def _eval_nseries(self, x, n):
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),
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"
if b.has(log(x)):
# we need to handle the log(x) singularity:
y = Dummy("y")
p = self.subs(log(x), -1 / y)
if not p.has(x):
p = p._eval_nseries(y, n=n)
p = p.subs(y, -1 / log(x))
return p
b = b._eval_nseries(x, n=n)
if b.has(log(x)):
# we need to handle the log(x) singularity:
y = Dummy("y")
self0 = 1 / b
p = self0.subs(log(x), -1 / y)
if not p.has(x):
p = p._eval_nseries(y, n=n)
p = p.subs(y, -1 / log(x))
return p
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 + rest) / prefactor
n2 = rest.getn()
if n2 is not None:
n = n2
term2 = collect(rest.as_leading_term(x), x)
k, l = C.Wild("k"), C.Wild("l")
r = term2.match(k * x**l)
# if term2 is NaN then r will not contain l
k = r.get(k, S.One)
l = r.get(l, S.Zero)
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)
if n2 is None:
# Append O(...) because it is not included in "r"
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)
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)
if e.has(x):
return exp(e * log(b))._eval_nseries(x, n=n)
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)
if n.is_Integer:
assert n.is_nonnegative
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.as_leading_term(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)
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)).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)
l.append(g)
r = Add(*l)
return r * b0**e + order
def do_integral(expr, prec, options):
func = expr.args[0]
x, xlow, xhigh = expr.args[1]
orig = mp.prec
oldmaxprec = options.get('maxprec', DEFAULT_MAXPREC)
options['maxprec'] = min(oldmaxprec, 2 * prec)
try:
mp.prec = prec + 5
xlow = as_mpmath(xlow, prec + 15, options)
xhigh = as_mpmath(xhigh, prec + 15, options)
# Integration is like summation, and we can phone home from
# the integrand function to update accuracy summation style
# Note that this accuracy is inaccurate, since it fails
# to account for the variable quadrature weights,
# but it is better than nothing
have_part = [False, False]
max_real_term = [MINUS_INF]
max_imag_term = [MINUS_INF]
def f(t):
re, im, re_acc, im_acc = evalf(func, mp.prec, {'subs': {x: t}})
have_part[0] = re or have_part[0]
have_part[1] = im or have_part[1]
max_real_term[0] = max(max_real_term[0], fastlog(re))
max_imag_term[0] = max(max_imag_term[0], fastlog(im))
if im:
return mpc(re or fzero, im)
return mpf(re or fzero)
if options.get('quad') == 'osc':
A = C.Wild('A', exclude=[x])
B = C.Wild('B', exclude=[x])
D = C.Wild('D')
m = func.match(C.cos(A * x + B) * D)
if not m:
m = func.match(C.sin(A * x + B) * D)
if not m:
raise ValueError(
"An integrand of the form sin(A*x+B)*f(x) "
"or cos(A*x+B)*f(x) is required for oscillatory quadrature"
)
period = as_mpmath(2 * S.Pi / m[A], prec + 15, options)
result = quadosc(f, [xlow, xhigh], period=period)
# XXX: quadosc does not do error detection yet
quadrature_error = MINUS_INF
else:
result, quadrature_error = quadts(f, [xlow, xhigh], error=1)
quadrature_error = fastlog(quadrature_error._mpf_)
finally:
options['maxprec'] = oldmaxprec
mp.prec = orig
if have_part[0]:
re = result.real._mpf_
if re == fzero:
re = mpf_shift(fone,
min(-prec, -max_real_term[0], -quadrature_error))
re_acc = -1
else:
re_acc = -max(max_real_term[0] - fastlog(re) - prec,
quadrature_error)
else:
re, re_acc = None, None
if have_part[1]:
im = result.imag._mpf_
if im == fzero:
im = mpf_shift(fone,
min(-prec, -max_imag_term[0], -quadrature_error))
im_acc = -1
else:
im_acc = -max(max_imag_term[0] - fastlog(im) - prec,
quadrature_error)
else:
im, im_acc = None, None
result = re, im, re_acc, im_acc
return result