Dear all,
I need to compute some larger powers of polynomials in characteristic
p>0. I've noticed that Sage does not do it very efficiently, as even
f^(p^n) takes a long time. I wrote the following code then:
================================
# write m in base n (as vector):
# [v0, v1, ..., vk] --> m = v0 + v1*n + ... + vk*n^k
def base_n(m,n):
if m == 0:
return []
tm, m0 = divmod(m,n)
return [ m0 ] + base_n(tm,n)
# powers of powers of p
def pol_p_power(f,p,k):
# computes f^(p^k)
pp=parent(f).characteristic()
if pp== p:
# right call!
res=0
for mon in f.monomials():
res+=f.monomial_coefficient(mon)^(p^k)*mon^(p^k)
return res
else:
# wrong p -- regular power
f^(p^k)
# compute f^n using the characteristic!
def pol_power(f,n):
P=parent(f)
p=P.characteristic()
v=base_n(n,p) # write n in base p
pwrs={0 : P(1), 1 : f} # save computed powers
res=1
for i in range(len(v)):
if v[i]!=0:
# need this term
if v[i] in pwrs:
# power computed before
res*=pol_p_power(pwrs[v[i]],p,i)
else:
# power not computed before -- save it!
pwrs[v[i]]=f^(v[i])
res*=pol_p_power(pwrs[v[i]],p,i)
return res
# compute maximum of a vector
def my_max(v):
res=v[0]
for i in range(1,len(v)):
if v[i] > res: res=v[i]
return res
# compute f^n using the characteristic
# in this one we precompute all necessary powers
def pol_power2(f,n):
P=parent(f)
p=P.characteristic()
v=base_n(n,p)
pwrs={0 : P(1), 1 : f} # save computed powers
M=my_max(v) # maximum power
for i in range(2,M+1):
# precompute all powers up to the maximum
pwrs[i]=pwrs[i-1]*f
for i in range(M+1):
# save some memory by keeping only the ones we need
if not(i in v):
del pwrs[i]
res=1
for i in range(len(v)):
if v[i]!=0:
res*=pol_p_power(pwrs[v[i]],p,i)
return res
================================
Here is a test:
================================
boole[~/comp/sage]$ sage
----------------------------------------------------------------------
| Sage Version 4.1.2, Release Date: 2009-10-13 |
| Type notebook() for the GUI, and license() for information. |
----------------------------------------------------------------------
sage: attach 'pol_power.sage'
sage: p=13
sage: F=GF(p)
sage: P.<x,y,z>=PolynomialRing(F,3)
sage: nterm=5
sage: pwr=10
sage: f=sum( [F.random_element()*x^randint(0,pwr)*y^randint(0,pwr)
*z^randint(0,pwr) for i in\
...: range(nterm)] )
sage: print "f = ", f
f = 2*x^10*y^6*z^3 - 2*x^6*y^3*z^9 - 5*x^3*y^2*z^9 + y^7*z^6 -
x^4*y^7
sage: ntry=20
sage: min=100
sage: max=1000
sage: for i in random_sublist(range(min,max),ntry/(max-min)):
....: print "i = ", i
....: %time a=pol_power(f,i)
....: %time b=pol_power2(f,i)
....: %time c=f^i # sage's builtin
....: a==c
....: b==c
....: print "######################"
....: print ""
....:
i = 101
CPU times: user 0.09 s, sys: 0.02 s, total: 0.11 s
Wall time: 0.11 s
CPU times: user 0.10 s, sys: 0.01 s, total: 0.11 s
Wall time: 0.11 s
CPU times: user 0.89 s, sys: 0.17 s, total: 1.06 s
Wall time: 1.07 s
_18 = True
_18 = True
######################
i = 137
CPU times: user 0.12 s, sys: 0.02 s, total: 0.14 s
Wall time: 0.14 s
CPU times: user 0.13 s, sys: 0.02 s, total: 0.14 s
Wall time: 0.14 s
CPU times: user 3.36 s, sys: 0.72 s, total: 4.08 s
Wall time: 4.08 s
_21 = True
_21 = True
######################
i = 138
CPU times: user 0.17 s, sys: 0.02 s, total: 0.19 s
Wall time: 0.19 s
CPU times: user 0.17 s, sys: 0.02 s, total: 0.19 s
Wall time: 0.19 s
CPU times: user 3.49 s, sys: 0.68 s, total: 4.17 s
Wall time: 4.17 s
_24 = True
_24 = True
######################
i = 310
CPU times: user 0.91 s, sys: 0.36 s, total: 1.27 s
Wall time: 1.27 s
CPU times: user 0.91 s, sys: 0.36 s, total: 1.27 s
Wall time: 1.27 s
CPU times: user 30.18 s, sys: 6.47 s, total: 36.65 s
Wall time: 36.65 s
_27 = True
_27 = True
######################
i = 312
CPU times: user 0.23 s, sys: 0.07 s, total: 0.30 s
Wall time: 0.30 s
CPU times: user 0.21 s, sys: 0.08 s, total: 0.29 s
Wall time: 0.29 s
CPU times: user 34.07 s, sys: 7.31 s, total: 41.39 s
Wall time: 41.40 s
_30 = True
_30 = True
######################
i = 417
CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s
Wall time: 0.01 s
CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s
Wall time: 0.01 s
CPU times: user 74.05 s, sys: 6.54 s, total: 80.58 s
Wall time: 80.62 s
_33 = True
_33 = True
######################
i = 456
CPU times: user 0.04 s, sys: 0.00 s, total: 0.04 s
Wall time: 0.04 s
CPU times: user 0.04 s, sys: 0.00 s, total: 0.05 s
Wall time: 0.05 s
CPU times: user 91.65 s, sys: 17.02 s, total: 108.66 s
Wall time: 108.69 s
_36 = True
_36 = True
######################
i = 491
CPU times: user 4.06 s, sys: 0.94 s, total: 5.00 s
Wall time: 5.00 s
CPU times: user 4.00 s, sys: 0.95 s, total: 4.95 s
Wall time: 4.95 s
CPU times: user 144.18 s, sys: 31.28 s, total: 175.46 s
Wall time: 175.48 s
_39 = True
_39 = True
######################
i = 513
CPU times: user 0.44 s, sys: 0.19 s, total: 0.63 s
Wall time: 0.63 s
CPU times: user 0.42 s, sys: 0.21 s, total: 0.62 s
Wall time: 0.62 s
CPU times: user 213.25 s, sys: 49.80 s, total: 263.05 s
Wall time: 263.12 s
_42 = True
_42 = True
######################
i = 522
CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 0.00 s
CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 0.00 s
CPU times: user 209.37 s, sys: 48.57 s, total: 257.93 s
Wall time: 257.99 s
_46 = True
_46 = True
######################
i = 603
CPU times: user 0.24 s, sys: 0.02 s, total: 0.27 s
Wall time: 0.27 s
CPU times: user 0.22 s, sys: 0.06 s, total: 0.28 s
Wall time: 0.29 s
CPU times: user 230.83 s, sys: 57.76 s, total: 288.60 s
Wall time: 288.69 s
_50 = True
_50 = True
######################
i = 662
CPU times: user 14.26 s, sys: 3.50 s, total: 17.76 s
Wall time: 17.76 s
CPU times: user 14.60 s, sys: 3.43 s, total: 18.03 s
Wall time: 18.03 s
CPU times: user 430.12 s, sys: 106.54 s, total: 536.66 s
Wall time: 536.77 s
_54 = True
_54 = True
######################
i = 688
CPU times: user 1.95 s, sys: 0.86 s, total: 2.81 s
Wall time: 2.81 s
CPU times: user 1.98 s, sys: 0.82 s, total: 2.80 s
Wall time: 2.80 s
CPU times: user 559.77 s, sys: 128.93 s, total: 688.70 s
Wall time: 688.82 s
_58 = True
_58 = True
######################
i = 697
CPU times: user 0.06 s, sys: 0.00 s, total: 0.06 s
Wall time: 0.06 s
CPU times: user 0.04 s, sys: 0.00 s, total: 0.04 s
Wall time: 0.04 s
CPU times: user 556.29 s, sys: 123.71 s, total: 680.00 s
Wall time: 680.15 s
_62 = True
_62 = True
######################
i = 758
CPU times: user 0.17 s, sys: 0.00 s, total: 0.18 s
Wall time: 0.18 s
CPU times: user 0.20 s, sys: 0.01 s, total: 0.21 s
Wall time: 0.21 s
CPU times: user 585.18 s, sys: 144.59 s, total: 729.77 s
Wall time: 729.88 s
_65 = True
_65 = True
######################
[snip]
================================
So, it seems that these powers could be improved a lot. My second
function did not show much improvement in these cases, but I think it
would in other (maybe extreme) examples.
Unfortunately, I don't know enough to suggest a patch, but hopefully
some one can find a quick way to improve those computations.
Best to all,
Luis
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