def dual_incidence_structure(self, algorithm=None):
"""
Wraps GAP Design's DualBlockDesign (see [1]). The dual of a block
design may not be a block design.
Also can be called with ``dual_design``.
.. NOTE:
The algorithm="gap" option requires GAP's Design package (included in
the gap_packages Sage spkg).
EXAMPLES::
sage: from sage.combinat.designs.block_design import BlockDesign
sage: D = BlockDesign(4, [[0,2],[1,2,3],[2,3]], test=False)
sage: D
Incidence structure with 4 points and 3 blocks
sage: D.dual_design()
Incidence structure with 3 points and 4 blocks
sage: print D.dual_design(algorithm="gap") # optional - gap_design
IncidenceStructure<points=[0, 1, 2], blocks=[[0], [0, 1, 2], [1], [1, 2]]>
sage: BD = IncidenceStructure(range(7),[[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]], name="FanoPlane")
sage: BD
Incidence structure with 7 points and 7 blocks
sage: print BD.dual_design(algorithm="gap") # optional - gap_design
IncidenceStructure<points=[0, 1, 2, 3, 4, 5, 6], blocks=[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]]>
sage: BD.dual_incidence_structure()
Incidence structure with 7 points and 7 blocks
REFERENCE:
- Soicher, Leonard, Design package manual, available at
http://www.gap-system.org/Manuals/pkg/design/htm/CHAP003.htm
"""
from sage.interfaces.gap import gap, GapElement
from sage.sets.set import Set
from sage.misc.flatten import flatten
from sage.combinat.designs.block_design import BlockDesign
from sage.misc.functional import transpose
if algorithm == "gap":
gap.load_package("design")
gD = self._gap_()
gap.eval("DD:=DualBlockDesign(" + gD + ")")
v = eval(gap.eval("DD.v"))
gblcks = eval(gap.eval("DD.blocks"))
gB = []
for b in gblcks:
gB.append([x - 1 for x in b])
return IncidenceStructure(range(v), gB, name=None, test=False)
pts = self.blocks()
M = transpose(self.incidence_matrix())
blks = self.points()
return IncidenceStructure(pts, blks, M, name=None, test=False)
def dual_incidence_structure(self, algorithm=None):
"""
Wraps GAP Design's DualBlockDesign (see [1]). The dual of a block
design may not be a block design.
Also can be called with ``dual_design``.
.. NOTE:
The algorithm="gap" option requires GAP's Design package (included in
the gap_packages Sage spkg).
EXAMPLES::
sage: from sage.combinat.designs.block_design import BlockDesign
sage: D = BlockDesign(4, [[0,2],[1,2,3],[2,3]], test=False)
sage: D
Incidence structure with 4 points and 3 blocks
sage: D.dual_design()
Incidence structure with 3 points and 4 blocks
sage: print D.dual_design(algorithm="gap") # optional - gap_design
IncidenceStructure<points=[0, 1, 2], blocks=[[0], [0, 1, 2], [1], [1, 2]]>
sage: BD = IncidenceStructure(range(7),[[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]], name="FanoPlane")
sage: BD
Incidence structure with 7 points and 7 blocks
sage: print BD.dual_design(algorithm="gap") # optional - gap_design
IncidenceStructure<points=[0, 1, 2, 3, 4, 5, 6], blocks=[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]]>
sage: BD.dual_incidence_structure()
Incidence structure with 7 points and 7 blocks
REFERENCE:
- Soicher, Leonard, Design package manual, available at
http://www.gap-system.org/Manuals/pkg/design/htm/CHAP003.htm
"""
from sage.interfaces.gap import gap, GapElement
from sage.sets.set import Set
from sage.misc.flatten import flatten
from sage.combinat.designs.block_design import BlockDesign
from sage.misc.functional import transpose
if algorithm=="gap":
gap.load_package("design")
gD = self._gap_()
gap.eval("DD:=DualBlockDesign("+gD+")")
v = eval(gap.eval("DD.v"))
gblcks = eval(gap.eval("DD.blocks"))
gB = []
for b in gblcks:
gB.append([x-1 for x in b])
return IncidenceStructure(range(v), gB, name=None, test=False)
pts = self.blocks()
M = transpose(self.incidence_matrix())
blks = self.points()
return IncidenceStructure(pts, blks, M, name=None, test=False)
def higher_level_UpGj(p, N, klist, m, modformsring, bound):
r"""
Returns a list ``[A_k]`` of square matrices over ``IntegerRing(p^m)``
parameterised by the weights k in ``klist``. The matrix `A_k` is the finite
square matrix which occurs on input p,k,N and m in Step 6 of Algorithm 2 in
[AGBL]_. Notational change from paper: In Step 1 following Wan we defined
j by `k = k_0 + j(p-1)` with `0 \le k_0 < p-1`. Here we replace j by
``kdiv`` so that we may use j as a column index for matrices.)
INPUT:
- ``p`` -- prime at least 5.
- ``N`` -- integer at least 2 and not divisible by p (level).
- ``klist`` -- list of integers all congruent modulo (p-1) (the weights).
- ``m`` -- positive integer.
- ``modformsring`` -- True or False.
- ``bound`` -- (even) positive integer.
OUTPUT:
- list of square matrices.
EXAMPLES::
sage: from sage.modular.overconvergent.hecke_series import higher_level_UpGj
sage: higher_level_UpGj(5,3,[4],2,true,6)
[
[ 1 0 0 0 0 0]
[ 0 1 0 0 0 0]
[ 0 7 0 0 0 0]
[ 0 5 10 20 0 0]
[ 0 7 20 0 20 0]
[ 0 1 24 0 20 0]
]
"""
t = cputime()
# Step 1
k0 = klist[0] % (p - 1)
n = floor(((p + 1) / (p - 1)) * (m + 1))
elldash = compute_elldash(p, N, k0, n)
elldashp = elldash * p
mdash = m + ceil(n / (p + 1))
verbose("done step 1", t)
t = cputime()
# Steps 2 and 3
e, Ep1 = higher_level_katz_exp(p, N, k0, m, mdash, elldash, elldashp,
modformsring, bound)
ell = dimension(transpose(e)[0].parent())
S = e[0, 0].parent()
verbose("done steps 2+3", t)
t = cputime()
# Step 4
R = Ep1.parent()
G = compute_G(p, Ep1)
Alist = []
verbose("done step 4a", t)
t = cputime()
for k in klist:
k = ZZ(k) # convert to sage integer
kdiv = k // (p - 1)
Gkdiv = G**kdiv
T = matrix(S, ell, elldash)
for i in xrange(ell):
ei = R(e[i].list())
Gkdivei = Gkdiv * ei
# act by G^kdiv
for j in xrange(0, elldash):
T[i, j] = Gkdivei[p * j]
verbose("done steps 4b and 5", t)
t = cputime()
# Step 6: solve T = AE using fact E is upper triangular.
# Warning: assumes that T = AE (rather than pT = AE) has
# a solution over Z/(p^mdash). This has always been the case in
# examples computed by the author, see Note 3.1.
A = matrix(S, ell, ell)
verbose("solving a square matrix problem of dimension %s" % ell)
verbose("elldash is %s" % elldash)
for i in xrange(0, ell):
Ti = T[i]
for j in xrange(0, ell):
ej = Ti.parent()([e[j][l] for l in xrange(0, elldash)])
ejleadpos = ej.nonzero_positions()[0]
lj = ZZ(ej[ejleadpos])
A[i, j] = S(ZZ(Ti[j]) / lj)
Ti = Ti - A[i, j] * ej
Alist.append(MatrixSpace(Zmod(p**m), ell, ell)(A))
verbose("done step 6", t)
return Alist
def higher_level_UpGj(p,N,klist,m,modformsring,bound):
r"""
Returns a list ``[A_k]`` of square matrices over ``IntegerRing(p^m)``
parameterised by the weights k in ``klist``. The matrix `A_k` is the finite
square matrix which occurs on input p,k,N and m in Step 6 of Algorithm 2 in
[AGBL]_. Notational change from paper: In Step 1 following Wan we defined
j by `k = k_0 + j(p-1)` with `0 \le k_0 < p-1`. Here we replace j by
``kdiv`` so that we may use j as a column index for matrices.)
INPUT:
- ``p`` -- prime at least 5.
- ``N`` -- integer at least 2 and not divisible by p (level).
- ``klist`` -- list of integers all congruent modulo (p-1) (the weights).
- ``m`` -- positive integer.
- ``modformsring`` -- True or False.
- ``bound`` -- (even) positive integer.
OUTPUT:
- list of square matrices.
EXAMPLES::
sage: from sage.modular.overconvergent.hecke_series import higher_level_UpGj
sage: higher_level_UpGj(5,3,[4],2,true,6)
[
[ 1 0 0 0 0 0]
[ 0 1 0 0 0 0]
[ 0 7 0 0 0 0]
[ 0 5 10 20 0 0]
[ 0 7 20 0 20 0]
[ 0 1 24 0 20 0]
]
"""
t = cputime()
# Step 1
k0 = klist[0] % (p-1)
n = floor(((p+1)/(p-1)) * (m+1))
elldash = compute_elldash(p,N,k0,n)
elldashp = elldash*p
mdash = m + ceil(n/(p+1))
verbose("done step 1",t)
t = cputime()
# Steps 2 and 3
e,Ep1 = higher_level_katz_exp(p,N,k0,m,mdash,elldash,elldashp,modformsring,bound)
ell = dimension(transpose(e)[0].parent())
S = e[0,0].parent()
verbose("done steps 2+3", t)
t = cputime()
# Step 4
R = Ep1.parent()
G = compute_G(p, Ep1)
Alist = []
verbose("done step 4a", t)
t = cputime()
for k in klist:
k = ZZ(k) # convert to sage integer
kdiv = k // (p-1)
Gkdiv = G**kdiv
T = matrix(S,ell,elldash)
for i in xrange(ell):
ei = R(e[i].list())
Gkdivei = Gkdiv*ei; # act by G^kdiv
for j in xrange(0, elldash):
T[i,j] = Gkdivei[p*j]
verbose("done steps 4b and 5", t)
t = cputime()
# Step 6: solve T = AE using fact E is upper triangular.
# Warning: assumes that T = AE (rather than pT = AE) has
# a solution over Z/(p^mdash). This has always been the case in
# examples computed by the author, see Note 3.1.
A = matrix(S,ell,ell)
verbose("solving a square matrix problem of dimension %s" % ell)
verbose("elldash is %s" % elldash)
for i in xrange(0,ell):
Ti = T[i]
for j in xrange(0,ell):
ej = Ti.parent()([e[j][l] for l in xrange(0,elldash)])
ejleadpos = ej.nonzero_positions()[0]
lj = ZZ(ej[ejleadpos])
A[i,j] = S(ZZ(Ti[j])/lj)
Ti = Ti - A[i,j]*ej
Alist.append(MatrixSpace(Zmod(p**m),ell,ell)(A))
verbose("done step 6", t)
return Alist
