You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
413 lines
15 KiB
413 lines
15 KiB
"""
|
|
Interval Arithmetic for plotting.
|
|
This module does not implement interval arithmetic accurately and
|
|
hence cannot be used for purposes other than plotting. If you want
|
|
to use interval arithmetic, use mpmath's interval arithmetic.
|
|
|
|
The module implements interval arithmetic using numpy and
|
|
python floating points. The rounding up and down is not handled
|
|
and hence this is not an accurate implementation of interval
|
|
arithmetic.
|
|
|
|
The module uses numpy for speed which cannot be achieved with mpmath.
|
|
"""
|
|
|
|
# Q: Why use numpy? Why not simply use mpmath's interval arithmetic?
|
|
# A: mpmath's interval arithmetic simulates a floating point unit
|
|
# and hence is slow, while numpy evaluations are orders of magnitude
|
|
# faster.
|
|
|
|
# Q: Why create a separate class for intervals? Why not use SymPy's
|
|
# Interval Sets?
|
|
# A: The functionalities that will be required for plotting is quite
|
|
# different from what Interval Sets implement.
|
|
|
|
# Q: Why is rounding up and down according to IEEE754 not handled?
|
|
# A: It is not possible to do it in both numpy and python. An external
|
|
# library has to used, which defeats the whole purpose i.e., speed. Also
|
|
# rounding is handled for very few functions in those libraries.
|
|
|
|
# Q Will my plots be affected?
|
|
# A It will not affect most of the plots. The interval arithmetic
|
|
# module based suffers the same problems as that of floating point
|
|
# arithmetic.
|
|
|
|
from sympy.core.logic import fuzzy_and
|
|
from sympy.simplify.simplify import nsimplify
|
|
|
|
from .interval_membership import intervalMembership
|
|
|
|
|
|
class interval:
|
|
""" Represents an interval containing floating points as start and
|
|
end of the interval
|
|
The is_valid variable tracks whether the interval obtained as the
|
|
result of the function is in the domain and is continuous.
|
|
- True: Represents the interval result of a function is continuous and
|
|
in the domain of the function.
|
|
- False: The interval argument of the function was not in the domain of
|
|
the function, hence the is_valid of the result interval is False
|
|
- None: The function was not continuous over the interval or
|
|
the function's argument interval is partly in the domain of the
|
|
function
|
|
|
|
A comparison between an interval and a real number, or a
|
|
comparison between two intervals may return ``intervalMembership``
|
|
of two 3-valued logic values.
|
|
"""
|
|
|
|
def __init__(self, *args, is_valid=True, **kwargs):
|
|
self.is_valid = is_valid
|
|
if len(args) == 1:
|
|
if isinstance(args[0], interval):
|
|
self.start, self.end = args[0].start, args[0].end
|
|
else:
|
|
self.start = float(args[0])
|
|
self.end = float(args[0])
|
|
elif len(args) == 2:
|
|
if args[0] < args[1]:
|
|
self.start = float(args[0])
|
|
self.end = float(args[1])
|
|
else:
|
|
self.start = float(args[1])
|
|
self.end = float(args[0])
|
|
|
|
else:
|
|
raise ValueError("interval takes a maximum of two float values "
|
|
"as arguments")
|
|
|
|
@property
|
|
def mid(self):
|
|
return (self.start + self.end) / 2.0
|
|
|
|
@property
|
|
def width(self):
|
|
return self.end - self.start
|
|
|
|
def __repr__(self):
|
|
return "interval(%f, %f)" % (self.start, self.end)
|
|
|
|
def __str__(self):
|
|
return "[%f, %f]" % (self.start, self.end)
|
|
|
|
def __lt__(self, other):
|
|
if isinstance(other, (int, float)):
|
|
if self.end < other:
|
|
return intervalMembership(True, self.is_valid)
|
|
elif self.start > other:
|
|
return intervalMembership(False, self.is_valid)
|
|
else:
|
|
return intervalMembership(None, self.is_valid)
|
|
|
|
elif isinstance(other, interval):
|
|
valid = fuzzy_and([self.is_valid, other.is_valid])
|
|
if self.end < other. start:
|
|
return intervalMembership(True, valid)
|
|
if self.start > other.end:
|
|
return intervalMembership(False, valid)
|
|
return intervalMembership(None, valid)
|
|
else:
|
|
return NotImplemented
|
|
|
|
def __gt__(self, other):
|
|
if isinstance(other, (int, float)):
|
|
if self.start > other:
|
|
return intervalMembership(True, self.is_valid)
|
|
elif self.end < other:
|
|
return intervalMembership(False, self.is_valid)
|
|
else:
|
|
return intervalMembership(None, self.is_valid)
|
|
elif isinstance(other, interval):
|
|
return other.__lt__(self)
|
|
else:
|
|
return NotImplemented
|
|
|
|
def __eq__(self, other):
|
|
if isinstance(other, (int, float)):
|
|
if self.start == other and self.end == other:
|
|
return intervalMembership(True, self.is_valid)
|
|
if other in self:
|
|
return intervalMembership(None, self.is_valid)
|
|
else:
|
|
return intervalMembership(False, self.is_valid)
|
|
|
|
if isinstance(other, interval):
|
|
valid = fuzzy_and([self.is_valid, other.is_valid])
|
|
if self.start == other.start and self.end == other.end:
|
|
return intervalMembership(True, valid)
|
|
elif self.__lt__(other)[0] is not None:
|
|
return intervalMembership(False, valid)
|
|
else:
|
|
return intervalMembership(None, valid)
|
|
else:
|
|
return NotImplemented
|
|
|
|
def __ne__(self, other):
|
|
if isinstance(other, (int, float)):
|
|
if self.start == other and self.end == other:
|
|
return intervalMembership(False, self.is_valid)
|
|
if other in self:
|
|
return intervalMembership(None, self.is_valid)
|
|
else:
|
|
return intervalMembership(True, self.is_valid)
|
|
|
|
if isinstance(other, interval):
|
|
valid = fuzzy_and([self.is_valid, other.is_valid])
|
|
if self.start == other.start and self.end == other.end:
|
|
return intervalMembership(False, valid)
|
|
if not self.__lt__(other)[0] is None:
|
|
return intervalMembership(True, valid)
|
|
return intervalMembership(None, valid)
|
|
else:
|
|
return NotImplemented
|
|
|
|
def __le__(self, other):
|
|
if isinstance(other, (int, float)):
|
|
if self.end <= other:
|
|
return intervalMembership(True, self.is_valid)
|
|
if self.start > other:
|
|
return intervalMembership(False, self.is_valid)
|
|
else:
|
|
return intervalMembership(None, self.is_valid)
|
|
|
|
if isinstance(other, interval):
|
|
valid = fuzzy_and([self.is_valid, other.is_valid])
|
|
if self.end <= other.start:
|
|
return intervalMembership(True, valid)
|
|
if self.start > other.end:
|
|
return intervalMembership(False, valid)
|
|
return intervalMembership(None, valid)
|
|
else:
|
|
return NotImplemented
|
|
|
|
def __ge__(self, other):
|
|
if isinstance(other, (int, float)):
|
|
if self.start >= other:
|
|
return intervalMembership(True, self.is_valid)
|
|
elif self.end < other:
|
|
return intervalMembership(False, self.is_valid)
|
|
else:
|
|
return intervalMembership(None, self.is_valid)
|
|
elif isinstance(other, interval):
|
|
return other.__le__(self)
|
|
|
|
def __add__(self, other):
|
|
if isinstance(other, (int, float)):
|
|
if self.is_valid:
|
|
return interval(self.start + other, self.end + other)
|
|
else:
|
|
start = self.start + other
|
|
end = self.end + other
|
|
return interval(start, end, is_valid=self.is_valid)
|
|
|
|
elif isinstance(other, interval):
|
|
start = self.start + other.start
|
|
end = self.end + other.end
|
|
valid = fuzzy_and([self.is_valid, other.is_valid])
|
|
return interval(start, end, is_valid=valid)
|
|
else:
|
|
return NotImplemented
|
|
|
|
__radd__ = __add__
|
|
|
|
def __sub__(self, other):
|
|
if isinstance(other, (int, float)):
|
|
start = self.start - other
|
|
end = self.end - other
|
|
return interval(start, end, is_valid=self.is_valid)
|
|
|
|
elif isinstance(other, interval):
|
|
start = self.start - other.end
|
|
end = self.end - other.start
|
|
valid = fuzzy_and([self.is_valid, other.is_valid])
|
|
return interval(start, end, is_valid=valid)
|
|
else:
|
|
return NotImplemented
|
|
|
|
def __rsub__(self, other):
|
|
if isinstance(other, (int, float)):
|
|
start = other - self.end
|
|
end = other - self.start
|
|
return interval(start, end, is_valid=self.is_valid)
|
|
elif isinstance(other, interval):
|
|
return other.__sub__(self)
|
|
else:
|
|
return NotImplemented
|
|
|
|
def __neg__(self):
|
|
if self.is_valid:
|
|
return interval(-self.end, -self.start)
|
|
else:
|
|
return interval(-self.end, -self.start, is_valid=self.is_valid)
|
|
|
|
def __mul__(self, other):
|
|
if isinstance(other, interval):
|
|
if self.is_valid is False or other.is_valid is False:
|
|
return interval(-float('inf'), float('inf'), is_valid=False)
|
|
elif self.is_valid is None or other.is_valid is None:
|
|
return interval(-float('inf'), float('inf'), is_valid=None)
|
|
else:
|
|
inters = []
|
|
inters.append(self.start * other.start)
|
|
inters.append(self.end * other.start)
|
|
inters.append(self.start * other.end)
|
|
inters.append(self.end * other.end)
|
|
start = min(inters)
|
|
end = max(inters)
|
|
return interval(start, end)
|
|
elif isinstance(other, (int, float)):
|
|
return interval(self.start*other, self.end*other, is_valid=self.is_valid)
|
|
else:
|
|
return NotImplemented
|
|
|
|
__rmul__ = __mul__
|
|
|
|
def __contains__(self, other):
|
|
if isinstance(other, (int, float)):
|
|
return self.start <= other and self.end >= other
|
|
else:
|
|
return self.start <= other.start and other.end <= self.end
|
|
|
|
def __rtruediv__(self, other):
|
|
if isinstance(other, (int, float)):
|
|
other = interval(other)
|
|
return other.__truediv__(self)
|
|
elif isinstance(other, interval):
|
|
return other.__truediv__(self)
|
|
else:
|
|
return NotImplemented
|
|
|
|
def __truediv__(self, other):
|
|
# Both None and False are handled
|
|
if not self.is_valid:
|
|
# Don't divide as the value is not valid
|
|
return interval(-float('inf'), float('inf'), is_valid=self.is_valid)
|
|
if isinstance(other, (int, float)):
|
|
if other == 0:
|
|
# Divide by zero encountered. valid nowhere
|
|
return interval(-float('inf'), float('inf'), is_valid=False)
|
|
else:
|
|
return interval(self.start / other, self.end / other)
|
|
|
|
elif isinstance(other, interval):
|
|
if other.is_valid is False or self.is_valid is False:
|
|
return interval(-float('inf'), float('inf'), is_valid=False)
|
|
elif other.is_valid is None or self.is_valid is None:
|
|
return interval(-float('inf'), float('inf'), is_valid=None)
|
|
else:
|
|
# denominator contains both signs, i.e. being divided by zero
|
|
# return the whole real line with is_valid = None
|
|
if 0 in other:
|
|
return interval(-float('inf'), float('inf'), is_valid=None)
|
|
|
|
# denominator negative
|
|
this = self
|
|
if other.end < 0:
|
|
this = -this
|
|
other = -other
|
|
|
|
# denominator positive
|
|
inters = []
|
|
inters.append(this.start / other.start)
|
|
inters.append(this.end / other.start)
|
|
inters.append(this.start / other.end)
|
|
inters.append(this.end / other.end)
|
|
start = max(inters)
|
|
end = min(inters)
|
|
return interval(start, end)
|
|
else:
|
|
return NotImplemented
|
|
|
|
def __pow__(self, other):
|
|
# Implements only power to an integer.
|
|
from .lib_interval import exp, log
|
|
if not self.is_valid:
|
|
return self
|
|
if isinstance(other, interval):
|
|
return exp(other * log(self))
|
|
elif isinstance(other, (float, int)):
|
|
if other < 0:
|
|
return 1 / self.__pow__(abs(other))
|
|
else:
|
|
if int(other) == other:
|
|
return _pow_int(self, other)
|
|
else:
|
|
return _pow_float(self, other)
|
|
else:
|
|
return NotImplemented
|
|
|
|
def __rpow__(self, other):
|
|
if isinstance(other, (float, int)):
|
|
if not self.is_valid:
|
|
#Don't do anything
|
|
return self
|
|
elif other < 0:
|
|
if self.width > 0:
|
|
return interval(-float('inf'), float('inf'), is_valid=False)
|
|
else:
|
|
power_rational = nsimplify(self.start)
|
|
num, denom = power_rational.as_numer_denom()
|
|
if denom % 2 == 0:
|
|
return interval(-float('inf'), float('inf'),
|
|
is_valid=False)
|
|
else:
|
|
start = -abs(other)**self.start
|
|
end = start
|
|
return interval(start, end)
|
|
else:
|
|
return interval(other**self.start, other**self.end)
|
|
elif isinstance(other, interval):
|
|
return other.__pow__(self)
|
|
else:
|
|
return NotImplemented
|
|
|
|
def __hash__(self):
|
|
return hash((self.is_valid, self.start, self.end))
|
|
|
|
|
|
def _pow_float(inter, power):
|
|
"""Evaluates an interval raised to a floating point."""
|
|
power_rational = nsimplify(power)
|
|
num, denom = power_rational.as_numer_denom()
|
|
if num % 2 == 0:
|
|
start = abs(inter.start)**power
|
|
end = abs(inter.end)**power
|
|
if start < 0:
|
|
ret = interval(0, max(start, end))
|
|
else:
|
|
ret = interval(start, end)
|
|
return ret
|
|
elif denom % 2 == 0:
|
|
if inter.end < 0:
|
|
return interval(-float('inf'), float('inf'), is_valid=False)
|
|
elif inter.start < 0:
|
|
return interval(0, inter.end**power, is_valid=None)
|
|
else:
|
|
return interval(inter.start**power, inter.end**power)
|
|
else:
|
|
if inter.start < 0:
|
|
start = -abs(inter.start)**power
|
|
else:
|
|
start = inter.start**power
|
|
|
|
if inter.end < 0:
|
|
end = -abs(inter.end)**power
|
|
else:
|
|
end = inter.end**power
|
|
|
|
return interval(start, end, is_valid=inter.is_valid)
|
|
|
|
|
|
def _pow_int(inter, power):
|
|
"""Evaluates an interval raised to an integer power"""
|
|
power = int(power)
|
|
if power & 1:
|
|
return interval(inter.start**power, inter.end**power)
|
|
else:
|
|
if inter.start < 0 and inter.end > 0:
|
|
start = 0
|
|
end = max(inter.start**power, inter.end**power)
|
|
return interval(start, end)
|
|
else:
|
|
return interval(inter.start**power, inter.end**power)
|