Source code for sympy.core.power

from math import log as _log

from sympify import _sympify
from cache import cacheit
from core import C
from singleton import S
from expr import Expr

from sympy import mpmath
from sympy.utilities.iterables import sift

[docs]def integer_nthroot(y, n): """ Return a tuple containing x = floor(y**(1/n)) and a boolean indicating whether the result is exact (that is, whether x**n == y). >>> from sympy import integer_nthroot >>> integer_nthroot(16,2) (4, True) >>> integer_nthroot(26,2) (5, False) """ y, n = int(y), int(n) if y < 0: raise ValueError("y must be nonnegative") if n < 1: raise ValueError("n must be positive") if y in (0, 1): return y, True if n == 1: return y, True if n == 2: x, rem = mpmath.libmp.sqrtrem(y) return int(x), not rem if n > y: return 1, False # Get initial estimate for Newton's method. Care must be taken to # avoid overflow try: guess = int(y**(1./n) + 0.5) except OverflowError: expt = _log(y, 2)/n if expt > 53: shift = int(expt - 53) guess = int(2.0**(expt-shift) + 1) << shift else: guess = int(2.0**expt) #print n if guess > 2**50: # Newton iteration xprev, x = -1, guess while 1: t = x**(n-1) #xprev, x = x, x - (t*x-y)//(n*t) xprev, x = x, ((n-1)*x + y//t)//n #print n, x-xprev, abs(x-xprev) < 2 if abs(x - xprev) < 2: break else: x = guess # Compensate t = x**n while t < y: x += 1 t = x**n while t > y: x -= 1 t = x**n return x, t == y
[docs]class Pow(Expr): is_Pow = True __slots__ = ['is_commutative'] @cacheit def __new__(cls, b, e, evaluate=True): b = _sympify(b) e = _sympify(e) if evaluate: if e is S.Zero: return S.One elif e is S.One: return b else: obj = b._eval_power(e) if obj is not None: return obj obj = Expr.__new__(cls, b, e) obj.is_commutative = (b.is_commutative and e.is_commutative) return obj @property def base(self): return self._args[0] @property def exp(self): return self._args[1] @classmethod def class_key(cls): return 3, 2, cls.__name__ def _eval_power(self, other): b, e = self.as_base_exp() if other.is_integer: return Pow(b, e * other) if b.is_nonnegative and (e.is_real or other.is_real): return Pow(b, e * other) if e.is_even and b.is_real: # hence b is pos and e is real return Pow(abs(b), e * other) if abs(e) < S.One and other.is_real: return Pow(b, e * other) def _eval_is_comparable(self): c1 = self.base.is_comparable if c1 is None: return c2 = self.exp.is_comparable if c2 is None: return return c1 and c2 def _eval_is_even(self): if self.exp.is_integer and self.exp.is_positive: if self.base.is_even: return True if self.base.is_integer: return False def _eval_is_positive(self): if self.base.is_positive: if self.exp.is_real: return True elif self.base.is_negative: if self.exp.is_even: return True if self.exp.is_odd: return False elif self.base.is_nonpositive: if self.exp.is_odd: return False def _eval_is_negative(self): if self.base.is_negative: if self.exp.is_odd: return True if self.exp.is_even: return False elif self.base.is_positive: if self.exp.is_real: return False elif self.base.is_nonnegative: if self.exp.is_real: return False elif self.base.is_nonpositive: if self.exp.is_even: return False elif self.base.is_real: if self.exp.is_even: return False def _eval_is_integer(self): c1 = self.base.is_integer c2 = self.exp.is_integer if c1 is None or c2 is None: return None if not c1: if self.exp.is_nonnegative: return False if c1 and c2: if self.exp.is_nonnegative or self.exp.is_positive: return True if self.exp.is_negative: return False def _eval_is_real(self): real_b = self.base.is_real if real_b is None: return real_e = self.exp.is_real if real_e is None: return if real_b and real_e: if self.base.is_positive: return True else: # negative or zero (or positive) if self.exp.is_integer: return True elif self.base.is_negative: if self.exp.is_Rational: return False im_b = self.base.is_imaginary im_e = self.exp.is_imaginary if im_b: if self.exp.is_integer: if self.exp.is_even: return True elif self.exp.is_odd: return False elif (self.exp in [S.ImaginaryUnit, -S.ImaginaryUnit] and self.base in [S.ImaginaryUnit, -S.ImaginaryUnit]): return True if real_b and im_e: c = self.exp.coeff(S.ImaginaryUnit) if c: ok = (c*C.log(self.base)/S.Pi).is_Integer if ok is not None: return ok def _eval_is_odd(self): if not (self.base.is_integer and self.exp.is_nonnegative): return return self.base.is_odd def _eval_is_bounded(self): if self.exp.is_negative: if self.base.is_infinitesimal: return False if self.base.is_unbounded: return True c1 = self.base.is_bounded if c1 is None: return c2 = self.exp.is_bounded if c2 is None: return if c1 and c2: if self.exp.is_nonnegative or self.base.is_nonzero: return True def _eval_subs(self, old, new): if self == old: return new if old.func is self.func and self.base == old.base: coeff1, terms1 = self.exp.as_coeff_mul() coeff2, terms2 = old.exp.as_coeff_mul() if terms1 == terms2: pow = coeff1/coeff2 if pow.is_Integer or self.base.is_commutative: return Pow(new, pow) # (x**(2*y)).subs(x**(3*y),z) -> z**(2/3) if old.func is C.exp: coeff1, terms1 = old.args[0].as_coeff_mul() coeff2, terms2 = (self.exp*C.log(self.base)).as_coeff_mul() if terms1 == terms2: pow = coeff1/coeff2 if pow.is_Integer or self.base.is_commutative: return Pow(new, pow) # (x**(2*y)).subs(x**(3*y),z) -> z**(2/3) return Pow(self.base._eval_subs(old, new), self.exp._eval_subs(old, new)) def as_base_exp(self): if self.base.is_Rational and self.base.p==1 and self.base is not S.Infinity: return 1/self.base, -self.exp return self.base, self.exp def _eval_conjugate(self): from sympy.functions.elementary.complexes import conjugate as c return c(self.base)**self.exp def _eval_expand_basic(self, deep=True, **hints): sargs, terms = self.args, [] for term in sargs: if hasattr(term, '_eval_expand_basic'): newterm = term._eval_expand_basic(deep=deep, **hints) else: newterm = term terms.append(newterm) return self.func(*terms) def _eval_expand_power_exp(self, deep=True, *args, **hints): """a**(n+m) -> a**n*a**m""" if deep: b = self.base.expand(deep=deep, **hints) e = self.exp.expand(deep=deep, **hints) else: b = self.base e = self.exp if e.is_Add and e.is_commutative: expr = [] for x in e.args: if deep: x = x.expand(deep=deep, **hints) expr.append(Pow(self.base, x)) return Mul(*expr) return Pow(b, e) def _eval_expand_power_base(self, deep=True, **hints): """(a*b)**n -> a**n * b**n""" force = hints.get('force', False) b, ewas = self.args if deep: e = self.exp.expand(deep=deep, **hints) else: e = self.exp if b.is_Mul: bargs = b.args if force or e.is_integer: nonneg = bargs other = [] elif ewas.is_Rational or len(bargs) == 2 and bargs[0] is S.NegativeOne: # the Rational exponent was already expanded automatically # if there is a negative Number * foo, foo must be unknown # or else it, too, would have automatically expanded; # sqrt(-Number*foo) -> sqrt(Number)*sqrt(-foo); then # sqrt(-foo) -> unchanged if foo is not positive else # -> I*sqrt(foo) # So...if we have a 2 arg Mul and the first is a Number # that number is -1 and there is nothing more than can # be done without the force=True hint nonneg= [] else: # this is just like what is happening automatically, except # that now we are doing it for an arbitrary exponent for which # no automatic expansion is done sifted = sift(b.args, lambda x: x.is_nonnegative) nonneg = sifted.get(True, []) other = sifted.get(None, []) neg = sifted.get(False, []) # make sure the Number gets pulled out if neg and neg[0].is_Number and neg[0] is not S.NegativeOne: nonneg.append(-neg[0]) neg[0] = S.NegativeOne # leave behind a negative sign oddneg = len(neg) % 2 if oddneg: other.append(S.NegativeOne) # negate all negatives and append to nonneg nonneg += [-n for n in neg] if nonneg: # then there's a new expression to return other = [Pow(Mul(*other), e)] if deep: return Mul(*([Pow(b.expand(deep=deep, **hints), e)\ for b in nonneg] + other)) else: return Mul(*([Pow(b, e) for b in nonneg] + other)) return Pow(b, e) def _eval_expand_mul(self, deep=True, **hints): sargs, terms = self.args, [] for term in sargs: if hasattr(term, '_eval_expand_mul'): newterm = term._eval_expand_mul(deep=deep, **hints) else: newterm = term terms.append(newterm) return self.func(*terms) def _eval_expand_multinomial(self, deep=True, **hints): """(a+b+..) ** n -> a**n + n*a**(n-1)*b + .., n is nonzero integer""" if deep: b = self.base.expand(deep=deep, **hints) e = self.exp.expand(deep=deep, **hints) else: b = self.base e = self.exp if b is None: base = self.base else: base = b if e is None: exp = self.exp else: exp = e if e is not None or b is not None: result = Pow(base, exp) if result.is_Pow: base, exp = result.base, result.exp else: return result else: result = None if exp.is_Rational and exp.p > 0 and base.is_Add: if not exp.is_Integer: n = Integer(exp.p // exp.q) if not n: return Pow(base, exp) else: radical, result = Pow(base, exp - n), [] for term in Add.make_args(Pow(base, n)._eval_expand_multinomial(deep=False)): result.append(term*radical) return Add(*result) n = int(exp) if base.is_commutative: order_terms, other_terms = [], [] for order in base.args: if order.is_Order: order_terms.append(order) else: other_terms.append(order) if order_terms: # (f(x) + O(x^n))^m -> f(x)^m + m*f(x)^{m-1} *O(x^n) f = Add(*other_terms) g = (f**(n-1)).expand() return (f*g).expand() + n*g*Add(*order_terms) if base.is_number: # Efficiently expand expressions of the form (a + b*I)**n # where 'a' and 'b' are real numbers and 'n' is integer. a, b = base.as_real_imag() if a.is_Rational and b.is_Rational: if not a.is_Integer: if not b.is_Integer: k = Pow(a.q * b.q, n) a, b = a.p*b.q, a.q*b.p else: k = Pow(a.q, n) a, b = a.p, a.q*b elif not b.is_Integer: k = Pow(b.q, n) a, b = a*b.q, b.p else: k = 1 a, b, c, d = int(a), int(b), 1, 0 while n: if n & 1: c, d = a*c-b*d, b*c+a*d n -= 1 a, b = a*a-b*b, 2*a*b n //= 2 I = S.ImaginaryUnit if k == 1: return c + I*d else: return Integer(c)/k + I*d/k p = other_terms # (x+y)**3 -> x**3 + 3*x**2*y + 3*x*y**2 + y**3 # in this particular example: # p = [x,y]; n = 3 # so now it's easy to get the correct result -- we get the # coefficients first: from sympy import multinomial_coefficients expansion_dict = multinomial_coefficients(len(p), n) # in our example: {(3, 0): 1, (1, 2): 3, (0, 3): 1, (2, 1): 3} # and now construct the expression. # An elegant way would be to use Poly, but unfortunately it is # slower than the direct method below, so it is commented out: #b = {} #for k in expansion_dict: # b[k] = Integer(expansion_dict[k]) #return Poly(b, *p).as_expr() from sympy.polys.polyutils import basic_from_dict result = basic_from_dict(expansion_dict, *p) return result else: if n == 2: return Add(*[f*g for f in base.args for g in base.args]) else: multi = (base**(n-1))._eval_expand_multinomial(deep=False) if multi.is_Add: return Add(*[f*g for f in base.args for g in multi.args]) else: return Add(*[f*multi for f in base.args]) elif exp.is_Rational and exp.p < 0 and base.is_Add and abs(exp.p) > exp.q: return 1 / Pow(base, -exp)._eval_expand_multinomial(deep=False) elif exp.is_Add and base.is_Number: # a + b a b # n --> n n , where n, a, b are Numbers coeff, tail = S.One, S.Zero for term in exp.args: if term.is_Number: coeff *= Pow(base, term) else: tail += term return coeff * Pow(base, tail) else: return result def _eval_expand_log(self, deep=True, **hints): sargs, terms = self.args, [] for term in sargs: if hasattr(term, '_eval_expand_log'): newterm = term._eval_expand_log(deep=deep, **hints) else: newterm = term terms.append(newterm) return self.func(*terms) def _eval_expand_complex(self, deep=True, **hints): re_part, im_part = self.as_real_imag(deep=deep, **hints) return re_part + im_part*S.ImaginaryUnit def as_real_imag(self, deep=True, **hints): from sympy.core.symbol import symbols from sympy.polys.polytools import poly from sympy.core.function import expand_multinomial if self.exp.is_Integer: exp = self.exp re, im = self.base.as_real_imag(deep=deep) a, b = symbols('a b', cls=Dummy) if exp >= 0: if re.is_Number and im.is_Number: # We can be more efficient in this case expr = expand_multinomial(self.base**exp) return expr.as_real_imag() expr = poly((a + b)**exp) # a = re, b = im; expr = (a + b*I)**exp else: mag = re**2 + im**2 re, im = re/mag, -im/mag if re.is_Number and im.is_Number: # We can be more efficient in this case expr = expand_multinomial((re + im*S.ImaginaryUnit)**-exp) return expr.as_real_imag() expr = poly((a + b)**-exp) # Terms with even b powers will be real r = [i for i in expr.terms() if not i[0][1] % 2] re_part = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r]) # Terms with odd b powers will be imaginary r = [i for i in expr.terms() if i[0][1] % 4 == 1] im_part1 = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r]) r = [i for i in expr.terms() if i[0][1] % 4 == 3] im_part3 = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r]) return (re_part.subs({a: re, b: S.ImaginaryUnit*im}), im_part1.subs({a: re, b: im}) + im_part3.subs({a: re, b: -im})) elif self.exp.is_Rational: # NOTE: This is not totally correct since for x**(p/q) with # x being imaginary there are actually q roots, but # only a single one is returned from here. re, im = self.base.as_real_imag(deep=deep) r = Pow(Pow(re, 2) + Pow(im, 2), S.Half) t = C.atan2(im, re) rp, tp = Pow(r, self.exp), t*self.exp return (rp*C.cos(tp), rp*C.sin(tp)) else: if deep: hints['complex'] = False return (C.re(self.expand(deep, complex=False)), C.im(self. expand(deep, **hints))) else: return (C.re(self), C.im(self)) def _eval_expand_trig(self, deep=True, **hints): sargs, terms = self.args, [] for term in sargs: if hasattr(term, '_eval_expand_trig'): newterm = term._eval_expand_trig(deep=deep, **hints) else: newterm = term terms.append(newterm) return self.func(*terms) def _eval_expand_func(self, deep=True, **hints): sargs, terms = self.args, [] for term in sargs: if hasattr(term, '_eval_expand_func'): newterm = term._eval_expand_func(deep=deep, **hints) else: newterm = term terms.append(newterm) return self.func(*terms) def _eval_derivative(self, s): dbase = self.base.diff(s) dexp = self.exp.diff(s) return self * (dexp * C.log(self.base) + dbase * self.exp/self.base) def _eval_evalf(self, prec): base, exp = self.as_base_exp() base = base._evalf(prec) if not exp.is_Integer: exp = exp._evalf(prec) if exp < 0 and not base.is_real: base = base.conjugate() / (base * base.conjugate())._evalf(prec) exp = -exp return Pow(base, exp).expand() def _eval_is_polynomial(self, syms): if self.exp.has(*syms): return False if self.base.has(*syms): return self.base._eval_is_polynomial(syms) and \ self.exp.is_Integer and \ self.exp >= 0 else: return True def _eval_is_rational_function(self, syms): if self.exp.has(*syms): return False if self.base.has(*syms): return self.base._eval_is_rational_function(syms) and \ self.exp.is_Integer else: return True def as_numer_denom(self): base, exp = self.as_base_exp() n, d = base.as_numer_denom() if d.is_negative and n.is_negative: n, d = -n, -d if exp.is_Integer: if exp.is_negative: n, d = d, n exp = -exp return Pow(n, exp), Pow(d, exp) elif exp.is_Rational or d.is_positive: if d.is_negative is None: # we won't split up the base if exp.is_negative: return S.One, Pow(base, -exp) else: return self, S.One if d.is_negative: n = -n d = -d c, t = exp.as_coeff_mul() if c.is_negative: n, d = d, n exp = -exp return Pow(n, exp), Pow(d, exp) else: c, t = exp.as_coeff_mul() if c.is_negative: return 1, base**-exp # unprocessed Float and NumberSymbol return self, S.One def matches(self, expr, repl_dict={}, evaluate=False): if evaluate: return self.subs(repl_dict).matches(expr, repl_dict) expr = _sympify(expr) b, e = expr.as_base_exp() # special case, pattern = 1 and expr.exp can match to 0 if expr is S.One: d = repl_dict.copy() d = self.exp.matches(S.Zero, d) if d is not None: return d d = repl_dict.copy() d = self.base.matches(b, d) if d is None: return None d = self.exp.subs(d).matches(e, d) if d is None: return Expr.matches(self, expr, repl_dict, evaluate) return d def _eval_nseries(self, x, n, logx): # NOTE! This function is an important part of the gruntz algorithm # for computing limits. It has to return a generalised power series # with coefficients in C(log, log(x)). In more detail: # It has to return an expression # c_0*x**e_0 + c_1*x**e_1 + ... (finitely many terms) # where e_i are numbers (not necessarily integers) and c_i are expression # involving only numbers, the log function, and log(x). from sympy import powsimp, collect, exp, log, O, ceiling b, e = self.args if e.is_Integer: if e > 0: # positive integer powers are easy to expand, e.g.: # sin(x)**4 = (x-x**3/3+...)**4 = ... return Pow(b._eval_nseries(x, n=n, logx=logx), e )._eval_expand_multinomial(deep = False) elif e is S.NegativeOne: # this is also easy to expand using the formula: # 1/(1 + x) = 1 - x + x**2 - x**3 ... # so we need to rewrite base to the form "1+x" b = b._eval_nseries(x, n=n, logx=logx) prefactor = b.as_leading_term(x) # express "rest" as: rest = 1 + k*x**l + ... + O(x**n) rest = ((b - prefactor)/prefactor)._eval_expand_mul() if rest == 0: # if prefactor == w**4 + x**2*w**4 + 2*x*w**4, we need to # factor the w**4 out using collect: return 1/collect(prefactor, x) if rest.is_Order: return 1/prefactor + rest/prefactor n2 = rest.getn() if n2 is not None: n = n2 # remove the O - powering this is slow if logx is not None: rest = rest.removeO() k, l = rest.leadterm(x) if l.is_Rational and l > 0: pass elif l.is_number and l > 0: l = l.evalf() else: raise NotImplementedError() terms = [1/prefactor] for m in xrange(1, ceiling(n/l)): new_term = terms[-1]*(-rest) if new_term.is_Pow: new_term = new_term._eval_expand_multinomial(deep = False) else: new_term = new_term._eval_expand_mul(deep = False) terms.append(new_term) # Append O(...), we know the order. if n2 is None or logx is not None: terms.append(O(x**n)) return powsimp(Add(*terms), deep=True, combine='exp') else: # negative powers are rewritten to the cases above, for example: # sin(x)**(-4) = 1/( sin(x)**4) = ... # and expand the denominator: denominator = (b**(-e))._eval_nseries(x, n=n, logx=logx) if 1/denominator == self: return self # now we have a type 1/f(x), that we know how to expand return (1/denominator)._eval_nseries(x, n=n, logx=logx) if e.has(Symbol): return exp(e*log(b))._eval_nseries(x, n=n, logx=logx) if b == x: return powsimp(self, deep=True, combine='exp') # work for b(x)**e where e is not an Integer and does not contain x # and hopefully has no other symbols def e2int(e): """return the integer value (if possible) of e and a flag indicating whether it is bounded or not.""" n = e.limit(x, 0) unbounded = n.is_unbounded if not unbounded: # XXX was int or floor intended? int used to behave like floor # so int(-Rational(1, 2)) returned -1 rather than int's 0 try: n = int(n) except TypeError: #well, the n is something more complicated (like 1+log(2)) try: n = int(n.evalf()) + 1 # XXX why is 1 being added? except TypeError: pass # hope that base allows this to be resolved n = _sympify(n) return n, unbounded order = O(x**n, x) ei, unbounded = e2int(e) b0 = b.limit(x, 0) if unbounded and (b0 is S.One or b0.has(Symbol)): # XXX what order if b0 is S.One: resid = (b - 1) if resid.is_positive: return S.Infinity elif resid.is_negative: return S.Zero raise ValueError('cannot determine sign of %s' % resid) return b0**ei if (b0 is S.Zero or b0.is_unbounded): if unbounded is not False: return b0**e # XXX what order if not ei.is_number: # if not, how will we proceed? raise ValueError('expecting numerical exponent but got %s' % ei) nuse = n - ei lt = b.compute_leading_term(x, logx=logx) # arg = sin(x); lt = x # XXX o is not used -- was this to be used as o and o2 below to compute a new e? o = order*lt**(1 - e) bs = b._eval_nseries(x, n=nuse, logx=logx) if bs.is_Add: bs = bs.removeO() if bs.is_Add: # bs -> lt + rest -> lt*(1 + (bs/lt - 1)) return ((Pow(lt, e)* Pow((bs/lt).expand(), e). nseries(x, n=nuse, logx=logx)).expand() + order) return bs**e + order # either b0 is bounded but neither 1 nor 0 or e is unbounded # b -> b0 + (b-b0) -> b0 * (1 + (b/b0-1)) o2 = order*(b0**-e) z = (b/b0 - 1) o = O(z, x) #r = self._compute_oseries3(z, o2, self.taylor_term) if o is S.Zero or o2 is S.Zero: unbounded = True else: if o.expr.is_number: e2 = log(o2.expr*x)/log(x) else: e2 = log(o2.expr)/log(o.expr) n, unbounded = e2int(e2) if unbounded: # requested accuracy gives infinite series, # order is probably nonpolynomial e.g. O(exp(-1/x), x). r = 1 + z else: l = [] g = None for i in xrange(n + 2): g = self.taylor_term(i, z, g) g = g.nseries(x, n=n, logx=logx) l.append(g) r = Add(*l) return r*b0**e + order def _eval_as_leading_term(self, x): if not self.exp.has(x): return Pow(self.base.as_leading_term(x), self.exp) return C.exp(self.exp * C.log(self.base)).as_leading_term(x) @cacheit def taylor_term(self, n, x, *previous_terms): # of (1+x)**e if n<0: return S.Zero x = _sympify(x) return C.binomial(self.exp, n) * Pow(x, n) def _sage_(self): return self.args[0]._sage_()**self.args[1]._sage_()
from add import Add from numbers import Integer from mul import Mul from symbol import Symbol, Dummy