def fixed_points_iterator(self, i=0):
"""Same as fixed_points() but acts as an interator.
See doc string for fixed_points().
NOTES: The iterator takes dramatically less time than
fixed_points() when i gets large. For example:
sage: ## TEST THE ITERATOR
sage: plist=[8,7,6,5,4,3,2,1]
sage: g = incidence_graph(plist)
sage: w = ig_weights(plist,plist)
sage: FP = fixed_points_iterator(g,w,len(plist)+1,len(plist))
sage: t0 = cputime()
sage: count=0
sage: for i,fp in enumerate(FP):
... if i%50000 == 0:
... print cputime()-t0,",",
...
sage: print "Total Time (iterator):",cputime()-t0
0.000598000000011 , 6.469392 , 12.936641 , 19.405214 ,
25.886539 , 32.357304 , 38.827102 , 45.297906 , Total Time
(iterator): 46.964347
Compare to 557.46 s for the same call to fixed_points()
"""
# conversion of names to class context
g = self.graph
w = self.weights
n = self.ic_size
if i == 0:
i = self.num_nodes
(pred,succ) = self.bounds[i-1]
node_weight = w[n+i-1]
pred_weight = w[pred]
weight_diff = node_weight - pred_weight
# base of induction
if i == 1:
pred_index_set = set(self.ic_fix[pred])
succ_index_set = set(self.ic_fix[succ])
diff_index_set = succ_index_set.difference(pred_index_set)
C = combinations_iterator( list(diff_index_set), weight_diff )
for c in C:
yield [self.ic_fix[pred] + c]
elif i > 1:
# get a fixed point associated to i-1
FPI = self.fixed_points_iterator(i-1)
for fp in FPI:
fpp = self.ic_fix + fp # tack on the initial data
pred_index_set = set(fpp[pred])
succ_index_set = set(fpp[succ])
diff_index_set = succ_index_set.difference(pred_index_set)
C = combinations_iterator( list(diff_index_set), weight_diff )
for c in C:
yield fp+[fpp[pred]+c]
else:
yield None
def fixed_points(self, i=0):
"""Returns a list of fixed points of the (projection of the) incidence
variety assoc. to g,w (where we project to include only nodes 1..i). This
allows us to build a list of fixed points for the whole incidence variety
inductively.
INPUT:
i = How many nodes to include
This defaults to i = num_nodes (or if i=0)
(from self)
g = incidence graph (result of calling incidence_graph(n,plist)
w = weights for the nodes (result of calling ig_weights(part,plist)
for the same plist)
n = length of init. chain (optional). defaults to (len(w)+1)/2
OUTPUT: [ fp_1, fp_2, ... ] where each fp_i is itself a list of
index sets, one for each node from 1..i. An index set is a list of
indices which determines the character of the $T$ action on the
fiber of the corresponding universal bundle at the fixed point
fp_i.
NOTES:
Tends to bog down around 8 levels of recursion, i.e.
p=[8,7,6,5,4,3,2,1] # part and plist will be the same here
g=incidence_graph(9,p)
w=ig_weights(p,p)
FP = fixed_points(g,w,9,7) # runs for 10 sec
FP = fixed_points(g,w,9,8) # runs for a LONG time
or...
sage: plist=[8,7,6,5,4,3,2,1]
sage: g = incidence_graph(plist)
sage: w = ig_weights(plist,plist)
sage: time FP2 = fixed_points(g,w,len(plist)+1,len(plist)-2)
sage: time FP1 = fixed_points(g,w,len(plist)+1,len(plist)-1)
sage: time FP0 = fixed_points(g,w,len(plist)+1,len(plist))
Time: CPU 0.54 s, Wall: 0.54 s
Time: CPU 9.14 s, Wall: 9.15 s
Time: CPU 557.46 s, Wall: 558.66 s
sage: len(FP2)
5040
sage: len(FP1)
40320
sage: len(FP0)
362880
"""
# conversion of names to class context
g = self.graph
w = self.weights
n = self.ic_size
# i defaults to 0 -- meaning i = num_nodes
if i == 0:
i = self.num_nodes
(pred,succ) = self.bounds[i-1]
node_weight = w[n+i-1]
pred_weight = w[pred]
weight_diff = node_weight - pred_weight
# base of induction
if i == 1:
pred_index_set = set(self.ic_fix[pred])
succ_index_set = set(self.ic_fix[succ])
diff_index_set = succ_index_set.difference(pred_index_set)
C = combinations_iterator( list(diff_index_set), weight_diff )
poss = [ [self.ic_fix[pred]+c] for c in C ]
return poss
elif i > 1:
# get the result for i-1
induct = self.fixed_points(i-1)
poss = []
for fp in induct:
fpp = self.ic_fix + fp # tack on the initial data
pred_index_set = set(fpp[pred])
succ_index_set = set(fpp[succ])
diff_index_set = succ_index_set.difference(pred_index_set)
C = combinations_iterator( list(diff_index_set), weight_diff )
newfps = [ fp+[fpp[pred]+c] for c in C ]
poss = poss + newfps
return poss
else:
return None