`numbers.Rational` and returns a new :class:`Rational` instance with value ``numerator/denominator``.
- If denominator is ``0``, it raises a :exc:`ZeroDivisionError`.
+ 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`.
The other version of the constructor expects a string.
The usual form for this instance is::
[sign] numerator ['/' denominator]
- where the optional ``sign`` may be either '+' or '-' and ``numerator`` and ``denominator`` (if present) are strings of decimal digits.
+ where the optional ``sign`` may be either '+' or '-' and the ``numerator`` and ``denominator`` (if present) are strings of decimal digits.
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.
+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, as such, inherits the functionalities of this class.
+ A polyhedron is a :class:`Domain` instance, and, therefore, inherits the functionalities of this class.
It is also a :class:`GeometricObject` instance.
.. attribute:: equalities
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.
+ 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.
.. attribute:: symbols
- The tuple of symbols present in the domain expression, sorted according to :meth:`Symbol.sortkey`.
+ The tuple of symbols present in the domain equations, sorted according to :meth:`Symbol.sortkey`.
.. attribute:: dimension
.. method:: __contains__(point)
- Return ``True`` if the :class:`Point` is contained within the domain.
+ Return ``True`` if the point is contained within the domain.
.. method:: faces()
Convert the domain to a sympy expression.
+.. _reference_operators:
+
Comparison and Logic Operators
------------------------------
-The following functions allow to create :class:`Polyhedron` or :class:`Domain` instances by comparison of :class:`LinExpr` instances:
+The following functions create :class:`Polyhedron` or :class:`Domain` instances using the comparisons of two or more :class:`LinExpr` instances:
.. function:: Lt(expr1, expr2[, expr3, ...])
.. 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, ...])
Create the polyhedron with constraints ``expr1 > expr2 > expr3 ...``.
-The following functions allow to combine :class:`Polyhedron` or :class:`Domain` instances using logic operators:
+The following functions combine :class:`Polyhedron` or :class:`Domain` instances using logic operators:
-.. function:: Or(domain1, domain2[, ...])
+.. function:: And(domain1, domain2[, ...])
- Create the union domain of domains given in arguments.
+ Create the intersection domain of the domains given in arguments.
-.. function:: And(domain1, domain2[, ...])
+.. function:: Or(domain1, domain2[, ...])
- Create the intersection domain of domains given in arguments.
+ 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
-----------------
.. class:: Point(coordinates)
- Create a point from a dictionnary or a sequence that maps symbols to their coordinates.
+ Create a point from a dictionary or a sequence that maps the symbols to their coordinates.
Coordinates must be rational numbers.
For example, the point ``(x: 1, y: 2)`` can be constructed using one of the following instructions:
>>> p = Point({x: 1, y: 2})
>>> p = Point([(x, 1), (y, 2)])
- :class:`Point` instances are hashable, and should be treated as immutable.
+ :class:`Point` instances are hashable and should be treated as immutable.
A point is a :class:`GeometricObject` instance.
The dimension of the point, i.e. the number of symbols present in it.
.. method:: coordinate(symbol)
- __getitem__(symbol)
+ __getitem__(symbol)
Return the coordinate value of the given symbol.
Raise :exc:`KeyError` if the symbol is not involved in the point.
.. method:: __add__(vector)
- Translate the point by a :class:`Vector` instance and return the resulting point.
+ Translate the point by a :class:`Vector` object and return the resulting point.
.. method:: __sub__(point)
- __sub__(vector)
+ __sub__(vector)
- The first version substract a point from another and return the resulting vector.
+ The first version substracts a point from another and returns the resulting vector.
The second version translates the point by the opposite vector of *vector* and returns the resulting point.
.. method:: __eq__(point)
.. class:: Vector(coordinates)
+ Vector(point1, point2)
- Create a point from a dictionnary or a sequence that maps symbols to their coordinates, similarly to :meth:`Point`.
- Coordinates must be rational numbers.
+ The first version creates a vector from a dictionary or a sequence that maps the symbols to their coordinates, similarly to :meth:`Point`.
+ The second version creates a vector between two points.
- :class:`Vector` instances are hashable, and should be treated as immutable.
+ :class:`Vector` instances are hashable and should be treated as immutable.
.. attribute:: symbols
The dimension of the point, i.e. the number of symbols present in it.
.. method:: coordinate(symbol)
- __getitem__(symbol)
+ __getitem__(symbol)
Return the coordinate value of the given symbol.
Raise :exc:`KeyError` if the symbol is not involved in the point.
Return ``True`` if not all coordinates are ``0``.
.. method:: __add__(point)
- __add__(vector)
+ __add__(vector)
The first version translates the point *point* to the vector and returns the resulting point.
The second version adds vector *vector* to the vector and returns the resulting vector.
.. method:: __sub__(point)
- __sub__(vector)
+ __sub__(vector)
The first version substracts a point from a vector and returns the resulting point.
The second version returns the difference vector between two vectors.
Return the opposite vector.
+ .. method:: __mul__(value)
+
+ Multiply the vector by a scalar value and return the resulting vector.
+
+ .. method:: __truediv__(value)
+
+ Divide the vector by a scalar value and return the resulting vector.
+
+ .. method:: __eq__(vector)
+
+ Test whether two vectors are equal.
+
.. method:: angle(vector)
Retrieve the angle required to rotate the vector into the vector passed in argument.
.. 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)
Compute the dot product of two vectors.
- .. method:: __eq__(vector)
-
- Test whether two vectors are equal.
-
- .. method:: __mul__(value)
-
- Multiply the vector by a scalar value and return the resulting vector.
-
- .. method:: __truediv__(value)
-
- Divide the vector by a scalar value and return the resulting vector.
-
.. method:: norm()
Return the norm of the vector.
.. method:: norm2()
- Return the square norm of the vector.
+ Return the squared norm of the vector.
.. method:: asunit()
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