import numbers
import re
-from collections import OrderedDict, defaultdict, Mapping
+from collections import defaultdict, Mapping, OrderedDict
from fractions import Fraction, gcd
__all__ = [
+ 'Dummy',
'LinExpr',
- 'Symbol', 'Dummy', 'symbols',
'Rational',
+ 'Symbol',
+ 'symbols',
]
class LinExpr:
"""
- This class implements linear expressions.
+ A linear expression consists of a list of coefficient-variable pairs
+ that capture the linear terms, plus a constant term. Linear expressions
+ are used to build constraints. They are temporary objects that typically
+ have short lifespans.
+
+ Linear expressions are generally built using overloaded operators. For
+ example, if x is a Symbol, then x + 1 is an instance of LinExpr.
+
+ LinExpr instances are hashable, and should be treated as immutable.
"""
def __new__(cls, coefficients=None, constant=0):
"""
- Create a new expression.
+ 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 + 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)
+
+ 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 + 2y + 1')
+
+ A linear expression with a single symbol of coefficient 1 and no
+ constant term is automatically subclassed as a Symbol instance. A
+ linear expression with no symbol, only a constant term, is
+ automatically subclassed as a Rational instance.
"""
if isinstance(coefficients, str):
if constant != 0:
symbol, coefficient = coefficients[0]
if coefficient == 1:
return symbol
- coefficients = [(symbol, Fraction(coefficient))
- for symbol, coefficient in coefficients if coefficient != 0]
+ coefficients = [(symbol_, Fraction(coefficient_))
+ for symbol_, coefficient_ in coefficients
+ if coefficient_ != 0]
coefficients.sort(key=lambda item: item[0].sortkey())
self = object().__new__(cls)
self._coefficients = OrderedDict(coefficients)
def coefficient(self, symbol):
"""
- Return the coefficient value of the given symbol.
+ Return the coefficient value of the given symbol, or 0 if the symbol
+ does not appear in the expression.
"""
if not isinstance(symbol, Symbol):
raise TypeError('symbol must be a Symbol instance')
- return Rational(self._coefficients.get(symbol, 0))
+ return self._coefficients.get(symbol, Fraction(0))
__getitem__ = coefficient
def coefficients(self):
"""
- Return a list of the coefficients of an expression
+ Iterate over the pairs (symbol, value) of linear terms in the
+ expression. The constant term is ignored.
"""
- for symbol, coefficient in self._coefficients.items():
- yield symbol, Rational(coefficient)
+ yield from self._coefficients.items()
@property
def constant(self):
"""
- Return the constant value of an expression.
+ The constant term of the expression.
"""
- return Rational(self._constant)
+ return self._constant
@property
def symbols(self):
"""
- Return a list of symbols in an expression.
+ The tuple of symbols present in the expression, sorted according to
+ Symbol.sortkey().
"""
return self._symbols
@property
def dimension(self):
"""
- Create and return a new linear expression from a string or a list of coefficients and a constant.
+ The dimension of the expression, i.e. the number of symbols present in
+ it.
"""
return self._dimension
def isconstant(self):
"""
- Return true if an expression is a constant.
+ Return True if the expression only consists of a constant term. In this
+ case, it is a Rational instance.
"""
return False
def issymbol(self):
"""
- Return true if an expression is a symbol.
+ Return True if an expression only consists of a symbol with coefficient
+ 1. In this case, it is a Symbol instance.
"""
return False
def values(self):
"""
- Return the coefficient and constant values of an expression.
+ Iterate over the coefficient values in the expression, and the constant
+ term.
"""
- for coefficient in self._coefficients.values():
- yield Rational(coefficient)
- yield Rational(self._constant)
+ yield from self._coefficients.values()
+ yield self._constant
def __bool__(self):
return True
@_polymorphic
def __add__(self, other):
"""
- Return the sum of two expressions.
+ Return the sum of two linear expressions.
"""
coefficients = defaultdict(Fraction, self._coefficients)
for symbol, coefficient in other._coefficients.items():
@_polymorphic
def __sub__(self, other):
"""
- Return the difference between two expressions.
+ Return the difference between two linear expressions.
"""
coefficients = defaultdict(Fraction, self._coefficients)
for symbol, coefficient in other._coefficients.items():
def __mul__(self, other):
"""
- Return the product of two expressions if other is a rational number.
+ Return the product of the linear expression by a rational.
"""
if isinstance(other, numbers.Rational):
- coefficients = ((symbol, coefficient * other)
+ coefficients = (
+ (symbol, coefficient * other)
for symbol, coefficient in self._coefficients.items())
constant = self._constant * other
return LinExpr(coefficients, constant)
__rmul__ = __mul__
def __truediv__(self, other):
+ """
+ Return the quotient of the linear expression by a rational.
+ """
if isinstance(other, numbers.Rational):
- coefficients = ((symbol, coefficient / other)
+ coefficients = (
+ (symbol, coefficient / other)
for symbol, coefficient in self._coefficients.items())
constant = self._constant / other
return LinExpr(coefficients, constant)
@_polymorphic
def __eq__(self, other):
"""
- Test whether two expressions are equal
+ Test whether two linear expressions are equal. Unlike methods
+ LinExpr.__lt__(), LinExpr.__le__(), LinExpr.__ge__(), LinExpr.__gt__(),
+ the result is a boolean value, not a polyhedron. To express that two
+ linear expressions are equal or not equal, use functions Eq() and Ne()
+ instead.
"""
- return isinstance(other, LinExpr) and \
- self._coefficients == other._coefficients and \
+ return self._coefficients == other._coefficients and \
self._constant == other._constant
- def __le__(self, other):
- from .polyhedra import Le
- return Le(self, other)
-
+ @_polymorphic
def __lt__(self, other):
- from .polyhedra import Lt
- return Lt(self, other)
+ from .polyhedra import Polyhedron
+ return Polyhedron([], [other - self - 1])
+
+ @_polymorphic
+ def __le__(self, other):
+ from .polyhedra import Polyhedron
+ return Polyhedron([], [other - self])
+ @_polymorphic
def __ge__(self, other):
- from .polyhedra import Ge
- return Ge(self, other)
+ from .polyhedra import Polyhedron
+ return Polyhedron([], [self - other])
+ @_polymorphic
def __gt__(self, other):
- from .polyhedra import Gt
- return Gt(self, other)
+ from .polyhedra import Polyhedron
+ return Polyhedron([], [self - other - 1])
def scaleint(self):
"""
- Multiply an expression by a scalar to make all coefficients integer values.
+ Return the expression multiplied by its lowest common denominator to
+ make all values integer.
"""
- lcm = functools.reduce(lambda a, b: a*b // gcd(a, b),
- [value.denominator for value in self.values()])
- return self * lcm
+ lcd = functools.reduce(lambda a, b: a*b // gcd(a, b),
+ [value.denominator for value in self.values()])
+ return self * lcd
def subs(self, symbol, expression=None):
"""
- Subsitute symbol by expression in equations and return the resulting
- expression.
+ Substitute the given symbol by an expression and return the resulting
+ expression. Raise TypeError if the resulting expression is not linear.
+
+ >>> x, y = symbols('x y')
+ >>> e = x + 2*y + 1
+ >>> e.subs(y, x - 1)
+ 3*x - 1
+
+ To perform multiple substitutions at once, pass a sequence or a
+ dictionary of (old, new) pairs to subs.
+
+ >>> e.subs({x: y, y: x})
+ 2*x + y + 1
"""
if expression is None:
- if isinstance(symbol, Mapping):
- symbol = symbol.items()
- substitutions = symbol
+ substitutions = dict(symbol)
else:
- substitutions = [(symbol, expression)]
- result = self
- for symbol, expression in substitutions:
+ substitutions = {symbol: expression}
+ for symbol in substitutions:
if not isinstance(symbol, Symbol):
raise TypeError('symbols must be Symbol instances')
- coefficients = [(othersymbol, coefficient)
- for othersymbol, coefficient in result._coefficients.items()
- if othersymbol != symbol]
- coefficient = result._coefficients.get(symbol, 0)
- constant = result._constant
- result = LinExpr(coefficients, constant) + coefficient*expression
+ result = Rational(self._constant)
+ for symbol, coefficient in self._coefficients.items():
+ expression = substitutions.get(symbol, symbol)
+ result += coefficient * expression
return result
@classmethod
return left / right
raise SyntaxError('invalid syntax')
- _RE_NUM_VAR = re.compile(r'(\d+|\))\s*([^\W\d_]\w*|\()')
+ _RE_NUM_VAR = re.compile(r'(\d+|\))\s*([^\W\d]\w*|\()')
@classmethod
def fromstring(cls, string):
"""
- Create an expression from a string.
+ Create an expression from a string. Raise SyntaxError if the string is
+ not properly formatted.
"""
- # add implicit multiplication operators, e.g. '5x' -> '5*x'
+ # Add implicit multiplication operators, e.g. '5x' -> '5*x'.
string = LinExpr._RE_NUM_VAR.sub(r'\1*\2', string)
tree = ast.parse(string, 'eval')
- return cls._fromast(tree)
+ expression = cls._fromast(tree)
+ if not isinstance(expression, cls):
+ raise SyntaxError('invalid syntax')
+ return expression
def __repr__(self):
string = ''
string += ' - {}'.format(-constant)
return string
- def _repr_latex_(self):
- string = ''
- for i, (symbol, coefficient) in enumerate(self.coefficients()):
- if coefficient == 1:
- if i != 0:
- string += ' + '
- elif coefficient == -1:
- string += '-' if i == 0 else ' - '
- elif i == 0:
- string += '{}'.format(coefficient._repr_latex_().strip('$'))
- elif coefficient > 0:
- string += ' + {}'.format(coefficient._repr_latex_().strip('$'))
- elif coefficient < 0:
- string += ' - {}'.format((-coefficient)._repr_latex_().strip('$'))
- string += '{}'.format(symbol._repr_latex_().strip('$'))
- constant = self.constant
- if len(string) == 0:
- string += '{}'.format(constant._repr_latex_().strip('$'))
- elif constant > 0:
- string += ' + {}'.format(constant._repr_latex_().strip('$'))
- elif constant < 0:
- string += ' - {}'.format((-constant)._repr_latex_().strip('$'))
- return '$${}$$'.format(string)
-
def _parenstr(self, always=False):
string = str(self)
if not always and (self.isconstant() or self.issymbol()):
return '({})'.format(string)
@classmethod
- def fromsympy(cls, expr):
+ def fromsympy(cls, expression):
"""
- Convert sympy object to an expression.
+ Create a linear expression from a SymPy expression. Raise TypeError is
+ the sympy expression is not linear.
"""
import sympy
coefficients = []
constant = 0
- for symbol, coefficient in expr.as_coefficients_dict().items():
+ for symbol, coefficient in expression.as_coefficients_dict().items():
coefficient = Fraction(coefficient.p, coefficient.q)
if symbol == sympy.S.One:
constant = coefficient
+ elif isinstance(symbol, sympy.Dummy):
+ # We cannot properly convert dummy symbols with respect to
+ # symbol equalities.
+ raise TypeError('cannot convert dummy symbols')
elif isinstance(symbol, sympy.Symbol):
symbol = Symbol(symbol.name)
coefficients.append((symbol, coefficient))
else:
- raise ValueError('non-linear expression: {!r}'.format(expr))
- return LinExpr(coefficients, constant)
+ raise TypeError('non-linear expression: {!r}'.format(
+ expression))
+ expression = LinExpr(coefficients, constant)
+ if not isinstance(expression, cls):
+ raise TypeError('cannot convert to a {} instance'.format(
+ cls.__name__))
+ return expression
def tosympy(self):
"""
- Return an expression as a sympy object.
+ Convert the linear expression to a SymPy expression.
"""
import sympy
- expr = 0
+ expression = 0
for symbol, coefficient in self.coefficients():
term = coefficient * sympy.Symbol(symbol.name)
- expr += term
- expr += self.constant
- return expr
+ expression += term
+ expression += self.constant
+ return expression
class Symbol(LinExpr):
+ """
+ Symbols are the basic components to build expressions and constraints.
+ They correspond to mathematical variables. Symbols are instances of
+ class LinExpr and inherit its functionalities.
+
+ Two instances of Symbol are equal if they have the same name.
+ """
+
+ __slots__ = (
+ '_name',
+ '_constant',
+ '_symbols',
+ '_dimension',
+ )
def __new__(cls, name):
"""
- Create and return a symbol from a string.
+ Return a symbol with the name string given in argument.
"""
if not isinstance(name, str):
raise TypeError('name must be a string')
+ node = ast.parse(name)
+ try:
+ name = node.body[0].value.id
+ except (AttributeError, SyntaxError):
+ raise SyntaxError('invalid syntax')
self = object().__new__(cls)
- self._name = name.strip()
- self._coefficients = {self: Fraction(1)}
+ self._name = name
self._constant = Fraction(0)
self._symbols = (self,)
self._dimension = 1
return self
+ @property
+ def _coefficients(self):
+ # This is not implemented as an attribute, because __hash__ is not
+ # callable in __new__ in class Dummy.
+ return {self: Fraction(1)}
+
@property
def name(self):
+ """
+ The name of the symbol.
+ """
return self._name
def __hash__(self):
return hash(self.sortkey())
def sortkey(self):
+ """
+ Return a sorting key for the symbol. It is useful to sort a list of
+ symbols in a consistent order, as comparison functions are overridden
+ (see the documentation of class LinExpr).
+
+ >>> sort(symbols, key=Symbol.sortkey)
+ """
return self.name,
def issymbol(self):
return True
def __eq__(self, other):
- return self.sortkey() == other.sortkey()
+ if isinstance(other, Symbol):
+ return self.sortkey() == other.sortkey()
+ return NotImplemented
def asdummy(self):
"""
- Return a symbol as a Dummy Symbol.
+ Return a new Dummy symbol instance with the same name.
"""
return Dummy(self.name)
- @classmethod
- def _fromast(cls, node):
- if isinstance(node, ast.Module) and len(node.body) == 1:
- return cls._fromast(node.body[0])
- elif isinstance(node, ast.Expr):
- return cls._fromast(node.value)
- elif isinstance(node, ast.Name):
- return Symbol(node.id)
- raise SyntaxError('invalid syntax')
-
def __repr__(self):
return self.name
- def _repr_latex_(self):
- return '$${}$$'.format(self.name)
- @classmethod
- def fromsympy(cls, expr):
- import sympy
- if isinstance(expr, sympy.Dummy):
- return Dummy(expr.name)
- elif isinstance(expr, sympy.Symbol):
- return Symbol(expr.name)
- else:
- raise TypeError('expr must be a sympy.Symbol instance')
+def symbols(names):
+ """
+ This function returns a tuple of symbols whose names are taken from a comma
+ or whitespace delimited string, or a sequence of strings. It is useful to
+ define several symbols at once.
+
+ >>> x, y = symbols('x y')
+ >>> x, y = symbols('x, y')
+ >>> x, y = symbols(['x', 'y'])
+ """
+ if isinstance(names, str):
+ names = names.replace(',', ' ').split()
+ return tuple(Symbol(name) for name in names)
class Dummy(Symbol):
"""
- This class returns a dummy symbol to ensure that no variables are repeated in an expression
+ A variation of 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.
+
+ Unlike Symbol, Dummy instances with the same name are not equal:
+
+ >>> x = Symbol('x')
+ >>> x1, x2 = Dummy('x'), Dummy('x')
+ >>> x == x1
+ False
+ >>> x1 == x2
+ False
+ >>> x1 == x1
+ True
"""
+
_count = 0
def __new__(cls, name=None):
"""
- Create and return a new dummy symbol.
+ Return a fresh dummy symbol with the name string given in argument.
"""
if name is None:
name = 'Dummy_{}'.format(Dummy._count)
- elif not isinstance(name, str):
- raise TypeError('name must be a string')
- self = object().__new__(cls)
+ self = super().__new__(cls, name)
self._index = Dummy._count
- self._name = name.strip()
- self._coefficients = {self: Fraction(1)}
- self._constant = Fraction(0)
- self._symbols = (self,)
- self._dimension = 1
Dummy._count += 1
return self
def __repr__(self):
return '_{}'.format(self.name)
- def _repr_latex_(self):
- return '$${}_{{{}}}$$'.format(self.name, self._index)
-
-
-def symbols(names):
- """
- Transform strings into instances of the Symbol class
- """
- if isinstance(names, str):
- names = names.replace(',', ' ').split()
- return tuple(Symbol(name) for name in names)
-
class Rational(LinExpr, Fraction):
"""
- This class represents integers and rational numbers of any size.
+ A particular case of linear expressions are rational values, i.e. linear
+ expressions consisting only of a constant term, with no symbol. They are
+ implemented by the Rational class, that inherits from both LinExpr and
+ fractions.Fraction classes.
"""
+ __slots__ = (
+ '_coefficients',
+ '_constant',
+ '_symbols',
+ '_dimension',
+ ) + Fraction.__slots__
+
def __new__(cls, numerator=0, denominator=None):
self = object().__new__(cls)
self._coefficients = {}
@property
def constant(self):
- """
- Return rational as a constant.
- """
return self
def isconstant(self):
- """
- Test whether a value is a constant.
- """
return True
def __bool__(self):
return '{!r}'.format(self.numerator)
else:
return '{!r}/{!r}'.format(self.numerator, self.denominator)
-
- def _repr_latex_(self):
- if self.denominator == 1:
- return '$${}$$'.format(self.numerator)
- elif self.numerator < 0:
- return '$$-\\frac{{{}}}{{{}}}$$'.format(-self.numerator,
- self.denominator)
- else:
- return '$$\\frac{{{}}}{{{}}}$$'.format(self.numerator,
- self.denominator)
-
- @classmethod
- def fromsympy(cls, expr):
- """
- Create a rational object from a sympy expression
- """
- import sympy
- if isinstance(expr, sympy.Rational):
- return Rational(expr.p, expr.q)
- elif isinstance(expr, numbers.Rational):
- return Rational(expr)
- else:
- raise TypeError('expr must be a sympy.Rational instance')
projects / linpy.git / blobdiff