def test_sets():
x = Integer(2)
y = Integer(3)
x1 = sympy.Integer(2)
y1 = sympy.Integer(3)
assert Interval(x, y) == Interval(x1, y1)
assert Interval(x1, y) == Interval(x1, y1)
assert Interval(x, y)._sympy_() == sympy.Interval(x1, y1)
assert sympify(sympy.Interval(x1, y1)) == Interval(x, y)
assert sympify(sympy.EmptySet()) == EmptySet()
assert sympy.EmptySet() == EmptySet()._sympy_()
assert FiniteSet(x, y) == FiniteSet(x1, y1)
assert FiniteSet(x1, y) == FiniteSet(x1, y1)
assert FiniteSet(x, y)._sympy_() == sympy.FiniteSet(x1, y1)
assert sympify(sympy.FiniteSet(x1, y1)) == FiniteSet(x, y)
x = Interval(1, 2)
y = Interval(2, 3)
x1 = sympy.Interval(1, 2)
y1 = sympy.Interval(2, 3)
assert Union(x, y) == Union(x1, y1)
assert Union(x1, y) == Union(x1, y1)
assert Union(x, y)._sympy_() == sympy.Union(x1, y1)
assert sympify(sympy.Union(x1, y1)) == Union(x, y)
assert Complement(x, y) == Complement(x1, y1)
assert Complement(x1, y) == Complement(x1, y1)
assert Complement(x, y)._sympy_() == sympy.Complement(x1, y1)
assert sympify(sympy.Complement(x1, y1)) == Complement(x, y)
def test_sets():
x = Integer(2)
y = Integer(3)
x1 = sympy.Integer(2)
y1 = sympy.Integer(3)
assert Interval(x, y) == Interval(x1, y1)
assert Interval(x1, y) == Interval(x1, y1)
assert Interval(x, y)._sympy_() == sympy.Interval(x1, y1)
assert sympify(sympy.Interval(x1, y1)) == Interval(x, y)
assert sympify(sympy.S.EmptySet) == EmptySet()
assert sympy.S.EmptySet == EmptySet()._sympy_()
assert sympify(sympy.S.UniversalSet) == UniversalSet()
assert sympy.S.UniversalSet == UniversalSet()._sympy_()
assert sympify(sympy.S.Reals) == Reals()
assert sympy.S.Reals == Reals()._sympy_()
assert sympify(sympy.S.Rationals) == Rationals()
assert sympy.S.Rationals == Rationals()._sympy_()
assert sympify(sympy.S.Integers) == Integers()
assert sympy.S.Integers == Integers()._sympy_()
assert FiniteSet(x, y) == FiniteSet(x1, y1)
assert FiniteSet(x1, y) == FiniteSet(x1, y1)
assert FiniteSet(x, y)._sympy_() == sympy.FiniteSet(x1, y1)
assert sympify(sympy.FiniteSet(x1, y1)) == FiniteSet(x, y)
x = Interval(1, 2)
y = Interval(2, 3)
x1 = sympy.Interval(1, 2)
y1 = sympy.Interval(2, 3)
assert Union(x, y) == Union(x1, y1)
assert Union(x1, y) == Union(x1, y1)
assert Union(x, y)._sympy_() == sympy.Union(x1, y1)
assert sympify(sympy.Union(x1, y1)) == Union(x, y)
assert Complement(x, y) == Complement(x1, y1)
assert Complement(x1, y) == Complement(x1, y1)
assert Complement(x, y)._sympy_() == sympy.Complement(x1, y1)
assert sympify(sympy.Complement(x1, y1)) == Complement(x, y)
def __sub__(self, subs):
'''
Return a new relation which is self sub subs.
'''
temp = sympy.Complement(self, subs)
l = []
for i in temp:
l.append(i)
if len(l) == 0:
l.append(('xxx', 'xxx'))
result = Relation(*l)
return result
def singularities(self):
return sp.Complement(sp.Reals, self.domain)
def maximum_2(max_f, max_solution, max_combination, max_feasible_interval,
max_interval, max3_f, max3_solution, max3_combination,
max3_feasible_interval, max3_interval):
# this function gets two arrays of maximum intervals and returns an overall maximum interval array
# the concept is simillar to the function maximum_f() above. Only the starting point is the first
# array, max_interval (instead the total feasible reagion of the constant ),
# and the loop is over the second array (max3_interval)
# this is used to combine intervals and maximums caculated by diffrent processors in case of parallel processing.
# loop over all inetrvals in max3_interval
for i_max3 in range(len(max3_interval)):
# temporary storage for intervals and their values
max2_f = []
max2_solution = []
max2_combination = []
max2_feasible_interval = []
max2_interval = []
# loop over all inetrvals in max_interval to be updated by max3_interval
for i_max in range(len(max_interval)):
# intersection of current maximum interval with each new maximum interval
inter1 = sp.Intersection(max_interval[i_max],
max3_interval[i_max3])
if inter1 == sp.EmptySet: # if there is no intersection keep current value
max2_f.append(max_f[i_max])
max2_solution.append(max_solution[i_max])
max2_combination.append(max_combination[i_max])
max2_feasible_interval.append(max_feasible_interval[i_max])
max2_interval.append(max_interval[i_max])
else: # if there is intesection
# check in which interval the new maximum is larger than the current one
exp1 = sp.sympify(max3_f[i_max3] - max_f[i_max] > 0)
exp1 = exp1.subs(
{'s': 'd'}
) # becuase sympy is stupid and sometimes considers same 's' two variables.
exp1 = exp1.subs({'d': 's'})
if exp1 == False:
sol1_set = sp.EmptySet
elif exp1 == True:
sol1_set = sp.UniversalSet
else:
sol1 = solve(exp1)
sol1_set = sol1.as_set()
new_max_inter = sp.Intersection(
inter1, sol1_set
) # intersection of interval that the new one is bigger and feasible
if new_max_inter != sp.EmptySet: # if not empty make a new interval and keep the new maximum value as its maximum
max2_f.append(max3_f[i_max3])
max2_solution.append(max3_solution[i_max3])
max2_combination.append(max3_combination[i_max3])
max2_feasible_interval.append(
max3_feasible_interval[i_max3])
max2_interval.append(new_max_inter)
max_comp_inter = sp.Complement(
max_interval[i_max],
new_max_inter) # complement of the updated interval
if max_comp_inter != sp.EmptySet: # if complemet is not emty, ratain the previous maximum value for this complement interval
max2_f.append(max_f[i_max])
max2_solution.append(max_solution[i_max])
max2_combination.append(max_combination[i_max])
max2_feasible_interval.append(
max_feasible_interval[i_max])
max2_interval.append(max_comp_inter)
else: # if there is no intersection between current interval and the interval that the new one is bigger ratin the current value
max2_f.append(max_f[i_max])
max2_solution.append(max_solution[i_max])
max2_combination.append(max_combination[i_max])
max2_feasible_interval.append(max_feasible_interval[i_max])
max2_interval.append(max_interval[i_max])
# move from temporary storage to permanent
max_f = max2_f.copy()
max_solution = max2_solution.copy()
max_combination = max2_combination.copy()
max_feasible_interval = max2_feasible_interval.copy()
max_interval = max2_interval.copy()
return max_f, max_solution, max_combination, max_feasible_interval, max_interval
def maximum_f(feasible_f, feasible_solution, feasible_combination,
feasible_interval, c2_exp, comm, p_n, p_i):
# this function gets the array of optimum values and their feasible regions and comparing all of them finds
# the maximum value for diffrent inetrvals of possible reagion of the constant.
# the maximum value can be a number or a function of the constant.
if c2_exp == []: # if there is no constants
sol1_set = sp.Interval(0, 0) # set inetval
else:
# convert constarint on constants to set. This is the initial interval to be divided later for diffrent maximums
sol1 = sp.solve(c2_exp)
sol1_set = sol1.as_set()
# here we define an array of intervals that have one interavl which is allowed region of the constant
# We also set the optimum function value for this iterval to be negative infinity (-sp.oo)
# later we update the arrays to have smaller inetrvals each with the optimum value for that interval
# Outputs:
# max_f: an array of maximum values for all the subintervals of possible interval of the constant
# max_solution: to store the variable values for each interval
# max_comb_id: keeping id for backward check
# max_intersect_id: keeping id for backward check
# max_opt_id: keeping id for backward check
# max_feasible_id: keeping id for backward check
# max_feasible_interval: list of intervals where the current maximum is feasible
# (can be larger than interval below sincs below is both feasible and maximum)
# max_interval: array of subintervals of the allowed region of the constant that has a distict maximum value save in max_f above.
max_f = [-sp.oo] # set the maximum value of function to negative infinity
max_solution = ["No answer"
] # to store the variable values for each maximum
max_combination = ["No answer"]
max_feasible_interval = ["No answer"]
max_interval = [
sol1_set
] # set array of interval to have one member which is the allowed reagion of constant
# t_p = t_print
# # used to caculate total caculate time
# start = time.time()
# loop over all feasible solutions
for i_f in range(len(feasible_f)):
# max_2* arrays are used as temporary storages of maximum intervals and at the end of the loop are
# copied to max* arrays
max2_f = []
max2_sol = []
max2_combination = []
max2_feasible_interval = []
max2_interval = []
# loop over all previous intervals of maximum values saved in max* arrays
for i_max in range(len(max_interval)):
# intersection of current optimum interval with each previously calculated maximum interval
inter1 = sp.Intersection(max_interval[i_max],
feasible_interval[i_f])
if inter1 == sp.EmptySet: # if there is no intersection retain the current interval and associated maximum
max2_f.append(max_f[i_max])
max2_sol.append(max_solution[i_max])
max2_combination.append(max_combination[i_max])
max2_feasible_interval.append(max_feasible_interval[i_max])
max2_interval.append(max_interval[i_max])
else: # if there is an intersection, find the interval that the new maximum is larger than the previous maximum
exp1 = (
feasible_f[i_f] - max_f[i_max]
) > 0 # make an expersion to solve to see if current optimum is larger than previously found optimum for this interval
exp1 = exp1.subs(
{'s': 'd'}
) # becuase sympy is stupid and sometimes considers same 's' two variables.
exp1 = exp1.subs({'d': 's'})
if exp1 == False: # new optimum is not larger anywhere
sol1_set = sp.EmptySet
elif exp1 == True: # new interval is always bigger
sol1_set = sp.UniversalSet
else: # if only larger on some interval, find the interval
sol1 = solve(exp1)
sol1_set = sol1.as_set()
new_max_inter = sp.Intersection(
inter1, sol1_set
) # define a new interval that is intersection of where the new optimum is larger and is feasible and whithin current maximum interval
if new_max_inter != sp.EmptySet: # if this new interval is not empty append this interval and its f value to the list of intervals for maximum
max2_f.append(feasible_f[i_f])
max2_sol.append(feasible_solution[i_f])
max2_combination.append(feasible_combination[i_f])
max2_feasible_interval.append(feasible_interval[i_f])
max2_interval.append(new_max_inter)
max_comp_inter = sp.Complement(
max_interval[i_max], new_max_inter
) # find the complement of the inetvarl above to retain the previous maxium value for the remaining interval
if max_comp_inter != sp.EmptySet: # if remaining is not empty set add it to the array of maximums and retain the previous value for the maximum
max2_f.append(max_f[i_max])
max2_sol.append(max_solution[i_max])
max2_combination.append(max_combination[i_max])
max2_feasible_interval.append(
max_feasible_interval[i_max])
max2_interval.append(max_comp_inter)
else: # if this new interval is empty retain the previous interval and its value as maximum
max2_f.append(max_f[i_max])
max2_sol.append(max_solution[i_max])
max2_combination.append(max_combination[i_max])
max2_feasible_interval.append(max_feasible_interval[i_max])
max2_interval.append(max_interval[i_max])
# copy from temporary to permanent storage for the next iteration
max_f = max2_f.copy()
max_solution = max2_sol.copy()
max_combination = max2_combination.copy()
max_feasible_interval = max2_feasible_interval.copy()
max_interval = max2_interval.copy()
# print progress
# t = time.time() - start
#
# # printing progress
# if t > t_p :
# perc = ((i_f+1)/(len(feasible_f)-1)*100)
# if perc > 0:
# t_left = t * (100 - perc) / perc
# print("process",p_i, "maximum", int(perc), "% Time to finish:" , str(datetime.timedelta(seconds=int(t_left)) ) )
# t_p = t_p + t_print
#
# # caculate and print run time
# end = time.time()
#
# print("process", p_i, " Maximum completed. Total time: ", str(datetime.timedelta(seconds=(end-start))) )
print("process", p_i, " Maximum completed.")
# Outputs:
# max_f: an array of maximum values for all the subintervals of possible interval of the constant
# max_solution: to store the variable values for each interval
# max_combination: keeping initial combination of constraints that lead to this.
# max_feasible_interval: list of intervals where the current maximum is feasible
# (can be larger than interval below sincs below is both feasible and maximum)
# max_interval: array of subintervals of the allowed region of the constant that has a distict maximum value save in max_f above.
return max_f, max_solution, max_combination, max_feasible_interval, max_interval