X-Git-Url: https://scm.cri.ensmp.fr/git/linpy.git/blobdiff_plain/a08ebc700e22f6aee8147cb5b5323a6c040b12db..dbe638f61f1927ee6f3dffaaf2dc34418725f50a:/doc/reference.rst diff --git a/doc/reference.rst b/doc/reference.rst index 8184c43..8e36056 100644 --- a/doc/reference.rst +++ b/doc/reference.rst @@ -2,6 +2,7 @@ Module Reference ================ + Symbols ------- @@ -12,7 +13,7 @@ They correspond to mathematical variables. 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 @@ -46,12 +47,12 @@ They correspond to mathematical variables. >>> 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. @@ -79,8 +80,8 @@ For example, if ``x`` is a :class:`Symbol`, then ``x + 1`` is an instance of :cl .. class:: LinExpr(coefficients=None, constant=0) 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: @@ -88,7 +89,7 @@ For example, if ``x`` is a :class:`Symbol`, then ``x + 1`` is an instance of :cl >>> 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 @@ -103,7 +104,7 @@ For example, if ``x`` is a :class:`Symbol`, then ``x + 1`` is an instance of :cl 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. @@ -161,9 +162,9 @@ For example, if ``x`` is a :class:`Symbol`, then ``x + 1`` is an instance of :cl 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. @@ -172,16 +173,15 @@ For example, if ``x`` is a :class:`Symbol`, then ``x + 1`` is an instance of :cl >>> x < y Le(x - y + 1, 0) - .. 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 @@ -203,7 +203,7 @@ For example, if ``x`` is a :class:`Symbol`, then ``x + 1`` is an instance of :cl .. 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() @@ -216,21 +216,22 @@ They are implemented by the :class:`Rational` class, that inherits from both :cl .. class:: Rational(numerator, denominator=1) 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. + 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. .. class:: Polyhedron(equalities, inequalities) @@ -260,7 +261,7 @@ This space can be unbounded. >>> square2 = Polyhedron('2 <= x <= 4, 2 <= y <= 4') >>> Polyhedron(square | square2) - 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 @@ -278,9 +279,19 @@ This space can be unbounded. 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, Ã  la Halbwachs. + Compute the *standard widening* of two polyhedra, Ã  la Halbwachs. + + In its current implementation, this method is slow and should not be used on large polyhedra. .. data:: Empty @@ -291,6 +302,7 @@ This space can be unbounded. The universe polyhedron, whose set of constraints is always satisfiable, i.e. is empty. + Domains ------- @@ -329,7 +341,7 @@ Unlike polyhedra, domains allow exact computation of union and complementary ope .. 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 @@ -435,7 +447,7 @@ Unlike polyhedra, domains allow exact computation of union and complementary ope .. 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() @@ -476,7 +488,7 @@ Unlike polyhedra, domains allow exact computation of union and complementary ope 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, ...]) @@ -493,7 +505,7 @@ The following functions allow to create :class:`Polyhedron` or :class:`Domain` i .. 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, ...]) @@ -503,15 +515,15 @@ The following functions allow to create :class:`Polyhedron` or :class:`Domain` i 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) @@ -545,7 +557,7 @@ 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: @@ -554,7 +566,7 @@ Geometric Objects >>> 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. @@ -567,7 +579,7 @@ Geometric Objects 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. @@ -590,12 +602,12 @@ Geometric Objects .. 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) @@ -604,11 +616,12 @@ Geometric Objects .. 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 @@ -619,7 +632,7 @@ Geometric Objects 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. @@ -641,13 +654,13 @@ Geometric Objects 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. @@ -656,6 +669,18 @@ Geometric Objects 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. @@ -664,31 +689,19 @@ 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) 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()