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""" The module contains implemented functions for interval arithmetic."""
from functools import reduce
from sympy.plotting.intervalmath import interval
from sympy.external import import_module
def Abs(x):
if isinstance(x, (int, float)):
return interval(abs(x))
elif isinstance(x, interval):
if x.start < 0 and x.end > 0:
return interval(0, max(abs(x.start), abs(x.end)), is_valid=x.is_valid)
else:
return interval(abs(x.start), abs(x.end))
else:
raise NotImplementedError
#Monotonic
def exp(x):
"""evaluates the exponential of an interval"""
np = import_module('numpy')
if isinstance(x, (int, float)):
return interval(np.exp(x), np.exp(x))
elif isinstance(x, interval):
return interval(np.exp(x.start), np.exp(x.end), is_valid=x.is_valid)
else:
raise NotImplementedError
#Monotonic
def log(x):
"""evaluates the natural logarithm of an interval"""
np = import_module('numpy')
if isinstance(x, (int, float)):
if x <= 0:
return interval(-np.inf, np.inf, is_valid=False)
else:
return interval(np.log(x))
elif isinstance(x, interval):
if not x.is_valid:
return interval(-np.inf, np.inf, is_valid=x.is_valid)
elif x.end <= 0:
return interval(-np.inf, np.inf, is_valid=False)
elif x.start <= 0:
return interval(-np.inf, np.inf, is_valid=None)
return interval(np.log(x.start), np.log(x.end))
else:
raise NotImplementedError
#Monotonic
def log10(x):
"""evaluates the logarithm to the base 10 of an interval"""
np = import_module('numpy')
if isinstance(x, (int, float)):
if x <= 0:
return interval(-np.inf, np.inf, is_valid=False)
else:
return interval(np.log10(x))
elif isinstance(x, interval):
if not x.is_valid:
return interval(-np.inf, np.inf, is_valid=x.is_valid)
elif x.end <= 0:
return interval(-np.inf, np.inf, is_valid=False)
elif x.start <= 0:
return interval(-np.inf, np.inf, is_valid=None)
return interval(np.log10(x.start), np.log10(x.end))
else:
raise NotImplementedError
#Monotonic
def atan(x):
"""evaluates the tan inverse of an interval"""
np = import_module('numpy')
if isinstance(x, (int, float)):
return interval(np.arctan(x))
elif isinstance(x, interval):
start = np.arctan(x.start)
end = np.arctan(x.end)
return interval(start, end, is_valid=x.is_valid)
else:
raise NotImplementedError
#periodic
def sin(x):
"""evaluates the sine of an interval"""
np = import_module('numpy')
if isinstance(x, (int, float)):
return interval(np.sin(x))
elif isinstance(x, interval):
if not x.is_valid:
return interval(-1, 1, is_valid=x.is_valid)
na, __ = divmod(x.start, np.pi / 2.0)
nb, __ = divmod(x.end, np.pi / 2.0)
start = min(np.sin(x.start), np.sin(x.end))
end = max(np.sin(x.start), np.sin(x.end))
if nb - na > 4:
return interval(-1, 1, is_valid=x.is_valid)
elif na == nb:
return interval(start, end, is_valid=x.is_valid)
else:
if (na - 1) // 4 != (nb - 1) // 4:
#sin has max
end = 1
if (na - 3) // 4 != (nb - 3) // 4:
#sin has min
start = -1
return interval(start, end)
else:
raise NotImplementedError
#periodic
def cos(x):
"""Evaluates the cos of an interval"""
np = import_module('numpy')
if isinstance(x, (int, float)):
return interval(np.sin(x))
elif isinstance(x, interval):
if not (np.isfinite(x.start) and np.isfinite(x.end)):
return interval(-1, 1, is_valid=x.is_valid)
na, __ = divmod(x.start, np.pi / 2.0)
nb, __ = divmod(x.end, np.pi / 2.0)
start = min(np.cos(x.start), np.cos(x.end))
end = max(np.cos(x.start), np.cos(x.end))
if nb - na > 4:
#differ more than 2*pi
return interval(-1, 1, is_valid=x.is_valid)
elif na == nb:
#in the same quadarant
return interval(start, end, is_valid=x.is_valid)
else:
if (na) // 4 != (nb) // 4:
#cos has max
end = 1
if (na - 2) // 4 != (nb - 2) // 4:
#cos has min
start = -1
return interval(start, end, is_valid=x.is_valid)
else:
raise NotImplementedError
def tan(x):
"""Evaluates the tan of an interval"""
return sin(x) / cos(x)
#Monotonic
def sqrt(x):
"""Evaluates the square root of an interval"""
np = import_module('numpy')
if isinstance(x, (int, float)):
if x > 0:
return interval(np.sqrt(x))
else:
return interval(-np.inf, np.inf, is_valid=False)
elif isinstance(x, interval):
#Outside the domain
if x.end < 0:
return interval(-np.inf, np.inf, is_valid=False)
#Partially outside the domain
elif x.start < 0:
return interval(-np.inf, np.inf, is_valid=None)
else:
return interval(np.sqrt(x.start), np.sqrt(x.end),
is_valid=x.is_valid)
else:
raise NotImplementedError
def imin(*args):
"""Evaluates the minimum of a list of intervals"""
np = import_module('numpy')
if not all(isinstance(arg, (int, float, interval)) for arg in args):
return NotImplementedError
else:
new_args = [a for a in args if isinstance(a, (int, float))
or a.is_valid]
if len(new_args) == 0:
if all(a.is_valid is False for a in args):
return interval(-np.inf, np.inf, is_valid=False)
else:
return interval(-np.inf, np.inf, is_valid=None)
start_array = [a if isinstance(a, (int, float)) else a.start
for a in new_args]
end_array = [a if isinstance(a, (int, float)) else a.end
for a in new_args]
return interval(min(start_array), min(end_array))
def imax(*args):
"""Evaluates the maximum of a list of intervals"""
np = import_module('numpy')
if not all(isinstance(arg, (int, float, interval)) for arg in args):
return NotImplementedError
else:
new_args = [a for a in args if isinstance(a, (int, float))
or a.is_valid]
if len(new_args) == 0:
if all(a.is_valid is False for a in args):
return interval(-np.inf, np.inf, is_valid=False)
else:
return interval(-np.inf, np.inf, is_valid=None)
start_array = [a if isinstance(a, (int, float)) else a.start
for a in new_args]
end_array = [a if isinstance(a, (int, float)) else a.end
for a in new_args]
return interval(max(start_array), max(end_array))
#Monotonic
def sinh(x):
"""Evaluates the hyperbolic sine of an interval"""
np = import_module('numpy')
if isinstance(x, (int, float)):
return interval(np.sinh(x), np.sinh(x))
elif isinstance(x, interval):
return interval(np.sinh(x.start), np.sinh(x.end), is_valid=x.is_valid)
else:
raise NotImplementedError
def cosh(x):
"""Evaluates the hyperbolic cos of an interval"""
np = import_module('numpy')
if isinstance(x, (int, float)):
return interval(np.cosh(x), np.cosh(x))
elif isinstance(x, interval):
#both signs
if x.start < 0 and x.end > 0:
end = max(np.cosh(x.start), np.cosh(x.end))
return interval(1, end, is_valid=x.is_valid)
else:
#Monotonic
start = np.cosh(x.start)
end = np.cosh(x.end)
return interval(start, end, is_valid=x.is_valid)
else:
raise NotImplementedError
#Monotonic
def tanh(x):
"""Evaluates the hyperbolic tan of an interval"""
np = import_module('numpy')
if isinstance(x, (int, float)):
return interval(np.tanh(x), np.tanh(x))
elif isinstance(x, interval):
return interval(np.tanh(x.start), np.tanh(x.end), is_valid=x.is_valid)
else:
raise NotImplementedError
def asin(x):
"""Evaluates the inverse sine of an interval"""
np = import_module('numpy')
if isinstance(x, (int, float)):
#Outside the domain
if abs(x) > 1:
return interval(-np.inf, np.inf, is_valid=False)
else:
return interval(np.arcsin(x), np.arcsin(x))
elif isinstance(x, interval):
#Outside the domain
if x.is_valid is False or x.start > 1 or x.end < -1:
return interval(-np.inf, np.inf, is_valid=False)
#Partially outside the domain
elif x.start < -1 or x.end > 1:
return interval(-np.inf, np.inf, is_valid=None)
else:
start = np.arcsin(x.start)
end = np.arcsin(x.end)
return interval(start, end, is_valid=x.is_valid)
def acos(x):
"""Evaluates the inverse cos of an interval"""
np = import_module('numpy')
if isinstance(x, (int, float)):
if abs(x) > 1:
#Outside the domain
return interval(-np.inf, np.inf, is_valid=False)
else:
return interval(np.arccos(x), np.arccos(x))
elif isinstance(x, interval):
#Outside the domain
if x.is_valid is False or x.start > 1 or x.end < -1:
return interval(-np.inf, np.inf, is_valid=False)
#Partially outside the domain
elif x.start < -1 or x.end > 1:
return interval(-np.inf, np.inf, is_valid=None)
else:
start = np.arccos(x.start)
end = np.arccos(x.end)
return interval(start, end, is_valid=x.is_valid)
def ceil(x):
"""Evaluates the ceiling of an interval"""
np = import_module('numpy')
if isinstance(x, (int, float)):
return interval(np.ceil(x))
elif isinstance(x, interval):
if x.is_valid is False:
return interval(-np.inf, np.inf, is_valid=False)
else:
start = np.ceil(x.start)
end = np.ceil(x.end)
#Continuous over the interval
if start == end:
return interval(start, end, is_valid=x.is_valid)
else:
#Not continuous over the interval
return interval(start, end, is_valid=None)
else:
return NotImplementedError
def floor(x):
"""Evaluates the floor of an interval"""
np = import_module('numpy')
if isinstance(x, (int, float)):
return interval(np.floor(x))
elif isinstance(x, interval):
if x.is_valid is False:
return interval(-np.inf, np.inf, is_valid=False)
else:
start = np.floor(x.start)
end = np.floor(x.end)
#continuous over the argument
if start == end:
return interval(start, end, is_valid=x.is_valid)
else:
#not continuous over the interval
return interval(start, end, is_valid=None)
else:
return NotImplementedError
def acosh(x):
"""Evaluates the inverse hyperbolic cosine of an interval"""
np = import_module('numpy')
if isinstance(x, (int, float)):
#Outside the domain
if x < 1:
return interval(-np.inf, np.inf, is_valid=False)
else:
return interval(np.arccosh(x))
elif isinstance(x, interval):
#Outside the domain
if x.end < 1:
return interval(-np.inf, np.inf, is_valid=False)
#Partly outside the domain
elif x.start < 1:
return interval(-np.inf, np.inf, is_valid=None)
else:
start = np.arccosh(x.start)
end = np.arccosh(x.end)
return interval(start, end, is_valid=x.is_valid)
else:
return NotImplementedError
#Monotonic
def asinh(x):
"""Evaluates the inverse hyperbolic sine of an interval"""
np = import_module('numpy')
if isinstance(x, (int, float)):
return interval(np.arcsinh(x))
elif isinstance(x, interval):
start = np.arcsinh(x.start)
end = np.arcsinh(x.end)
return interval(start, end, is_valid=x.is_valid)
else:
return NotImplementedError
def atanh(x):
"""Evaluates the inverse hyperbolic tangent of an interval"""
np = import_module('numpy')
if isinstance(x, (int, float)):
#Outside the domain
if abs(x) >= 1:
return interval(-np.inf, np.inf, is_valid=False)
else:
return interval(np.arctanh(x))
elif isinstance(x, interval):
#outside the domain
if x.is_valid is False or x.start >= 1 or x.end <= -1:
return interval(-np.inf, np.inf, is_valid=False)
#partly outside the domain
elif x.start <= -1 or x.end >= 1:
return interval(-np.inf, np.inf, is_valid=None)
else:
start = np.arctanh(x.start)
end = np.arctanh(x.end)
return interval(start, end, is_valid=x.is_valid)
else:
return NotImplementedError
#Three valued logic for interval plotting.
def And(*args):
"""Defines the three valued ``And`` behaviour for a 2-tuple of
three valued logic values"""
def reduce_and(cmp_intervala, cmp_intervalb):
if cmp_intervala[0] is False or cmp_intervalb[0] is False:
first = False
elif cmp_intervala[0] is None or cmp_intervalb[0] is None:
first = None
else:
first = True
if cmp_intervala[1] is False or cmp_intervalb[1] is False:
second = False
elif cmp_intervala[1] is None or cmp_intervalb[1] is None:
second = None
else:
second = True
return (first, second)
return reduce(reduce_and, args)
def Or(*args):
"""Defines the three valued ``Or`` behaviour for a 2-tuple of
three valued logic values"""
def reduce_or(cmp_intervala, cmp_intervalb):
if cmp_intervala[0] is True or cmp_intervalb[0] is True:
first = True
elif cmp_intervala[0] is None or cmp_intervalb[0] is None:
first = None
else:
first = False
if cmp_intervala[1] is True or cmp_intervalb[1] is True:
second = True
elif cmp_intervala[1] is None or cmp_intervalb[1] is None:
second = None
else:
second = False
return (first, second)
return reduce(reduce_or, args)
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