Source code for sympy.concrete.summations

from sympy.core.add import Add
from sympy.core.basic import C
from sympy.core.containers import Tuple
from sympy.core.expr import Expr
from sympy.core.function import Derivative
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, Wild)
from sympy.core.sympify import sympify
from sympy.concrete.gosper import gosper_sum
from sympy.functions.elementary.piecewise import piecewise_fold, Piecewise
from sympy.polys import apart, PolynomialError
from sympy.solvers import solve


[docs]class Sum(Expr): """Represents unevaluated summation. ``Sum`` represents a finite or infinite series, with the first argument being the general form of terms in the series, and the second argument being ``(dummy_variable, start, end)``, with ``dummy_variable`` taking all integer values from ``start`` through ``end``. In accordance with long-standing mathematical convention, the end term is included in the summation. Finite sums =========== For finite sums (and sums with symbolic limits assumed to be finite) we follow the summation convention described by Karr [1], especially definition 3 of section 1.4. The sum: .. math:: \sum_{m \leq i < n} f(i) has *the obvious meaning* for `m < n`, namely: .. math:: \sum_{m \leq i < n} f(i) = f(m) + f(m+1) + \ldots + f(n-2) + f(n-1) with the upper limit value `f(n)` excluded. The sum over an empty set is zero if and only if `m = n`: .. math:: \sum_{m \leq i < n} f(i) = 0 \quad \mathrm{for} \quad m = n Finally, for all other sums over empty sets we assume the following definition: .. math:: \sum_{m \leq i < n} f(i) = - \sum_{n \leq i < m} f(i) \quad \mathrm{for} \quad m > n It is important to note that Karr defines all sums with the upper limit being exclusive. This is in contrast to the usual mathematical notation, but does not affect the summation convention. Indeed we have: .. math:: \sum_{m \leq i < n} f(i) = \sum_{i = m}^{n - 1} f(i) where the difference in notation is intentional to emphasize the meaning, with limits typeset on the top being inclusive. Examples ======== >>> from sympy.abc import i, k, m, n, x >>> from sympy import Sum, factorial, oo >>> Sum(k,(k,1,m)) Sum(k, (k, 1, m)) >>> Sum(k,(k,1,m)).doit() m**2/2 + m/2 >>> Sum(k**2,(k,1,m)) Sum(k**2, (k, 1, m)) >>> Sum(k**2,(k,1,m)).doit() m**3/3 + m**2/2 + m/6 >>> Sum(x**k,(k,0,oo)) Sum(x**k, (k, 0, oo)) >>> Sum(x**k,(k,0,oo)).doit() Piecewise((1/(-x + 1), Abs(x) < 1), (Sum(x**k, (k, 0, oo)), True)) >>> Sum(x**k/factorial(k),(k,0,oo)).doit() exp(x) An example showing that the symbolic result of a summation is still valid for seemingly nonsensical values of the limits. Then the Karr convention allows us to give a perfectly valid interpretation to those sums by interchanging the limits according to the above rules: >>> S = Sum(i, (i,1,n)).doit() >>> S n**2/2 + n/2 >>> S.subs(n, -4) 6 >>> Sum(i, (i, 1, -4)).doit() 6 >>> Sum(-i, (i, -3, 0)).doit() 6 An explicit example of the Karr summation convention: >>> S1 = Sum(i**2, (i, m, m+n-1)).doit() >>> S1 m**2*n + m*n**2 - m*n + n**3/3 - n**2/2 + n/6 >>> S2 = Sum(i**2, (i, m+n, m-1)).doit() >>> S2 -m**2*n - m*n**2 + m*n - n**3/3 + n**2/2 - n/6 >>> S1 + S2 0 >>> S3 = Sum(i, (i, m, m-1)).doit() >>> S3 0 See Also ======== summation Product, product References ========== .. [1] Michael Karr, "Summation in Finite Terms", Journal of the ACM, Volume 28 Issue 2, April 1981, Pages 305-350 http://dl.acm.org/citation.cfm?doid=322248.322255 .. [2] http://en.wikipedia.org/wiki/Summation#Capital-sigma_notation .. [3] http://en.wikipedia.org/wiki/Empty_sum """ __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("Summation variables must be given") limits, sign = _process_limits(*symbols) # Only limits with lower and upper bounds are supported; the indefinite Sum # is not supported if any(len(l) != 3 or None in l for l in limits): raise ValueError('Sum 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 function(self): return self._args[0] @property def limits(self): return self._args[1:] @property
[docs] def variables(self): """Return a list of the summation variables >>> from sympy import Sum >>> from sympy.abc import x, i >>> Sum(x**i, (i, 1, 3)).variables [i] """ return [l[0] for l in self.limits]
@property def free_symbols(self): from sympy.integrals.integrals import _free_symbols if self.function.is_zero: return set() return _free_symbols(self) @property
[docs] def is_zero(self): """A Sum is only zero if its function is zero or if all terms cancel out. This only answers whether the summand zero.""" return self.function.is_zero
@property
[docs] def is_number(self): """ Return True if the Sum will result in a number, else False. Sums are a special case since they contain symbols that can be replaced with numbers. Whether the integral can be done or not is another issue. But answering whether the final result is a number is not difficult. Examples ======== >>> from sympy import Sum >>> from sympy.abc import x, y >>> Sum(x, (y, 1, x)).is_number False >>> Sum(1, (y, 1, x)).is_number False >>> Sum(0, (y, 1, x)).is_number True >>> Sum(x, (y, 1, 2)).is_number False >>> Sum(x, (y, 1, 1)).is_number False >>> Sum(x, (x, 1, 2)).is_number True >>> Sum(x*y, (x, 1, 2), (y, 1, 3)).is_number True """ return self.function.is_zero or not self.free_symbols
def as_dummy(self): from sympy.integrals.integrals import _as_dummy return _as_dummy(self) def doit(self, **hints): if hints.get('deep', True): f = self.function.doit(**hints) else: f = self.function for limit in self.limits: i, a, b = limit dif = b - a if dif.is_Integer and dif < 0: a, b = b + 1, a - 1 f = -f f = eval_sum(f, (i, a, b)) if f is None: return self if hints.get('deep', True): # eval_sum could return partially unevaluated # result with Piecewise. In this case we won't # doit() recursively. if not isinstance(f, Piecewise): return f.doit(**hints) return f def _eval_adjoint(self): return Sum(self.function.adjoint(), *self.limits) def _eval_conjugate(self): return Sum(self.function.conjugate(), *self.limits) def _eval_derivative(self, x): """ Differentiate wrt x as long as x is not in the free symbols of any of the upper or lower limits. Sum(a*b*x, (x, 1, a)) can be differentiated wrt x or b but not `a` since the value of the sum is discontinuous in `a`. In a case involving a limit variable, the unevaluated derivative is returned. """ # diff already confirmed that x is in the free symbols of self, but we # don't want to differentiate wrt any free symbol in the upper or lower # limits # XXX remove this test for free_symbols when the default _eval_derivative is in if x not in self.free_symbols: return S.Zero # get limits and the function f, limits = self.function, list(self.limits) limit = limits.pop(-1) if limits: # f is the argument to a Sum f = Sum(f, *limits) if len(limit) == 3: _, a, b = limit if x in a.free_symbols or x in b.free_symbols: return None df = Derivative(f, x, **{'evaluate': True}) rv = Sum(df, limit) if limit[0] not in df.free_symbols: rv = rv.doit() return rv else: return NotImplementedError('Lower and upper bound expected.') def _eval_summation(self, f, x): return None def _eval_transpose(self): return Sum(self.function.transpose(), *self.limits)
[docs] def euler_maclaurin(self, m=0, n=0, eps=0, eval_integral=True): """ Return an Euler-Maclaurin approximation of self, where m is the number of leading terms to sum directly and n is the number of terms in the tail. With m = n = 0, this is simply the corresponding integral plus a first-order endpoint correction. Returns (s, e) where s is the Euler-Maclaurin approximation and e is the estimated error (taken to be the magnitude of the first omitted term in the tail): >>> from sympy.abc import k, a, b >>> from sympy import Sum >>> Sum(1/k, (k, 2, 5)).doit().evalf() 1.28333333333333 >>> s, e = Sum(1/k, (k, 2, 5)).euler_maclaurin() >>> s -log(2) + 7/20 + log(5) >>> from sympy import sstr >>> print sstr((s.evalf(), e.evalf()), full_prec=True) (1.26629073187415, 0.0175000000000000) The endpoints may be symbolic: >>> s, e = Sum(1/k, (k, a, b)).euler_maclaurin() >>> s -log(a) + log(b) + 1/(2*b) + 1/(2*a) >>> e Abs(-1/(12*b**2) + 1/(12*a**2)) If the function is a polynomial of degree at most 2n+1, the Euler-Maclaurin formula becomes exact (and e = 0 is returned): >>> Sum(k, (k, 2, b)).euler_maclaurin() (b**2/2 + b/2 - 1, 0) >>> Sum(k, (k, 2, b)).doit() b**2/2 + b/2 - 1 With a nonzero eps specified, the summation is ended as soon as the remainder term is less than the epsilon. """ m = int(m) n = int(n) f = self.function assert len(self.limits) == 1 i, a, b = self.limits[0] if a > b: a, b = b + 1, a - 1 f = -f s = S.Zero if m: for k in range(m): term = f.subs(i, a + k) if (eps and term and abs(term.evalf(3)) < eps): return s, abs(term) s += term a += m x = Dummy('x') I = C.Integral(f.subs(i, x), (x, a, b)) if eval_integral: I = I.doit() s += I def fpoint(expr): if b is S.Infinity: return expr.subs(i, a), 0 return expr.subs(i, a), expr.subs(i, b) fa, fb = fpoint(f) iterm = (fa + fb)/2 g = f.diff(i) for k in xrange(1, n + 2): ga, gb = fpoint(g) term = C.bernoulli(2*k)/C.factorial(2*k)*(gb - ga) if (eps and term and abs(term.evalf(3)) < eps) or (k > n): break s += term g = g.diff(i, 2, simplify=False) return s + iterm, abs(term)
def _eval_subs(self, old, new): from sympy.integrals.integrals import _eval_subs return _eval_subs(self, old, new)
[docs]def summation(f, *symbols, **kwargs): r""" Compute the summation of f with respect to symbols. The notation for symbols is similar to the notation used in Integral. summation(f, (i, a, b)) computes the sum of f with respect to i from a to b, i.e., :: b ____ \ ` summation(f, (i, a, b)) = ) f /___, i = a If it cannot compute the sum, it returns an unevaluated Sum object. Repeated sums can be computed by introducing additional symbols tuples:: >>> from sympy import summation, oo, symbols, log >>> i, n, m = symbols('i n m', integer=True) >>> summation(2*i - 1, (i, 1, n)) n**2 >>> summation(1/2**i, (i, 0, oo)) 2 >>> summation(1/log(n)**n, (n, 2, oo)) Sum(log(n)**(-n), (n, 2, oo)) >>> summation(i, (i, 0, n), (n, 0, m)) m**3/6 + m**2/2 + m/3 >>> from sympy.abc import x >>> from sympy import factorial >>> summation(x**n/factorial(n), (n, 0, oo)) exp(x) See Also ======== Sum Product, product """ return Sum(f, *symbols, **kwargs).doit(deep=False)
def telescopic_direct(L, R, n, limits): """Returns the direct summation of the terms of a telescopic sum L is the term with lower index R is the term with higher index n difference between the indexes of L and R For example: >>> from sympy.concrete.summations import telescopic_direct >>> from sympy.abc import k, a, b >>> telescopic_direct(1/k, -1/(k+2), 2, (k, a, b)) -1/(b + 2) - 1/(b + 1) + 1/(a + 1) + 1/a """ (i, a, b) = limits s = 0 for m in xrange(n): s += L.subs(i, a + m) + R.subs(i, b - m) return s def telescopic(L, R, limits): '''Tries to perform the summation using the telescopic property return None if not possible ''' (i, a, b) = limits if L.is_Add or R.is_Add: return None # We want to solve(L.subs(i, i + m) + R, m) # First we try a simple match since this does things that # solve doesn't do, e.g. solve(f(k+m)-f(k), m) fails k = Wild("k") sol = (-R).match(L.subs(i, i + k)) s = None if sol and k in sol: s = sol[k] if not (s.is_Integer and L.subs(i, i + s) == -R): #sometimes match fail(f(x+2).match(-f(x+k))->{k: -2 - 2x})) s = None # But there are things that match doesn't do that solve # can do, e.g. determine that 1/(x + m) = 1/(1 - x) when m = 1 if s is None: m = Dummy('m') try: sol = solve(L.subs(i, i + m) + R, m) or [] except NotImplementedError: return None sol = [si for si in sol if si.is_Integer and (L.subs(i, i + si) + R).expand().is_zero] if len(sol) != 1: return None s = sol[0] if s < 0: return telescopic_direct(R, L, abs(s), (i, a, b)) elif s > 0: return telescopic_direct(L, R, s, (i, a, b)) def eval_sum(f, limits): from sympy.concrete.delta import deltasummation, _has_simple_delta from sympy.functions import KroneckerDelta (i, a, b) = limits if f is S.Zero: return S.Zero if i not in f.free_symbols: return f*(b - a + 1) if a == b: return f.subs(i, a) if f.has(KroneckerDelta) and _has_simple_delta(f, limits[0]): return deltasummation(f, limits) dif = b - a definite = dif.is_Integer # Doing it directly may be faster if there are very few terms. if definite and (dif < 100): return eval_sum_direct(f, (i, a, b)) # Try to do it symbolically. Even when the number of terms is known, # this can save time when b-a is big. # We should try to transform to partial fractions value = eval_sum_symbolic(f.expand(), (i, a, b)) if value is not None: return value # Do it directly if definite: return eval_sum_direct(f, (i, a, b)) def eval_sum_direct(expr, limits): (i, a, b) = limits dif = b - a return Add(*[expr.subs(i, a + j) for j in xrange(dif + 1)]) def eval_sum_symbolic(f, limits): (i, a, b) = limits if not f.has(i): return f*(b - a + 1) # Linearity if f.is_Mul: L, R = f.as_two_terms() if not L.has(i): sR = eval_sum_symbolic(R, (i, a, b)) if sR: return L*sR if not R.has(i): sL = eval_sum_symbolic(L, (i, a, b)) if sL: return R*sL try: f = apart(f, i) # see if it becomes an Add except PolynomialError: pass if f.is_Add: L, R = f.as_two_terms() lrsum = telescopic(L, R, (i, a, b)) if lrsum: return lrsum lsum = eval_sum_symbolic(L, (i, a, b)) rsum = eval_sum_symbolic(R, (i, a, b)) if None not in (lsum, rsum): return lsum + rsum # Polynomial terms with Faulhaber's formula n = Wild('n') result = f.match(i**n) if result is not None: n = result[n] if n.is_Integer: if n >= 0: return ((C.bernoulli(n + 1, b + 1) - C.bernoulli(n + 1, a))/(n + 1)).expand() elif a.is_Integer and a >= 1: if n == -1: return C.harmonic(b) - C.harmonic(a - 1) else: return C.harmonic(b, abs(n)) - C.harmonic(a - 1, abs(n)) if not (a.has(S.Infinity, S.NegativeInfinity) or b.has(S.Infinity, S.NegativeInfinity)): # Geometric terms c1 = C.Wild('c1', exclude=[i]) c2 = C.Wild('c2', exclude=[i]) c3 = C.Wild('c3', exclude=[i]) e = f.match(c1**(c2*i + c3)) if e is not None: p = (c1**c3).subs(e) q = (c1**c2).subs(e) r = p*(q**a - q**(b + 1))/(1 - q) l = p*(b - a + 1) return Piecewise((l, Eq(q, S.One)), (r, True)) r = gosper_sum(f, (i, a, b)) if not r in (None, S.NaN): return r return eval_sum_hyper(f, (i, a, b)) def _eval_sum_hyper(f, i, a): """ Returns (res, cond). Sums from a to oo. """ from sympy.functions import hyper from sympy.simplify import hyperexpand, hypersimp, fraction, simplify from sympy.polys.polytools import Poly, factor if a != 0: return _eval_sum_hyper(f.subs(i, i + a), i, 0) if f.subs(i, 0) == 0: if simplify(f.subs(i, Dummy('i', integer=True, positive=True))) == 0: return S(0), True return _eval_sum_hyper(f.subs(i, i + 1), i, 0) hs = hypersimp(f, i) if hs is None: return None numer, denom = fraction(factor(hs)) top, topl = numer.as_coeff_mul(i) bot, botl = denom.as_coeff_mul(i) ab = [top, bot] factors = [topl, botl] params = [[], []] for k in range(2): for fac in factors[k]: mul = 1 if fac.is_Pow: mul = fac.exp fac = fac.base if not mul.is_Integer: return None p = Poly(fac, i) if p.degree() != 1: return None m, n = p.all_coeffs() ab[k] *= m**mul params[k] += [n/m]*mul # Add "1" to numerator parameters, to account for implicit n! in # hypergeometric series. ap = params[0] + [1] bq = params[1] x = ab[0]/ab[1] h = hyper(ap, bq, x) return f.subs(i, 0)*hyperexpand(h), h.convergence_statement def eval_sum_hyper(f, (i, a, b)): from sympy import oo, And if b != oo: if a == -oo: res = _eval_sum_hyper(f.subs(i, -i), i, -b) if res is not None: return Piecewise(res, (Sum(f, (i, a, b)), True)) else: res1 = _eval_sum_hyper(f, i, a) res2 = _eval_sum_hyper(f, i, b + 1) if res1 is None or res2 is None: return None (res1, cond1), (res2, cond2) = res1, res2 cond = And(cond1, cond2) if cond is False: return None return Piecewise((res1 - res2, cond), (Sum(f, (i, a, b)), True)) if a == -oo: res1 = _eval_sum_hyper(f.subs(i, -i), i, 1) res2 = _eval_sum_hyper(f, i, 0) if res1 is None or res2 is None: return None res1, cond1 = res1 res2, cond2 = res2 cond = And(cond1, cond2) if cond is False: return None return Piecewise((res1 + res2, cond), (Sum(f, (i, a, b)), True)) # Now b == oo, a != -oo res = _eval_sum_hyper(f, i, a) if res is not None: r, c = res if c is False: if r.is_number: if f.is_positive or f.is_zero: return S.Infinity elif f.is_negative: return S.NegativeInfinity return None return Piecewise(res, (Sum(f, (i, a, b)), True))