X-Git-Url: https://scm.cri.ensmp.fr/git/linpy.git/blobdiff_plain/7f4790ef28ae6335966de93be23a8be9320d8cde..a16251fd5fb481e97f05fd488ad718ba2147396b:/doc/reference.rst diff --git a/doc/reference.rst b/doc/reference.rst index 4dcfbdc..4a2c419 100644 --- a/doc/reference.rst +++ b/doc/reference.rst @@ -1,7 +1,12 @@ +.. _reference: + Module Reference ================ + +.. _reference_symbols: + Symbols ------- @@ -67,6 +72,8 @@ This is achieved using ``Dummy('x')``. True +.. _reference_linexprs: + Linear Expressions ------------------ @@ -77,12 +84,12 @@ Linear expressions are generally built using overloaded operators. 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) @@ -95,7 +102,7 @@ For example, if ``x`` is a :class:`Symbol`, then ``x + 1`` is an instance of :cl 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. @@ -103,7 +110,7 @@ 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. @@ -148,11 +155,11 @@ For example, if ``x`` is a :class:`Symbol`, then ``x + 1`` is an instance of :cl .. method:: __mul__(value) - Return the product of the linear expression as a rational. + Return the product of the linear expression by a rational. .. method:: __truediv__(value) - Return the quotient of the linear expression as a rational. + Return the quotient of the linear expression by a rational. .. method:: __eq__(expr) @@ -161,24 +168,23 @@ 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. >>> 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` if the resulting expression is not linear. @@ -203,7 +209,7 @@ 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() @@ -214,7 +220,7 @@ Apart from :mod:`Symbol`, a particular case of linear expressions are rational v 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`. @@ -227,6 +233,9 @@ They are implemented by the :class:`Rational` class, that inherits from both :cl See the documentation of :class:`fractions.Fraction` for more information and examples. + +.. _reference_polyhedra: + Polyhedra --------- @@ -234,31 +243,34 @@ A *convex polyhedron* (or simply "polyhedron") is the space defined by a system This space can be unbounded. .. 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. @@ -278,9 +290,19 @@ 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 @@ -291,32 +313,35 @@ This space can be unbounded. 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. @@ -329,7 +354,7 @@ 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 +460,7 @@ 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() @@ -473,6 +498,8 @@ Unlike polyhedra, domains allow exact computation of union and complementary ope Convert the domain to a sympy expression. +.. _reference_operators: + Comparison and Logic Operators ------------------------------ @@ -493,7 +520,7 @@ The following functions create :class:`Polyhedron` or :class:`Domain` instances .. 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, ...]) @@ -505,19 +532,21 @@ The following functions create :class:`Polyhedron` or :class:`Domain` instances 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 ----------------- @@ -567,7 +596,7 @@ 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. @@ -590,10 +619,10 @@ 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 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. @@ -604,9 +633,10 @@ Geometric Objects .. class:: Vector(coordinates) + Vector(point1, point2) - Create a point from a dictionary or a sequence that maps the symbols to their coordinates, similar 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. @@ -619,7 +649,7 @@ 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 +671,13 @@ 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 +686,18 @@ 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 +706,19 @@ 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()