def real_root(arg, n=None):
"""Return the real nth-root of arg if possible. If n is omitted then
all instances of (-n)**(1/odd) will be changed to -n**(1/odd); this
will only create a real root of a principle root -- the presence of
other factors may cause the result to not be real.
Examples
========
>>> from sympy import root, real_root, Rational
>>> from sympy.abc import x, n
>>> real_root(-8, 3)
-2
>>> root(-8, 3)
2*(-1)**(1/3)
>>> real_root(_)
-2
If one creates a non-principle root and applies real_root, the
result will not be real (so use with caution):
>>> root(-8, 3, 2)
-2*(-1)**(2/3)
>>> real_root(_)
-2*(-1)**(2/3)
See Also
========
sympy.polys.rootoftools.RootOf
sympy.core.power.integer_nthroot
root, sqrt
"""
if n is not None:
try:
n = as_int(n)
arg = sympify(arg)
if arg.is_positive or arg.is_negative:
rv = root(arg, n)
else:
raise ValueError
except ValueError:
return root(arg, n)*C.Piecewise(
(S.One, ~C.Equality(C.im(arg), 0)),
(C.Pow(S.NegativeOne, S.One/n)**(2*C.floor(n/2)), C.And(
C.Equality(n % 2, 1),
arg < 0)),
(S.One, True))
else:
rv = sympify(arg)
n1pow = Transform(lambda x: -(-x.base)**x.exp,
lambda x:
x.is_Pow and
x.base.is_negative and
x.exp.is_Rational and
x.exp.p == 1 and x.exp.q % 2)
return rv.xreplace(n1pow)
def _print_Product(self, expr):
func = expr.term
pretty_func = self._print(func)
horizontal_chr = xobj('_', 1)
corner_chr = xobj('_', 1)
vertical_chr = xobj('|', 1)
if self._use_unicode:
# use unicode corners
horizontal_chr = xobj('-', 1)
corner_chr = u'\u252c'
func_height = pretty_func.height()
first = True
max_upper = 0
sign_height = 0
for lim in expr.limits:
width = (func_height + 2) * 5 // 3 - 2
sign_lines = []
sign_lines.append(corner_chr + (horizontal_chr * width) +
corner_chr)
for i in range(func_height + 1):
sign_lines.append(vertical_chr + (' ' * width) + vertical_chr)
pretty_sign = stringPict('')
pretty_sign = prettyForm(*pretty_sign.stack(*sign_lines))
pretty_upper = self._print(lim[2])
pretty_lower = self._print(C.Equality(lim[0], lim[1]))
max_upper = max(max_upper, pretty_upper.height())
if first:
sign_height = pretty_sign.height()
pretty_sign = prettyForm(*pretty_sign.above(pretty_upper))
pretty_sign = prettyForm(*pretty_sign.below(pretty_lower))
if first:
pretty_func.baseline = 0
first = False
height = pretty_sign.height()
padding = stringPict('')
padding = prettyForm(*padding.stack(*[' '] * (height - 1)))
pretty_sign = prettyForm(*pretty_sign.right(padding))
pretty_func = prettyForm(*pretty_sign.right(pretty_func))
#pretty_func.baseline = 0
pretty_func.baseline = max_upper + sign_height // 2
return pretty_func
def _print_Product(self, expr):
func = expr.term
pretty_func = self._print(func)
horizontal_chr = xobj('_', 1)
corner_chr = xobj('_', 1)
vertical_chr = xobj('|', 1)
if self._use_unicode:
# use unicode corners
horizontal_chr = xobj('-', 1)
corner_chr = u'\u252c'
func_height = pretty_func.height()
width = (func_height + 2) * 5 // 3 - 2
sign_lines = []
sign_lines.append(corner_chr + (horizontal_chr * width) + corner_chr)
for i in range(func_height + 1):
sign_lines.append(vertical_chr + (' ' * width) + vertical_chr)
pretty_sign = stringPict('')
pretty_sign = prettyForm(*pretty_sign.stack(*sign_lines))
pretty_upper = self._print(expr.upper)
pretty_lower = self._print(C.Equality(expr.index, expr.lower))
pretty_sign = prettyForm(*pretty_sign.above(pretty_upper))
pretty_sign = prettyForm(*pretty_sign.below(pretty_lower))
height = pretty_sign.height()
padding = stringPict('')
padding = prettyForm(*padding.stack(*[' '] * (height - 1)))
pretty_sign = prettyForm(*pretty_sign.right(padding))
pretty_func.baseline = 0
pretty_func = prettyForm(*pretty_sign.right(pretty_func))
return pretty_func
def _print_Sum(self, expr):
ascii_mode = not self._use_unicode
def asum(hrequired, lower, upper, use_ascii):
def adjust(s, wid=None, how='<^>'):
if not wid or len(s) > wid:
return s
need = wid - len(s)
if how == '<^>' or how == "<" or how not in list('<^>'):
return s + ' ' * need
half = need // 2
lead = ' ' * half
if how == ">":
return " " * need + s
return lead + s + ' ' * (need - len(lead))
h = max(hrequired, 2)
d = h // 2
wrequired = max(lower, upper)
w = d + 1
more = hrequired % 2
lines = []
if use_ascii:
lines.append("_" * (w) + ' ')
lines.append("\%s`" % (' ' * (w - 1)))
for i in range(1, d):
lines.append('%s\\%s' % (' ' * i, ' ' * (w - i)))
if more:
lines.append('%s)%s' % (' ' * (d), ' ' * (w - d)))
for i in reversed(range(1, d)):
lines.append('%s/%s' % (' ' * i, ' ' * (w - i)))
lines.append("/" + "_" * (w - 1) + ',')
return d, h + more, lines, 0
else:
w = w + more
d = d + more
vsum = vobj('sum', 4)
lines.append("_" * (w))
for i in range(0, d):
lines.append('%s%s%s' % (' ' * i, vsum[2], ' ' *
(w - i - 1)))
for i in reversed(range(0, d)):
lines.append('%s%s%s' % (' ' * i, vsum[4], ' ' *
(w - i - 1)))
lines.append(vsum[8] * (w))
return d, h + 2 * more, lines, more
f = expr.function
prettyF = self._print(f)
if f.is_Add: # add parens
prettyF = prettyForm(*prettyF.parens())
H = prettyF.height() + 2
# \sum \sum \sum ...
first = True
max_upper = 0
sign_height = 0
for lim in expr.limits:
if len(lim) == 3:
prettyUpper = self._print(lim[2])
prettyLower = self._print(C.Equality(lim[0], lim[1]))
elif len(lim) == 2:
prettyUpper = self._print("")
prettyLower = self._print(C.Equality(lim[0], lim[1]))
elif len(lim) == 1:
prettyUpper = self._print("")
prettyLower = self._print(lim[0])
max_upper = max(max_upper, prettyUpper.height())
# Create sum sign based on the height of the argument
d, h, slines, adjustment = asum(H, prettyLower.width(),
prettyUpper.width(), ascii_mode)
prettySign = stringPict('')
prettySign = prettyForm(*prettySign.stack(*slines))
if first:
sign_height = prettySign.height()
prettySign = prettyForm(*prettySign.above(prettyUpper))
prettySign = prettyForm(*prettySign.below(prettyLower))
if first:
# change F baseline so it centers on the sign
prettyF.baseline -= d - (prettyF.height() // 2 -
prettyF.baseline) - adjustment
first = False
# put padding to the right
pad = stringPict('')
pad = prettyForm(*pad.stack(*[' '] * h))
prettySign = prettyForm(*prettySign.right(pad))
# put the present prettyF to the right
prettyF = prettyForm(*prettySign.right(prettyF))
prettyF.baseline = max_upper + sign_height // 2
return prettyF
