Source code for sympy.concrete.products

from sympy.core import C, Expr, Mul, S, sympify
from sympy.functions.elementary.piecewise import piecewise_fold
from sympy.polys import quo, roots
from sympy.simplify import powsimp

[docs]class Product(Expr): """Represents unevaluated product. """ __slots__ = ['is_commutative'] def __new__(cls, function, *symbols, **assumptions): from sympy.integrals.integrals import _process_limits # Any embedded piecewise functions need to be brought out to the # top level so that integration can go into piecewise mode at the # earliest possible moment. function = piecewise_fold(sympify(function)) if function is S.NaN: return S.NaN if not symbols: raise ValueError("Product variables must be given") limits, sign = _process_limits(*symbols) # Only limits with lower and upper bounds are supported; the indefinite # Product is not supported if any(len(l) != 3 or None in l for l in limits): raise ValueError('Product requires values for lower and upper bounds.') obj = Expr.__new__(cls, **assumptions) arglist = [sign*function] arglist.extend(limits) obj._args = tuple(arglist) obj.is_commutative = function.is_commutative # limits already checked return obj @property def term(self): return self._args[0] function = term @property def limits(self): return self._args[1:] @property
[docs] def variables(self): """Return a list of the product variables >>> from sympy import Product >>> from sympy.abc import x, i >>> Product(x**i, (i, 1, 3)).variables [i] """ return [l[0] for l in self.limits]
[docs] def free_symbols(self): """ This method returns the symbols that will affect the value of the Product when evaluated. This is useful if one is trying to determine whether a product depends on a certain symbol or not. >>> from sympy import Product >>> from sympy.abc import x, y >>> Product(x, (x, y, 1)).free_symbols set([y]) """ from sympy.concrete.summations import _free_symbols if self.function.is_zero or self.function == 1: return set() return _free_symbols(self.function, self.limits)
[docs] def is_zero(self): """A Product is zero only if its term is zero. """ return self.term.is_zero
[docs] def is_number(self): """ Return True if the Product will result in a number, else False. sympy considers anything that will result in a number to have is_number == True. >>> from sympy import log, Product >>> from sympy.abc import x, y, z >>> log(2).is_number True >>> Product(x, (x, 1, 2)).is_number True >>> Product(y, (x, 1, 2)).is_number False >>> Product(1, (x, y, z)).is_number True >>> Product(2, (x, y, z)).is_number False """ return self.function.is_zero or self.function == 1 or not self.free_symbols
def doit(self, **hints): f = g = self.function for index, limit in enumerate(self.limits): i, a, b = limit dif = b - a if dif.is_Integer and dif < 0: a, b = b, a g = self._eval_product(f, (i, a, b)) if g is None: return Product(powsimp(f), *self.limits[index:]) else: f = g if hints.get('deep', True): return f.doit(**hints) else: return powsimp(f) def _eval_product(self, term, limits): from sympy import summation (k, a, n) = limits if k not in term.free_symbols: return term**(n - a + 1) if a == n: return term.subs(k, a) dif = n - a if dif.is_Integer: return Mul(*[term.subs(k, a + i) for i in range(dif + 1)]) elif term.is_polynomial(k): poly = term.as_poly(k) A = B = Q = S.One all_roots = roots(poly, multiple=True) for r in all_roots: A *= C.RisingFactorial(a-r, n-a+1) Q *= n - r if len(all_roots) < poly.degree(): arg = quo(poly, Q.as_poly(k)) B = Product(arg, (k, a, n)).doit() return poly.LC()**(n-a+1) * A * B elif term.is_Add: p, q = term.as_numer_denom() p = self._eval_product(p, (k, a, n)) q = self._eval_product(q, (k, a, n)) return p / q elif term.is_Mul: exclude, include = [], [] for t in term.args: p = self._eval_product(t, (k, a, n)) if p is not None: exclude.append(p) else: include.append(t) if not exclude: return None else: arg = term._new_rawargs(*include) A = Mul(*exclude) B = Product(arg, (k, a, n)).doit() return A * B elif term.is_Pow: if not term.base.has(k): s = summation(term.exp, (k, a, n)) return term.base**s elif not term.exp.has(k): p = self._eval_product(term.base, (k, a, n)) if p is not None: return p**term.exp elif isinstance(term, Product): evaluated = term.doit() f = self._eval_product(evaluated, limits) if f is None: return Product(evaluated, limits) else: return f
[docs]def product(*args, **kwargs): r""" Compute the product. The notation for symbols is similiar to the notation used in Sum or Integral. product(f, (i, a, b)) computes the product of f with respect to i from a to b, i.e., :: b _____ product(f(n), (i, a, b)) = | | f(n) | | i = a If it cannot compute the product, it returns an unevaluated Product object. Repeated products can be computed by introducing additional symbols tuples:: >>> from sympy import product, symbols >>> i, n, m, k = symbols('i n m k', integer=True) >>> product(i, (i, 1, k)) k! >>> product(m, (i, 1, k)) m**k >>> product(i, (i, 1, k), (k, 1, n)) Product(k!, (k, 1, n)) """ prod = Product(*args, **kwargs) if isinstance(prod, Product): return prod.doit(deep=False) else: return prod