def _compute_eigenvector(s, l, m, gam, verbose=False, min_nmax=8):
r"""
Compute the eigenvector and the eigenvalue corresponding to (s, l, m, gam)
INPUT:
- ``s`` -- integer; the spin weight
- ``l`` -- non-negative integer; the harmonic degree
- ``m`` -- integer within the range ``[-l, l]``; the azimuthal number
- ``gam`` -- spheroidicity parameter
- ``verbose`` -- (default: ``False``) determines whether some details of the
computation are printed out
- ``min_nmax`` -- (default: 8) integer; floor for the evaluation of the
parameter ``nmax``, which sets the highest degree of the spherical
harmonic expansion as ``l+nmax``.
"""
nmax = ceil(abs(3 * gam / 2 - gam * gam /
250)) + min_nmax # FIXME : improve the estimate of nmax
if nmax % 2 == 0:
nmax += 1
lmin = max(abs(s), abs(m))
nmin = min(l - lmin, nmax)
size = nmax + nmin + 1
mat = matrix(RDF, size, size)
for i in range(1, size + 1):
mat[i - 1, i - 1] = -kHat(s, l - nmin - 1 + i, m, gam)
if i > 2:
mat[i - 1, i - 3] = -k2(s, l - nmin - 3 + i, m, gam)
if i > 1:
mat[i - 1, i - 2] = -kTilde2(s, l - nmin + i - 2, m, gam)
if (i < size):
mat[i - 1, i] = -kTilde2(s, l - nmin + i - 1, m, gam)
if (i < size - 1):
mat[i - 1, i + 1] = -k2(s, l - nmin + i + -1, m, gam)
if verbose:
print("nmax: {}".format(nmax))
print("lmin: {}".format(lmin))
print("nmin: {}".format(nmin))
print("size: {}".format(size))
# show(mat)
# Computation of the eigenvalues and eigenvectors:
evlist = mat.eigenvectors_right() # list of triples, each triple being
# (eigenvalue, [eigenvector], 1)
sevlist = sorted(evlist, key=lambda x: x[0], reverse=True)
egval, egvec, mult = sevlist[-(nmin + 1)]
egvec = egvec[0] # since egvec is the single-element list [eigenvector]
if verbose:
print("eigenvalue: {}".format(egval))
print("eigenvector: {}".format(egvec))
check = mat * egvec - egval * egvec
print("check: {}".format(check))
lamb = egval - s * (s + 1) - 2 * m * gam + gam * gam
return lamb, egvec, nmin, nmax, lmin
def neighbor(self, pt, d):
r"""
Return the neighbors of the point pt in direction d.
INPUT:
- ``pt`` - tuple, point in Z^d
- ``direction`` - integer, possible values are 1, 2, ..., d and -1,
-2, ..., -d.
EXAMPLES:
sage: from slabbe import BondPercolationSample
sage: S = BondPercolationSample(0.5,2)
sage: S.neighbor((2,3),1)
(3, 3)
sage: S.neighbor((2,3),2)
(2, 4)
sage: S.neighbor((2,3),-1)
(1, 3)
sage: S.neighbor((2,3),-2)
(2, 2)
"""
R = xrange(self._dimension)
a = sgn(d)
d = abs(d)
return tuple(pt[k]+a if k==d-1 else pt[k] for k in R)
def neighbor(self, pt, d):
r"""
Return the neighbors of the point pt in direction d.
INPUT:
- ``pt`` - tuple, point in Z^d
- ``direction`` - integer, possible values are 1, 2, ..., d and -1,
-2, ..., -d.
EXAMPLES:
sage: from slabbe import BondPercolationSample
sage: S = BondPercolationSample(0.5,2)
sage: S.neighbor((2,3),1)
(3, 3)
sage: S.neighbor((2,3),2)
(2, 4)
sage: S.neighbor((2,3),-1)
(1, 3)
sage: S.neighbor((2,3),-2)
(2, 2)
"""
R = xrange(self._dimension)
a = sgn(d)
d = abs(d)
return tuple(pt[k] + a if k == d - 1 else pt[k] for k in R)
def discrepancy(self):
r"""
Return the discrepancy of the word.
This is a distance to the euclidean line defined in [T1980]_.
EXAMPLES::
sage: from slabbe.finite_word import discrepancy
sage: w = words.ChristoffelWord(5,8)
sage: w
word: 0010010100101
sage: discrepancy(w)
12/13
::
sage: for c in w.conjugates(): print c, discrepancy(c)
0010010100101 12/13
0100101001010 7/13
1001010010100 10/13
0010100101001 10/13
0101001010010 7/13
1010010100100 12/13
0100101001001 8/13
1001010010010 9/13
0010100100101 11/13
0101001001010 6/13
1010010010100 11/13
0100100101001 9/13
1001001010010 8/13
REFERENCES:
.. [T1980] R., Tijdeman. The chairman assignment problem. Discrete
Mathematics 32, no 3 (1980): 323-30. doi:10.1016/0012-365X(80)90269-1.
"""
length = self.length()
freq = {
a: QQ((v, length))
for (a, v) in self.evaluation_dict().iteritems()
}
C = Counter()
M = 0
for i, a in enumerate(self):
C[a] += 1
MM = max(abs(freq[b] * (i + 1) - C[b]) for b in freq)
M = max(M, MM)
return M
def __init__(self, v, mod=None):
r"""
Constructor.
Return the Christoffel set of edges.
EXAMPLES::
sage: from slabbe import ChristoffelGraph
sage: ChristoffelGraph((2,5))
Christoffel set of edges for normal vector v=(2, 5)
"""
self._v = vector(v)
if mod:
self._sum_v = mod
else:
self._sum_v = sum(abs(a) for a in v)
DiscreteSubset.__init__(self, dimension=len(v), edge_predicate=self.has_edge)
def __init__(self, v, mod=None):
r"""
Constructor.
Return the Christoffel set of edges.
EXAMPLES::
sage: from slabbe import ChristoffelGraph
sage: ChristoffelGraph((2,5))
Christoffel set of edges for normal vector v=(2, 5)
"""
self._v = vector(v)
if mod:
self._sum_v = mod
else:
self._sum_v = sum(abs(a) for a in v)
DiscreteSubset.__init__(self,
dimension=len(v),
edge_predicate=self.has_edge)
def my_log(number):
return floor(log(abs(number), 10.))