def module_composition_factors(self, algorithm=None):
r"""
Returns a list of triples consisting of [base field, dimension,
irreducibility], for each of the Meataxe composition factors
modules. The algorithm="verbose" option returns more information, but
in Meataxe notation.
EXAMPLES::
sage: F=GF(3);MS=MatrixSpace(F,4,4)
sage: M=MS(0)
sage: M[0,1]=1;M[1,2]=1;M[2,3]=1;M[3,0]=1
sage: G = MatrixGroup([M])
sage: G.module_composition_factors()
[(Finite Field of size 3, 1, True),
(Finite Field of size 3, 1, True),
(Finite Field of size 3, 2, True)]
sage: F = GF(7); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[0,1],[-1,0]]),MS([[1,1],[2,3]])]
sage: G = MatrixGroup(gens)
sage: G.module_composition_factors()
[(Finite Field of size 7, 2, True)]
Type "G.module_composition_factors(algorithm='verbose')" to get a
more verbose version.
For more on MeatAxe notation, see
http://www.gap-system.org/Manuals/doc/htm/ref/CHAP067.htm
"""
from sage.misc.sage_eval import sage_eval
F = self.base_ring()
if not (F.is_finite()):
raise NotImplementedError, "Base ring must be finite."
q = F.cardinality()
gens = self.gens()
n = self.degree()
MS = MatrixSpace(F, n, n)
mats = [] # initializing list of mats by which the gens act on self
W = self.matrix_space().row_space()
for g in gens:
p = MS(g.matrix())
m = p.rows()
mats.append(m)
mats_str = str(gap([[list(r) for r in m] for m in mats]))
gap.eval("M:=GModuleByMats(" + mats_str + ", GF(" + str(q) + "))")
gap.eval("MCFs := MTX.CompositionFactors( M )")
N = eval(gap.eval("Length(MCFs)"))
if algorithm == "verbose":
print gap.eval('MCFs') + "\n"
L = []
for i in range(1, N + 1):
gap.eval("MCF := MCFs[%s]" % i)
L.append(
tuple([
sage_eval(gap.eval("MCF.field")),
eval(gap.eval("MCF.dimension")),
sage_eval(gap.eval("MCF.IsIrreducible"))
]))
return sorted(L)
def module_composition_factors(self, algorithm=None):
r"""
Return a list of triples consisting of [base field, dimension,
irreducibility], for each of the Meataxe composition factors
modules. The ``algorithm="verbose"`` option returns more information,
but in Meataxe notation.
EXAMPLES::
sage: F=GF(3);MS=MatrixSpace(F,4,4)
sage: M=MS(0)
sage: M[0,1]=1;M[1,2]=1;M[2,3]=1;M[3,0]=1
sage: G = MatrixGroup([M])
sage: G.module_composition_factors()
[(Finite Field of size 3, 1, True),
(Finite Field of size 3, 1, True),
(Finite Field of size 3, 2, True)]
sage: F = GF(7); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[0,1],[-1,0]]),MS([[1,1],[2,3]])]
sage: G = MatrixGroup(gens)
sage: G.module_composition_factors()
[(Finite Field of size 7, 2, True)]
Type ``G.module_composition_factors(algorithm='verbose')`` to get a
more verbose version.
For more on MeatAxe notation, see
http://www.gap-system.org/Manuals/doc/ref/chap69.html
"""
from sage.misc.sage_eval import sage_eval
F = self.base_ring()
if not(F.is_finite()):
raise NotImplementedError("Base ring must be finite.")
q = F.cardinality()
gens = self.gens()
n = self.degree()
MS = MatrixSpace(F,n,n)
mats = [] # initializing list of mats by which the gens act on self
W = self.matrix_space().row_space()
for g in gens:
p = MS(g.matrix())
m = p.rows()
mats.append(m)
mats_str = str(gap([[list(r) for r in m] for m in mats]))
gap.eval("M:=GModuleByMats("+mats_str+", GF("+str(q)+"))")
gap.eval("MCFs := MTX.CompositionFactors( M )")
N = eval(gap.eval("Length(MCFs)"))
if algorithm == "verbose":
print(gap.eval('MCFs') + "\n")
L = []
for i in range(1,N+1):
gap.eval("MCF := MCFs[%s]"%i)
L.append(tuple([sage_eval(gap.eval("MCF.field")),
eval(gap.eval("MCF.dimension")),
sage_eval(gap.eval("MCF.IsIrreducible")) ]))
return sorted(L)
def _element_constructor_(self, x):
r"""
Convert ``x`` into ``self``.
EXAMPLES::
sage: R = ShuffleAlgebra(QQ,'xy')
sage: x, y = R.gens()
sage: R(3) # indirect doctest
3*B[word: ]
sage: R(x)
B[word: x]
"""
P = x.parent()
if isinstance(P, ShuffleAlgebra):
if P is self:
return x
if not (P is self.base_ring()):
return self.element_class(self, x.monomial_coefficients())
# ok, not a shuffle algebra element (or should not be viewed as one).
if isinstance(x, basestring):
from sage.misc.sage_eval import sage_eval
return sage_eval(x,locals=self.gens_dict())
R = self.base_ring()
# coercion via base ring
x = R(x)
if x == 0:
return self.element_class(self,{})
else:
return self.from_base_ring_from_one_basis(x)
def spherical_bessel_Y(n, var, algorithm="maxima"):
r"""
Returns the spherical Bessel function of the second kind for
integers n -1.
Reference: AS 10.1.9 page 437 and AS 10.1.15 page 439.
EXAMPLES::
sage: x = PolynomialRing(QQ, 'x').gen()
sage: spherical_bessel_Y(2,x)
-((3/x^2 - 1)*cos(x) + 3*sin(x)/x)/x
"""
if algorithm == "scipy":
import scipy.special
ans = str(scipy.special.sph_yn(int(n), float(var)))
ans = ans.replace("(", "")
ans = ans.replace(")", "")
ans = ans.replace("j", "*I")
return sage_eval(ans)
elif algorithm == 'maxima':
_init()
return meval("spherical_bessel_y(%s,%s)" % (ZZ(n), var))
else:
raise ValueError, "unknown algorithm '%s'" % algorithm
def gen_laguerre(n, a, x):
"""
Returns the generalized Laguerre polynomial for integers `n > -1`.
Typically, a = 1/2 or a = -1/2.
REFERENCE:
- table on page 789 in AS.
EXAMPLES::
sage: x = PolynomialRing(QQ, 'x').gen()
sage: gen_laguerre(2,1,x)
1/2*x^2 - 3*x + 3
sage: gen_laguerre(2,1/2,x)
1/2*x^2 - 5/2*x + 15/8
sage: gen_laguerre(2,-1/2,x)
1/2*x^2 - 3/2*x + 3/8
sage: gen_laguerre(2,0,x)
1/2*x^2 - 2*x + 1
sage: gen_laguerre(3,0,x)
-1/6*x^3 + 3/2*x^2 - 3*x + 1
"""
from sage.functions.all import sqrt
_init()
return sage_eval(maxima.eval("gen_laguerre(%s,%s,x)" % (ZZ(n), a)), locals={"x": x})
def right_coset_representatives(self):
r"""
Return the right coset representatives of ``self``.
EXAMPLES::
sage: W = ReflectionGroup(['A',2]) # optional - gap3
sage: for w in W: # optional - gap3
....: rcr = w.right_coset_representatives() # optional - gap3
....: print("%s %s"%(w.reduced_word(), # optional - gap3
....: [v.reduced_word() for v in rcr])) # optional - gap3
[] [[], [2], [1], [2, 1], [1, 2], [1, 2, 1]]
[2] [[], [2], [1]]
[1] [[], [1], [1, 2]]
[1, 2] [[]]
[2, 1] [[]]
[1, 2, 1] [[], [2], [2, 1]]
"""
from sage.combinat.root_system.reflection_group_complex import _gap_return
W = self.parent()
T = W.reflections()
T_fix = [i + 1 for i in T.keys() if self.fix_space().is_subspace(T[i].fix_space())]
S = str(
gap3(
"ReducedRightCosetRepresentatives(%s,ReflectionSubgroup(%s,%s))"
% (W._gap_group._name, W._gap_group._name, T_fix)
)
)
return sage_eval(_gap_return(S, coerce_obj="W"), locals={"self": self, "W": W})
def _element_constructor_(self, x):
r"""
Convert ``x`` into ``self``.
EXAMPLES::
sage: R = ShuffleAlgebra(QQ,'xy')
sage: x, y = R.gens()
sage: R(3) # indirect doctest
3*B[word: ]
sage: R(x)
B[word: x]
"""
P = x.parent()
if isinstance(P, ShuffleAlgebra):
if P is self:
return x
if not (P is self.base_ring()):
return self.element_class(self, x.monomial_coefficients())
# ok, not a shuffle algebra element (or should not be viewed as one).
if isinstance(x, basestring):
from sage.misc.sage_eval import sage_eval
return sage_eval(x, locals=self.gens_dict())
R = self.base_ring()
# coercion via base ring
x = R(x)
if x == 0:
return self.element_class(self, {})
else:
return self.from_base_ring_from_one_basis(x)
def spherical_bessel_Y(n,var, algorithm="maxima"):
r"""
Returns the spherical Bessel function of the second kind for
integers n -1.
Reference: AS 10.1.9 page 437 and AS 10.1.15 page 439.
EXAMPLES::
sage: x = PolynomialRing(QQ, 'x').gen()
sage: spherical_bessel_Y(2,x)
-((3/x^2 - 1)*cos(x) + 3*sin(x)/x)/x
"""
if algorithm=="scipy":
import scipy.special
ans = str(scipy.special.sph_yn(int(n),float(var)))
ans = ans.replace("(","")
ans = ans.replace(")","")
ans = ans.replace("j","*I")
return sage_eval(ans)
elif algorithm == 'maxima':
_init()
return meval("spherical_bessel_y(%s,%s)"%(ZZ(n),var))
else:
raise ValueError, "unknown algorithm '%s'"%algorithm
def eigenvalues( self, index_list):
evs = DB.find( { 'owner_id': self.__id,
'data_type': 'ev',
'index': { '$in': [ str(l) for l in index_list]}
})
loc_f = self.__field.gens_dict()
return dict( (eval(ev['index']),sage_eval( ev['data'], locals = loc_f)) for ev in evs)
def eigenvalues( self, index_list):
evs = DB.find( { 'owner_id': self.__id,
'data_type': 'ev',
'index': { '$in': [ str(l) for l in index_list]}
})
loc_f = self.__field.gens_dict()
return dict( (eval(ev['index']),sage_eval( ev['data'], locals = loc_f)) for ev in evs)
def legendre_P(n, x):
"""
Returns the Legendre polynomial of the first kind for integers
`n > -1`.
REFERENCE:
- AS 22.5.35 page 779.
EXAMPLES::
sage: P.<t> = QQ[]
sage: legendre_P(2,t)
3/2*t^2 - 1/2
sage: legendre_P(3, 1.1)
1.67750000000000
sage: legendre_P(3, 1 + I)
7/2*I - 13/2
sage: legendre_P(3, MatrixSpace(ZZ, 2)([1, 2, -4, 7]))
[-179 242]
[-484 547]
sage: legendre_P(3, GF(11)(5))
8
"""
_init()
return sage_eval(maxima.eval("legendre_p(%s,x)" % ZZ(n)), locals={"x": x})
def hermite(n, x):
"""
Returns the Hermite polynomial for integers `n > -1`.
REFERENCE:
- AS 22.5.40 and 22.5.41, page 779.
EXAMPLES::
sage: x = PolynomialRing(QQ, 'x').gen()
sage: hermite(2,x)
4*x^2 - 2
sage: hermite(3,x)
8*x^3 - 12*x
sage: hermite(3,2)
40
sage: S.<y> = PolynomialRing(RR)
sage: hermite(3,y)
8.00000000000000*y^3 - 12.0000000000000*y
sage: R.<x,y> = QQ[]
sage: hermite(3,y^2)
8*y^6 - 12*y^2
sage: w = var('w')
sage: hermite(3,2*w)
8*(8*w^2 - 3)*w
"""
_init()
return sage_eval(maxima.eval("hermite(%s,x)" % ZZ(n)), locals={"x": x})
def legendre_P(n, x):
"""
Returns the Legendre polynomial of the first kind for integers
`n > -1`.
REFERENCE:
- AS 22.5.35 page 779.
EXAMPLES::
sage: P.<t> = QQ[]
sage: legendre_P(2,t)
3/2*t^2 - 1/2
sage: legendre_P(3, 1.1)
1.67750000000000
sage: legendre_P(3, 1 + I)
7/2*I - 13/2
sage: legendre_P(3, MatrixSpace(ZZ, 2)([1, 2, -4, 7]))
[-179 242]
[-484 547]
sage: legendre_P(3, GF(11)(5))
8
"""
_init()
return sage_eval(maxima.eval('legendre_p(%s,x)' % ZZ(n)), locals={'x': x})
def hermite(n, x):
"""
Returns the Hermite polynomial for integers `n > -1`.
REFERENCE:
- AS 22.5.40 and 22.5.41, page 779.
EXAMPLES::
sage: x = PolynomialRing(QQ, 'x').gen()
sage: hermite(2,x)
4*x^2 - 2
sage: hermite(3,x)
8*x^3 - 12*x
sage: hermite(3,2)
40
sage: S.<y> = PolynomialRing(RR)
sage: hermite(3,y)
8.00000000000000*y^3 - 12.0000000000000*y
sage: R.<x,y> = QQ[]
sage: hermite(3,y^2)
8*y^6 - 12*y^2
sage: w = var('w')
sage: hermite(3,2*w)
8*(8*w^2 - 3)*w
"""
_init()
return sage_eval(maxima.eval('hermite(%s,x)' % ZZ(n)), locals={'x': x})
def eval(self, s, globals=None, locals=None):
r"""
EXAMPLES::
sage: html.eval('<hr>')
<html><font color='black'><hr></font></html>
''
"""
if globals is None:
globals = {}
if locals is None:
locals = {}
s = str(s)
s = math_parse(s)
t = ''
while len(s) > 0:
i = s.find('<sage>')
if i == -1:
t += s
break
j = s.find('</sage>')
if j == -1:
t += s
break
t += s[:i] + '<script type="math/tex">%s</script>'%\
latex(sage_eval(s[6+i:j], locals=locals))
s = s[j+7:]
print "<html><font color='black'>%s</font></html>"%t
return ''
def _element_constructor_(self, x):
"""
Construct a complex number.
EXAMPLES::
sage: CC((1,2)) # indirect doctest
1.00000000000000 + 2.00000000000000*I
"""
if not isinstance(x, (real_mpfr.RealNumber, tuple)):
if isinstance(x, complex_double.ComplexDoubleElement):
return complex_number.ComplexNumber(self, x.real(), x.imag())
elif isinstance(x, str):
# TODO: this is probably not the best and most
# efficient way to do this. -- Martin Albrecht
return complex_number.ComplexNumber(self,
sage_eval(x.replace(' ',''), locals={"I":self.gen(),"i":self.gen()}))
late_import()
if isinstance(x, NumberFieldElement_quadratic) and list(x.parent().polynomial()) == [1, 0, 1]:
(re, im) = list(x)
return complex_number.ComplexNumber(self, re, im)
try:
return self(x.sage())
except (AttributeError, TypeError):
pass
try:
return x._complex_mpfr_field_( self )
except AttributeError:
pass
return complex_number.ComplexNumber(self, x)
def eval(self, s, globals=None, locals=None):
r"""
EXAMPLES::
sage: html.eval('<hr>')
<html><font color='black'><hr></font></html>
''
"""
if globals is None:
globals = {}
if locals is None:
locals = {}
s = str(s)
s = math_parse(s)
t = ''
while len(s) > 0:
i = s.find('<sage>')
if i == -1:
t += s
break
j = s.find('</sage>')
if j == -1:
t += s
break
t += s[:i] + '<span class="math">%s</span>'%\
latex(sage_eval(s[6+i:j], locals=locals))
s = s[j+7:]
print "<html><font color='black'>%s</font></html>"%t
return ''
def right_coset_representatives(self):
r"""
Return the right coset representatives of ``self``.
EXAMPLES::
sage: W = ReflectionGroup(['A',2]) # optional - gap3
sage: for w in W: # optional - gap3
....: rcr = w.right_coset_representatives() # optional - gap3
....: print("%s %s"%(w.reduced_word(), # optional - gap3
....: [v.reduced_word() for v in rcr])) # optional - gap3
[] [[], [2], [1], [2, 1], [1, 2], [1, 2, 1]]
[2] [[], [2], [1]]
[1] [[], [1], [1, 2]]
[1, 2] [[]]
[2, 1] [[]]
[1, 2, 1] [[], [2], [2, 1]]
"""
from sage.combinat.root_system.reflection_group_complex import _gap_return
W = self.parent()
T = W.reflections()
T_fix = [i + 1 for i in T.keys()
if self.fix_space().is_subspace(T[i].fix_space())]
S = str(gap3('ReducedRightCosetRepresentatives(%s,ReflectionSubgroup(%s,%s))' % (W._gap_group._name, W._gap_group._name, T_fix)))
return sage_eval(_gap_return(S, coerce_obj='W'),
locals={'self': self, 'W': W})
def gen_laguerre(n, a, x):
"""
Returns the generalized Laguerre polynomial for integers `n > -1`.
Typically, a = 1/2 or a = -1/2.
REFERENCE:
- table on page 789 in AS.
EXAMPLES::
sage: x = PolynomialRing(QQ, 'x').gen()
sage: gen_laguerre(2,1,x)
1/2*x^2 - 3*x + 3
sage: gen_laguerre(2,1/2,x)
1/2*x^2 - 5/2*x + 15/8
sage: gen_laguerre(2,-1/2,x)
1/2*x^2 - 3/2*x + 3/8
sage: gen_laguerre(2,0,x)
1/2*x^2 - 2*x + 1
sage: gen_laguerre(3,0,x)
-1/6*x^3 + 3/2*x^2 - 3*x + 1
"""
from sage.functions.all import sqrt
_init()
return sage_eval(maxima.eval('gen_laguerre(%s,%s,x)' % (ZZ(n), a)),
locals={'x': x})
def ultraspherical(n, a, x):
"""
Returns the ultraspherical (or Gegenbauer) polynomial for integers
`n > -1`.
Computed using Maxima.
REFERENCE:
- AS 22.5.27
EXAMPLES::
sage: x = PolynomialRing(QQ, 'x').gen()
sage: ultraspherical(2,3/2,x)
15/2*x^2 - 3/2
sage: ultraspherical(2,1/2,x)
3/2*x^2 - 1/2
sage: ultraspherical(1,1,x)
2*x
sage: t = PolynomialRing(RationalField(),"t").gen()
sage: gegenbauer(3,2,t)
32*t^3 - 12*t
"""
_init()
return sage_eval(maxima.eval('ultraspherical(%s,%s,x)' % (ZZ(n), a)),
locals={'x': x})
def best_known_covering_design_www(v, k, t, verbose=False):
r"""
Return the best known `(v, k, t)` covering design from an online database.
This uses the La Jolla Covering Repository, a database
available at `<https://ljcr.dmgordon.org/cover.html>`_
INPUT:
- ``v`` -- integer, the size of the point set for the design
- ``k`` -- integer, the number of points per block
- ``t`` -- integer, the size of sets covered by the blocks
- ``verbose`` -- bool (default: ``False``), print verbose message
OUTPUT:
A :class:`CoveringDesign` object representing the ``(v, k, t)``-covering
design with smallest number of blocks available in the database.
EXAMPLES::
sage: from sage.combinat.designs.covering_design import ( # optional - internet
....: best_known_covering_design_www)
sage: C = best_known_covering_design_www(7, 3, 2) # optional - internet
sage: print(C) # optional - internet
C(7, 3, 2) = 7
Method: lex covering
Submitted on: 1996-12-01 00:00:00
0 1 2
0 3 4
0 5 6
1 3 5
1 4 6
2 3 6
2 4 5
A ValueError is raised if the ``(v, k, t)`` parameters are not
found in the database.
"""
v = int(v)
k = int(k)
t = int(t)
param = "?v=%s&k=%s&t=%s" % (v, k, t)
url = "https://ljcr.dmgordon.org/cover/get_cover.php" + param
if verbose:
print("Looking up the bounds at %s" % url)
f = urlopen(url)
try:
s = bytes_to_str(f.read())
finally:
f.close()
if 'covering not in database' in s: # not found
str = "no (%d, %d, %d) covering design in database\n" % (v, k, t)
raise ValueError(str)
return sage_eval(s)
def gen_laguerre(n, a, x):
"""
Returns the generalized Laguerre polynomial for integers `n > -1`.
Typically, `a = 1/2` or `a = -1/2`.
REFERENCES:
- Table on page 789 in [ASHandbook]_.
EXAMPLES::
sage: x = PolynomialRing(QQ, 'x').gen()
sage: gen_laguerre(2,1,x)
1/2*x^2 - 3*x + 3
sage: gen_laguerre(2,1/2,x)
1/2*x^2 - 5/2*x + 15/8
sage: gen_laguerre(2,-1/2,x)
1/2*x^2 - 3/2*x + 3/8
sage: gen_laguerre(2,0,x)
1/2*x^2 - 2*x + 1
sage: gen_laguerre(3,0,x)
-1/6*x^3 + 3/2*x^2 - 3*x + 1
"""
_init()
return sage_eval(maxima.eval('gen_laguerre(%s,%s,x)' % (ZZ(n), a)),
locals={'x': x})
def gen_laguerre(n, a, x):
"""
Returns the generalized Laguerre polynomial for integers `n > -1`.
Typically, `a = 1/2` or `a = -1/2`.
REFERENCES:
- Table on page 789 in [ASHandbook]_.
EXAMPLES::
sage: x = PolynomialRing(QQ, 'x').gen()
sage: gen_laguerre(2,1,x)
1/2*x^2 - 3*x + 3
sage: gen_laguerre(2,1/2,x)
1/2*x^2 - 5/2*x + 15/8
sage: gen_laguerre(2,-1/2,x)
1/2*x^2 - 3/2*x + 3/8
sage: gen_laguerre(2,0,x)
1/2*x^2 - 2*x + 1
sage: gen_laguerre(3,0,x)
-1/6*x^3 + 3/2*x^2 - 3*x + 1
"""
_init()
return sage_eval(maxima.eval("gen_laguerre(%s,%s,x)" % (ZZ(n), a)), locals={"x": x})
def _element_constructor_(self, x):
"""
EXAMPLES::
sage: CC((1,2))
1.00000000000000 + 2.00000000000000*I
"""
if not isinstance(x, (real_mpfr.RealNumber, tuple)):
if isinstance(x, complex_double.ComplexDoubleElement):
return complex_number.ComplexNumber(self, x.real(), x.imag())
elif isinstance(x, str):
# TODO: this is probably not the best and most
# efficient way to do this. -- Martin Albrecht
return complex_number.ComplexNumber(self,
sage_eval(x.replace(' ',''), locals={"I":self.gen(),"i":self.gen()}))
late_import()
if isinstance(x, NumberFieldElement_quadratic) and list(x.parent().polynomial()) == [1, 0, 1]:
(re, im) = list(x)
return complex_number.ComplexNumber(self, re, im)
try:
return self(x.sage())
except:
pass
try:
return x._complex_mpfr_field_( self )
except AttributeError:
pass
return complex_number.ComplexNumber(self, x)
def ultraspherical(n, a, x):
"""
Returns the ultraspherical (or Gegenbauer) polynomial for integers
`n > -1`.
Computed using Maxima.
REFERENCE:
- AS 22.5.27
EXAMPLES::
sage: x = PolynomialRing(QQ, 'x').gen()
sage: ultraspherical(2,3/2,x)
15/2*x^2 - 3/2
sage: ultraspherical(2,1/2,x)
3/2*x^2 - 1/2
sage: ultraspherical(1,1,x)
2*x
sage: t = PolynomialRing(RationalField(),"t").gen()
sage: gegenbauer(3,2,t)
32*t^3 - 12*t
"""
_init()
return sage_eval(maxima.eval("ultraspherical(%s,%s,x)" % (ZZ(n), a)), locals={"x": x})
def right_coset_representatives(self, J):
r"""
Return the right coset representatives of ``self`` for the
parabolic subgroup generated by the simple reflections in ``J``.
EXAMPLES::
sage: W = ReflectionGroup(["A",3]) # optional - gap3
sage: for J in Subsets([1,2,3]): W.right_coset_representatives(J) # optional - gap3
[(), (2,5)(3,9)(4,6)(8,11)(10,12), (1,4)(2,8)(3,5)(7,10)(9,11),
(1,7)(2,4)(5,6)(8,10)(11,12), (1,2,10)(3,6,5)(4,7,8)(9,12,11),
(1,4,6)(2,3,11)(5,8,9)(7,10,12), (1,6,4)(2,11,3)(5,9,8)(7,12,10),
(1,7)(2,6)(3,9)(4,5)(8,12)(10,11),
(1,10,2)(3,5,6)(4,8,7)(9,11,12), (1,2,3,12)(4,5,10,11)(6,7,8,9),
(1,5,9,10)(2,12,8,6)(3,4,7,11), (1,6)(2,9)(3,8)(5,11)(7,12),
(1,8)(2,7)(3,6)(4,10)(9,12), (1,10,9,5)(2,6,8,12)(3,11,7,4),
(1,12,3,2)(4,11,10,5)(6,9,8,7), (1,3)(2,12)(4,10)(5,11)(6,8)(7,9),
(1,5,12)(2,9,4)(3,10,8)(6,7,11), (1,8,11)(2,5,7)(3,12,4)(6,10,9),
(1,11,8)(2,7,5)(3,4,12)(6,9,10), (1,12,5)(2,4,9)(3,8,10)(6,11,7),
(1,3,7,9)(2,11,6,10)(4,8,5,12), (1,9,7,3)(2,10,6,11)(4,12,5,8),
(1,11)(3,10)(4,9)(5,7)(6,12), (1,9)(2,8)(3,7)(4,11)(5,10)(6,12)]
[(), (2,5)(3,9)(4,6)(8,11)(10,12), (1,4)(2,8)(3,5)(7,10)(9,11),
(1,2,10)(3,6,5)(4,7,8)(9,12,11), (1,4,6)(2,3,11)(5,8,9)(7,10,12),
(1,6,4)(2,11,3)(5,9,8)(7,12,10), (1,2,3,12)(4,5,10,11)(6,7,8,9),
(1,5,9,10)(2,12,8,6)(3,4,7,11), (1,6)(2,9)(3,8)(5,11)(7,12),
(1,3)(2,12)(4,10)(5,11)(6,8)(7,9),
(1,5,12)(2,9,4)(3,10,8)(6,7,11), (1,3,7,9)(2,11,6,10)(4,8,5,12)]
[(), (2,5)(3,9)(4,6)(8,11)(10,12), (1,7)(2,4)(5,6)(8,10)(11,12),
(1,4,6)(2,3,11)(5,8,9)(7,10,12),
(1,7)(2,6)(3,9)(4,5)(8,12)(10,11),
(1,10,2)(3,5,6)(4,8,7)(9,11,12), (1,2,3,12)(4,5,10,11)(6,7,8,9),
(1,10,9,5)(2,6,8,12)(3,11,7,4), (1,12,3,2)(4,11,10,5)(6,9,8,7),
(1,8,11)(2,5,7)(3,12,4)(6,10,9), (1,12,5)(2,4,9)(3,8,10)(6,11,7),
(1,11)(3,10)(4,9)(5,7)(6,12)]
[(), (1,4)(2,8)(3,5)(7,10)(9,11), (1,7)(2,4)(5,6)(8,10)(11,12),
(1,2,10)(3,6,5)(4,7,8)(9,12,11), (1,6,4)(2,11,3)(5,9,8)(7,12,10),
(1,10,2)(3,5,6)(4,8,7)(9,11,12), (1,5,9,10)(2,12,8,6)(3,4,7,11),
(1,8)(2,7)(3,6)(4,10)(9,12), (1,12,3,2)(4,11,10,5)(6,9,8,7),
(1,3)(2,12)(4,10)(5,11)(6,8)(7,9),
(1,11,8)(2,7,5)(3,4,12)(6,9,10), (1,9,7,3)(2,10,6,11)(4,12,5,8)]
[(), (2,5)(3,9)(4,6)(8,11)(10,12), (1,4,6)(2,3,11)(5,8,9)(7,10,12),
(1,2,3,12)(4,5,10,11)(6,7,8,9)]
[(), (1,4)(2,8)(3,5)(7,10)(9,11), (1,2,10)(3,6,5)(4,7,8)(9,12,11),
(1,6,4)(2,11,3)(5,9,8)(7,12,10), (1,5,9,10)(2,12,8,6)(3,4,7,11),
(1,3)(2,12)(4,10)(5,11)(6,8)(7,9)]
[(), (1,7)(2,4)(5,6)(8,10)(11,12), (1,10,2)(3,5,6)(4,8,7)(9,11,12),
(1,12,3,2)(4,11,10,5)(6,9,8,7)]
[()]
"""
from sage.combinat.root_system.reflection_group_complex import _gap_return
J_inv = [self._index_set_inverse[j] + 1 for j in J]
S = str(
gap3(
"ReducedRightCosetRepresentatives(%s,ReflectionSubgroup(%s,%s))"
% (self._gap_group._name, self._gap_group._name, J_inv)
)
)
return sage_eval(_gap_return(S), locals={"self": self})
def best_known_covering_design_www(v, k, t, verbose=False):
r"""
Gives the best known `(v,k,t)` covering design, using the database
available at `<http://www.ccrwest.org/>`_
INPUeT:
- ``v`` -- integer, the size of the point set for the design
- ``k`` -- integer, the number of points per block
- ``t`` -- integer, the size of sets covered by the blocks
- ``verbose`` -- bool (default=``False``), print verbose message
OUTPUT:
A :class:`CoveringDesign` object representing the ``(v,k,t)``-covering
design with smallest number of blocks available in the database.
EXAMPLES::
sage: from sage.combinat.designs.covering_design import best_known_covering_design_www
sage: C = best_known_covering_design_www(7, 3, 2) # optional - internet
sage: print(C) # optional - internet
C(7,3,2) = 7
Method: lex covering
Submitted on: 1996-12-01 00:00:00
0 1 2
0 3 4
0 5 6
1 3 5
1 4 6
2 3 6
2 4 5
This function raises a ValueError if the ``(v,k,t)`` parameters are not
found in the database.
"""
# import compatible with py2 and py3
from six.moves.urllib.request import urlopen
from sage.misc.sage_eval import sage_eval
v = int(v)
k = int(k)
t = int(t)
param = ("?v=%s&k=%s&t=%s" % (v, k, t))
url = "http://www.ccrwest.org/cover/get_cover.php" + param
if verbose:
print("Looking up the bounds at %s" % url)
f = urlopen(url)
s = f.read()
f.close()
if 'covering not in database' in s: #not found
str = "no (%d,%d,%d) covering design in database\n" % (v, k, t)
raise ValueError(str)
return sage_eval(s)
def best_known_covering_design_www(v, k, t, verbose=False):
r"""
Gives the best known `(v,k,t)` covering design, using the database
available at `<http://www.ccrwest.org/>`_
INPUeT:
- ``v`` -- integer, the size of the point set for the design
- ``k`` -- integer, the number of points per block
- ``t`` -- integer, the size of sets covered by the blocks
- ``verbose`` -- bool (default=``False``), print verbose message
OUTPUT:
A :class:`CoveringDesign` object representing the ``(v,k,t)``-covering
design with smallest number of blocks available in the database.
EXAMPLES::
sage: from sage.combinat.designs.covering_design import best_known_covering_design_www
sage: C = best_known_covering_design_www(7, 3, 2) # optional - internet
sage: print(C) # optional - internet
C(7,3,2) = 7
Method: lex covering
Submitted on: 1996-12-01 00:00:00
0 1 2
0 3 4
0 5 6
1 3 5
1 4 6
2 3 6
2 4 5
This function raises a ValueError if the ``(v,k,t)`` parameters are not
found in the database.
"""
# import compatible with py2 and py3
from six.moves.urllib.request import urlopen
from sage.misc.sage_eval import sage_eval
v = int(v)
k = int(k)
t = int(t)
param = ("?v=%s&k=%s&t=%s"%(v,k,t))
url = "http://www.ccrwest.org/cover/get_cover.php"+param
if verbose:
print("Looking up the bounds at %s" % url)
f = urlopen(url)
s = f.read()
f.close()
if 'covering not in database' in s: #not found
str = "no (%d,%d,%d) covering design in database\n"%(v,k,t)
raise ValueError(str)
return sage_eval(s)
def __init__( self, doc):
self.__collection = doc.get( 'collection')
self.__name = doc.get( 'name')
weight = doc.get( 'weight')
self.__weight = sage_eval( weight) if weight else weight
field = doc.get( 'field')
R = PolynomialRing( IntegerRing(), name = 'x')
self.__field = sage_eval( field, locals = R.gens_dict()) if field else field
self.__explicit_formula = doc.get( 'explicit_formula')
self.__type = doc.get( 'type')
self.__is_eigenform = doc.get( 'is_eigenform')
self.__is_integral = doc.get( 'is_integral')
self.__representation = doc.get( 'representation')
self.__id = doc.get( '_id')
def __init__( self, doc):
self.__collection = doc.get( 'collection')
self.__name = doc.get( 'name')
weight = doc.get( 'weight')
self.__weight = sage_eval( weight) if weight else weight
field = doc.get( 'field')
R = PolynomialRing( IntegerRing(), name = 'x')
self.__field = sage_eval( field, locals = R.gens_dict()) if field else field
self.__explicit_formula = doc.get( 'explicit_formula')
self.__type = doc.get( 'type')
self.__is_eigenform = doc.get( 'is_eigenform')
self.__is_integral = doc.get( 'is_integral')
self.__representation = doc.get( 'representation')
self.__id = doc.get( '_id')
def right_coset_representatives(self, J):
r"""
Return the right coset representatives of ``self`` for the
parabolic subgroup generated by the simple reflections in ``J``.
EXAMPLES::
sage: W = ReflectionGroup(["A",3]) # optional - gap3
sage: for J in Subsets([1,2,3]): W.right_coset_representatives(J) # optional - gap3
[(), (2,5)(3,9)(4,6)(8,11)(10,12), (1,4)(2,8)(3,5)(7,10)(9,11),
(1,7)(2,4)(5,6)(8,10)(11,12), (1,2,10)(3,6,5)(4,7,8)(9,12,11),
(1,4,6)(2,3,11)(5,8,9)(7,10,12), (1,6,4)(2,11,3)(5,9,8)(7,12,10),
(1,7)(2,6)(3,9)(4,5)(8,12)(10,11),
(1,10,2)(3,5,6)(4,8,7)(9,11,12), (1,2,3,12)(4,5,10,11)(6,7,8,9),
(1,5,9,10)(2,12,8,6)(3,4,7,11), (1,6)(2,9)(3,8)(5,11)(7,12),
(1,8)(2,7)(3,6)(4,10)(9,12), (1,10,9,5)(2,6,8,12)(3,11,7,4),
(1,12,3,2)(4,11,10,5)(6,9,8,7), (1,3)(2,12)(4,10)(5,11)(6,8)(7,9),
(1,5,12)(2,9,4)(3,10,8)(6,7,11), (1,8,11)(2,5,7)(3,12,4)(6,10,9),
(1,11,8)(2,7,5)(3,4,12)(6,9,10), (1,12,5)(2,4,9)(3,8,10)(6,11,7),
(1,3,7,9)(2,11,6,10)(4,8,5,12), (1,9,7,3)(2,10,6,11)(4,12,5,8),
(1,11)(3,10)(4,9)(5,7)(6,12), (1,9)(2,8)(3,7)(4,11)(5,10)(6,12)]
[(), (2,5)(3,9)(4,6)(8,11)(10,12), (1,4)(2,8)(3,5)(7,10)(9,11),
(1,2,10)(3,6,5)(4,7,8)(9,12,11), (1,4,6)(2,3,11)(5,8,9)(7,10,12),
(1,6,4)(2,11,3)(5,9,8)(7,12,10), (1,2,3,12)(4,5,10,11)(6,7,8,9),
(1,5,9,10)(2,12,8,6)(3,4,7,11), (1,6)(2,9)(3,8)(5,11)(7,12),
(1,3)(2,12)(4,10)(5,11)(6,8)(7,9),
(1,5,12)(2,9,4)(3,10,8)(6,7,11), (1,3,7,9)(2,11,6,10)(4,8,5,12)]
[(), (2,5)(3,9)(4,6)(8,11)(10,12), (1,7)(2,4)(5,6)(8,10)(11,12),
(1,4,6)(2,3,11)(5,8,9)(7,10,12),
(1,7)(2,6)(3,9)(4,5)(8,12)(10,11),
(1,10,2)(3,5,6)(4,8,7)(9,11,12), (1,2,3,12)(4,5,10,11)(6,7,8,9),
(1,10,9,5)(2,6,8,12)(3,11,7,4), (1,12,3,2)(4,11,10,5)(6,9,8,7),
(1,8,11)(2,5,7)(3,12,4)(6,10,9), (1,12,5)(2,4,9)(3,8,10)(6,11,7),
(1,11)(3,10)(4,9)(5,7)(6,12)]
[(), (1,4)(2,8)(3,5)(7,10)(9,11), (1,7)(2,4)(5,6)(8,10)(11,12),
(1,2,10)(3,6,5)(4,7,8)(9,12,11), (1,6,4)(2,11,3)(5,9,8)(7,12,10),
(1,10,2)(3,5,6)(4,8,7)(9,11,12), (1,5,9,10)(2,12,8,6)(3,4,7,11),
(1,8)(2,7)(3,6)(4,10)(9,12), (1,12,3,2)(4,11,10,5)(6,9,8,7),
(1,3)(2,12)(4,10)(5,11)(6,8)(7,9),
(1,11,8)(2,7,5)(3,4,12)(6,9,10), (1,9,7,3)(2,10,6,11)(4,12,5,8)]
[(), (2,5)(3,9)(4,6)(8,11)(10,12), (1,4,6)(2,3,11)(5,8,9)(7,10,12),
(1,2,3,12)(4,5,10,11)(6,7,8,9)]
[(), (1,4)(2,8)(3,5)(7,10)(9,11), (1,2,10)(3,6,5)(4,7,8)(9,12,11),
(1,6,4)(2,11,3)(5,9,8)(7,12,10), (1,5,9,10)(2,12,8,6)(3,4,7,11),
(1,3)(2,12)(4,10)(5,11)(6,8)(7,9)]
[(), (1,7)(2,4)(5,6)(8,10)(11,12), (1,10,2)(3,5,6)(4,8,7)(9,11,12),
(1,12,3,2)(4,11,10,5)(6,9,8,7)]
[()]
"""
from sage.combinat.root_system.reflection_group_element import _gap_return
J_inv = [self._index_set_inverse[j] + 1 for j in J]
S = str(
gap3(
'ReducedRightCosetRepresentatives(%s,ReflectionSubgroup(%s,%s))'
% (self._gap_group._name, self._gap_group._name, J_inv)))
return sage_eval(_gap_return(S), locals={'self': self})
def __call__(self, x, im=None):
"""
Construct an element.
EXAMPLES::
sage: CIF(2) # indirect doctest
2
sage: CIF(CIF.0)
1*I
sage: CIF('1+I')
1 + 1*I
sage: CIF(2,3)
2 + 3*I
sage: CIF(pi, e)
3.141592653589794? + 2.718281828459046?*I
sage: ComplexIntervalField(100)(CIF(RIF(2,3)))
3.?
"""
if im is None:
if isinstance(x, complex_interval.ComplexIntervalFieldElement):
if x.parent() is self:
return x
else:
return complex_interval.ComplexIntervalFieldElement(
self, x)
elif isinstance(x, complex_double.ComplexDoubleElement):
return complex_interval.ComplexIntervalFieldElement(
self, x.real(), x.imag())
elif isinstance(x, str):
# TODO: this is probably not the best and most
# efficient way to do this. -- Martin Albrecht
return complex_interval.ComplexIntervalFieldElement(
self,
sage_eval(x.replace(' ', ''),
locals={
"I": self.gen(),
"i": self.gen()
}))
late_import()
if isinstance(x, NumberFieldElement_quadratic) and list(
x.parent().polynomial()) == [1, 0, 1]:
(re, im) = list(x)
return complex_interval.ComplexIntervalFieldElement(
self, re, im)
try:
return x._complex_mpfi_(self)
except AttributeError:
pass
try:
return x._complex_mpfr_field_(self)
except AttributeError:
pass
return complex_interval.ComplexIntervalFieldElement(self, x, im)
def load(filename):
with open(filename,'r') as f:
if f.next() != "# base_ring\n":
raise IOError("Not a valid file for describing a DeforSys")
base_ring = sage_eval(f.next()[:-1])
f.next()
if f.next()!="# Variables\n":
raise IOError("Not a valid file for describing a DeforSys")
variables = [f.next()[:-1]]
while f.next() != "\n":
variables += [f.next()[:-1]]
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
ring = PolynomialRing(base_ring, variables)
local_eval = {st : va for st,va in zip(variables,ring.gens())}
if f.next()!="# Equations\n":
raise IOError("Not a valid file for describing a DeforSys")
eqs = [sage_eval(f.next()[:-1],locals = local_eval)]
while f.next() != "\n":
eqs += [sage_eval(f.next()[:-1], locals = local_eval)]
if f.next()!="# Inequations\n":
raise IOError("Not a valid file for describing a DeforSys")
ineqs = [sage_eval(f.next()[:-1], locals = local_eval)]
while f.next() != "\n":
ineqs += [sage_eval(f.next()[:-1], locals = local_eval)]
ineqs = set(ineqs)
if f.next()!="# Projecions\n":
raise IOError("Not a valid file for describing a DeforSys")
[ke,val] = f.next()[:-1].split(':')
proj = {sage_eval(ke, locals = local_eval):sage_eval(val, locals = local_eval)}
while f.next() != "\n":
[ke,val] = f.next()[:-1].split(':')
proj[sage_eval(ke, locals = local_eval)] = sage_eval(val, locals = local_eval)
if f.next()!="# # Initial Data\n":
raise IOError("Not a valid file for describing a DeforSys")
initial = sage_eval(f.next(), locals = local_eval)
return DeforSystem(eqs,ineqs,proj,init = initial)
def bessel_Y(nu,z,algorithm="maxima", prec=53):
r"""
Implements the "Y-Bessel function", or "Bessel function of the 2nd
kind", with index (or "order") nu and argument z.
.. note::
Currently only prec=53 is supported.
Defn::
cos(pi n)*bessel_J(nu, z) - bessel_J(-nu, z)
-------------------------------------------------
sin(nu*pi)
if nu is not an integer and by taking a limit otherwise.
Sometimes bessel_Y(n,z) is denoted Y_n(z) in the literature.
This is computed using Maxima by default.
EXAMPLES::
sage: bessel_Y(2,1.1,"scipy")
-1.4314714939...
sage: bessel_Y(2,1.1)
-1.4314714939590...
sage: bessel_Y(3.001,2.1)
-1.0299574976424...
TESTS::
sage: bessel_Y(2,1.1, algorithm="pari")
Traceback (most recent call last):
...
NotImplementedError: The Y-Bessel function is only implemented for the maxima and scipy algorithms
"""
if algorithm=="scipy":
if prec != 53:
raise ValueError, "for the scipy algorithm the precision must be 53"
import scipy.special
ans = str(scipy.special.yv(float(nu),complex(real(z),imag(z))))
ans = ans.replace("(","")
ans = ans.replace(")","")
ans = ans.replace("j","*I")
ans = sage_eval(ans)
return real(ans) if z in RR else ans
elif algorithm == "maxima":
if prec != 53:
raise ValueError, "for the maxima algorithm the precision must be 53"
return RR(maxima.eval("bessel_y(%s,%s)"%(float(nu),float(z))))
elif algorithm == "pari":
raise NotImplementedError, "The Y-Bessel function is only implemented for the maxima and scipy algorithms"
else:
raise ValueError, "unknown algorithm '%s'"%algorithm
def bessel_Y(nu, z, algorithm="maxima", prec=53):
r"""
Implements the "Y-Bessel function", or "Bessel function of the 2nd
kind", with index (or "order") nu and argument z.
.. note::
Currently only prec=53 is supported.
Defn::
cos(pi n)*bessel_J(nu, z) - bessel_J(-nu, z)
-------------------------------------------------
sin(nu*pi)
if nu is not an integer and by taking a limit otherwise.
Sometimes bessel_Y(n,z) is denoted Y_n(z) in the literature.
This is computed using Maxima by default.
EXAMPLES::
sage: bessel_Y(2,1.1,"scipy")
-1.4314714939...
sage: bessel_Y(2,1.1)
-1.4314714939590...
sage: bessel_Y(3.001,2.1)
-1.0299574976424...
TESTS::
sage: bessel_Y(2,1.1, algorithm="pari")
Traceback (most recent call last):
...
NotImplementedError: The Y-Bessel function is only implemented for the maxima and scipy algorithms
"""
if algorithm == "scipy":
if prec != 53:
raise ValueError, "for the scipy algorithm the precision must be 53"
import scipy.special
ans = str(scipy.special.yv(float(nu), complex(real(z), imag(z))))
ans = ans.replace("(", "")
ans = ans.replace(")", "")
ans = ans.replace("j", "*I")
ans = sage_eval(ans)
return real(ans) if z in RR else ans
elif algorithm == "maxima":
if prec != 53:
raise ValueError, "for the maxima algorithm the precision must be 53"
return RR(maxima.eval("bessel_y(%s,%s)" % (float(nu), float(z))))
elif algorithm == "pari":
raise NotImplementedError, "The Y-Bessel function is only implemented for the maxima and scipy algorithms"
else:
raise ValueError, "unknown algorithm '%s'" % algorithm
def format_oplanes(prev_R, curr_R, oplanes):
"""Format database version of oplanes to match fixed_loci here"""
new_oplanes = []
for op in oplanes:
oideal = []
for oi in op['OIDEAL']:
eval_oi = sage_eval(oi, locals=prev_R.gens_dict())
oideal.append(eval_oi)
oideal = prev_R.ideal(*oideal).change_ring(curr_R)
new_oplanes.append(oideal)
return new_oplanes
def _expression_from_dict(expr):
from sage.misc.sage_eval import sage_eval
from sage.calculus.calculus import var
# assume we have checked the dictionary already
argset = [
arg.decode() if isinstance(arg, bytes) else arg
for arg in expr['arguments']
]
return sage_eval(expr['expression'],
locals=dict([(arg, eval('var("' + arg + '")',
{'var': var}))
for arg in argset]))
def Fourier_coefficients( self, det_list):
fcs = DB.find( { 'owner_id': self.__id,
'data_type': 'fc',
'det': { '$in': [ str(d) for d in det_list]}
})
P = PolynomialRing( self.__field, names = 'x,y')
loc = P.gens_dict()
loc.update ( self.__field.gens_dict())
return dict( (Integer(fcd['det']),
dict( (tuple( eval(f)), sage_eval( fcd['data'][f], locals = loc))
for f in fcd['data'] ))
for fcd in fcs)
def Fourier_coefficients( self, det_list):
fcs = DB.find( { 'owner_id': self.__id,
'data_type': 'fc',
'det': { '$in': [ str(d) for d in det_list]}
})
P = PolynomialRing( self.__field, names = 'x,y')
loc = P.gens_dict()
loc.update ( self.__field.gens_dict())
return dict( (Integer(fcd['det']),
dict( (tuple( eval(f)), sage_eval( fcd['data'][f], locals = loc))
for f in fcd['data'] ))
for fcd in fcs)
def number_of_automorphisms__souvigner(self):
"""
Uses the Souvigner code to compute the number of automorphisms.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1,1])
sage: Q.number_of_automorphisms__souvigner() # optional -- souvigner
3840
sage: 2^5 * factorial(5)
3840
"""
## Write an input text file
F_filename = '/tmp/tmp_isom_input' + str(random()) + ".txt"
F = open(F_filename, 'w')
#F = tempfile.NamedTemporaryFile(prefix='tmp_auto_input', suffix=".txt") ## This fails because the Souvigner code doesn't like random filenames (hyphens are bad...)!
F.write("#1 \n")
n = self.dim()
F.write(str(n) + "x0 \n") ## Use the lower-triangular form
for i in range(n):
for j in range(i+1):
if i == j:
F.write(str(2 * self[i,j]) + " ")
else:
F.write(str(self[i,j]) + " ")
F.write("\n")
F.flush()
#print "Input filename = ", F.name
#os.system("less " + F.name)
## Call the Souvigner automorphism code
souvigner_auto_path = os.path.join(SAGE_LOCAL,'bin','Souvigner_AUTO') ## FIX THIS LATER!!!
G1 = tempfile.NamedTemporaryFile(prefix='tmp_auto_ouput', suffix=".txt")
#print "Output filename = ", G1.name
os.system(souvigner_auto_path + " '" + F.name + "' > '" + G1.name +"'")
## Read the output
G2 = open(G1.name, 'r')
for line in G2:
if line.startswith("|Aut| = "):
num_of_autos = sage_eval(line.lstrip("|Aut| = "))
F.close()
G1.close()
G2.close()
os.system("rm -f " + F_filename)
#os.system("rm -f " + G1.name)
return num_of_autos
## Raise and error if we're here:
raise RuntimeError("Oops! There is a problem...")
def format_srideal(prev_R, curr_R, srideal):
"""Format database version of srideal to match srideal here"""
terms = srideal.replace('{', '').replace('}', '').replace('D',
'x').split(',')
new_srideal = []
for t in terms:
new_term = []
for tt in t.split('*'):
eval_tt = sage_eval(tt, locals=prev_R.gens_dict())
new_term.append(eval_tt)
new_term = prev_R.ideal(*new_term).change_ring(curr_R)
new_srideal.append(new_term)
return new_srideal
def _element_constructor_(self, x, y=1, coerce=True):
"""
Construct an element of this fraction field.
EXAMPLES::
sage: F = QQ['x,y'].fraction_field()
sage: F._element_constructor_(1)
1
sage: F._element_constructor_(F.gen(0)/F.gen(1))
x/y
sage: F._element_constructor_('1 + x/y')
(x + y)/y
::
sage: K = ZZ['x,y'].fraction_field()
sage: x,y = K.gens()
::
sage: F._element_constructor_(x/y)
x/y
"""
Element = self._element_class
if isinstance(x, Element):
if x.parent() is self:
return x
else:
return Element(self, x.numerator(), x.denominator())
elif isinstance(x, basestring):
try:
from sage.misc.sage_eval import sage_eval
x = sage_eval(x, self.gens_dict_recursive())
y = sage_eval(str(y), self.gens_dict_recursive())
return Element(self, x, y, coerce=coerce, reduce=True)
except NameError, e:
raise TypeError, "unable to convert string"
def _element_constructor_(self, x, y=1, coerce=True):
"""
Construct an element of this fraction field.
EXAMPLES::
sage: F = QQ['x,y'].fraction_field()
sage: F._element_constructor_(1)
1
sage: F._element_constructor_(F.gen(0)/F.gen(1))
x/y
sage: F._element_constructor_('1 + x/y')
(x + y)/y
::
sage: K = ZZ['x,y'].fraction_field()
sage: x,y = K.gens()
::
sage: F._element_constructor_(x/y)
x/y
"""
Element = self._element_class
if isinstance(x, Element):
if x.parent() is self:
return x
else:
return Element(self, x.numerator(), x.denominator())
elif isinstance(x, basestring):
try:
from sage.misc.sage_eval import sage_eval
x = sage_eval(x, self.gens_dict_recursive())
y = sage_eval(str(y), self.gens_dict_recursive())
return Element(self, x, y, coerce=coerce, reduce=True)
except NameError, e:
raise TypeError, "unable to convert string"
def repr_short_to_parent(s):
r"""
Helper method for the growth group factory, which converts a short
representation string to a parent.
INPUT:
- ``s`` -- a string, short representation of a parent.
OUTPUT:
A parent.
The possible short representations are shown in the examples below.
EXAMPLES::
sage: from sage.rings.asymptotic.misc import repr_short_to_parent
sage: repr_short_to_parent('ZZ')
Integer Ring
sage: repr_short_to_parent('QQ')
Rational Field
sage: repr_short_to_parent('SR')
Symbolic Ring
sage: repr_short_to_parent('NN')
Non negative integer semiring
TESTS::
sage: repr_short_to_parent('abcdef')
Traceback (most recent call last):
...
ValueError: Cannot create a parent out of 'abcdef'.
> *previous* NameError: name 'abcdef' is not defined
"""
from sage.misc.sage_eval import sage_eval
try:
P = sage_eval(s)
except Exception as e:
raise combine_exceptions(
ValueError("Cannot create a parent out of '%s'." % (s,)), e)
from sage.misc.lazy_import import LazyImport
if type(P) is LazyImport:
P = P._get_object()
from sage.structure.parent import is_Parent
if not is_Parent(P):
raise ValueError("'%s' does not describe a parent." % (s,))
return P
def hypergeometric_U(alpha, beta, x, algorithm="pari", prec=53):
r"""
Default is a wrap of PARI's hyperu(alpha,beta,x) function.
Optionally, algorithm = "scipy" can be used.
The confluent hypergeometric function `y = U(a,b,x)` is
defined to be the solution to Kummer's differential equation
.. math::
xy'' + (b-x)y' - ay = 0.
This satisfies `U(a,b,x) \sim x^{-a}`, as
`x\rightarrow \infty`, and is sometimes denoted
``x^{-a}2_F_0(a,1+a-b,-1/x)``. This is not the same as Kummer's
`M`-hypergeometric function, denoted sometimes as
``_1F_1(alpha,beta,x)``, though it satisfies the same DE that
`U` does.
.. warning::
In the literature, both are called "Kummer confluent
hypergeometric" functions.
EXAMPLES::
sage: hypergeometric_U(1,1,1,"scipy")
0.596347362323...
sage: hypergeometric_U(1,1,1)
0.59634736232319...
sage: hypergeometric_U(1,1,1,"pari",70)
0.59634736232319407434...
"""
if algorithm == "scipy":
if prec != 53:
raise ValueError("for the scipy algorithm the precision must be 53")
import scipy.special
ans = str(scipy.special.hyperu(float(alpha), float(beta), float(x)))
ans = ans.replace("(", "")
ans = ans.replace(")", "")
ans = ans.replace("j", "*I")
return sage_eval(ans)
elif algorithm == "pari":
from sage.libs.pari.all import pari
R = RealField(prec)
return R(pari(R(alpha)).hyperu(R(beta), R(x), precision=prec))
else:
raise ValueError("unknown algorithm '%s'" % algorithm)
def best_known_covering_design_www(v, k, t, verbose=False):
r"""
Gives the best known (v,k,t) covering design, using database at
\url{http://www.ccrwest.org/}.
INPUT:
v -- integer, the size of the point set for the design
k -- integer, the number of points per block
t -- integer, the size of sets covered by the blocks
verbose -- bool (default=False), print verbose message
OUTPUT:
CoveringDesign -- (v,k,t) covering design with smallest number of blocks
EXAMPLES:
sage: C = best_known_covering_design_www(7, 3, 2) # optional -- requires internet
sage: print C # optional -- requires internet
C(7,3,2) = 7
Method: lex covering
Submitted on: 1996-12-01 00:00:00
0 1 2
0 3 4
0 5 6
1 3 5
1 4 6
2 3 6
2 4 5
This function raises a ValueError if the (v,k,t) parameters are
not found in the database.
"""
v = int(v)
k = int(k)
t = int(t)
param = ("?v=%s&k=%s&t=%s" % (v, k, t))
url = "http://www.ccrwest.org/cover/get_cover.php" + param
if verbose:
print "Looking up the bounds at %s" % url
f = urllib.urlopen(url)
s = f.read()
f.close()
if 'covering not in database' in s: #not found
str = "no (%d,%d,%d) covering design in database\n" % (v, k, t)
raise ValueError, str
return sage_eval(s)
def _element_constructor_(self, f, prec=infinity, check=True):
"""
Coerce object to this power series ring.
Returns a new instance unless the parent of f is self, in which
case f is returned (since f is immutable).
INPUT:
- ``f`` - object, e.g., a power series ring element
- ``prec`` - (default: infinity); truncation precision
for coercion
- ``check`` - bool (default: True), whether to verify
that the coefficients, etc., coerce in correctly.
EXAMPLES::
sage: R.<t> = PowerSeriesRing(ZZ)
sage: R(t+O(t^5)) # indirect doctest
t + O(t^5)
sage: R(13)
13
sage: R(2/3)
Traceback (most recent call last):
...
TypeError: no conversion of this rational to integer
sage: R([1,2,3])
1 + 2*t + 3*t^2
sage: S.<w> = PowerSeriesRing(QQ)
sage: R(w + 3*w^2 + O(w^3))
t + 3*t^2 + O(t^3)
sage: x = polygen(QQ,'x')
sage: R(x + x^2 + x^3 + x^5, 3)
t + t^2 + O(t^3)
"""
if isinstance(
f,
power_series_ring_element.PowerSeries) and f.parent() is self:
if prec >= f.prec():
return f
f = f.truncate(prec)
elif isinstance(f, MagmaElement) and str(f.Type()) == 'RngSerPowElt':
v = sage_eval(f.Eltseq())
return self(v) * (self.gen(0)**f.Valuation())
return self.element_class(self, f, prec, check=check)
def hypergeometric_U(alpha, beta, x, algorithm="pari", prec=53):
r"""
Default is a wrap of PARI's hyperu(alpha,beta,x) function.
Optionally, algorithm = "scipy" can be used.
The confluent hypergeometric function `y = U(a,b,x)` is
defined to be the solution to Kummer's differential equation
.. math::
xy'' + (b-x)y' - ay = 0.
This satisfies `U(a,b,x) \sim x^{-a}`, as
`x\rightarrow \infty`, and is sometimes denoted
``x^{-a}2_F_0(a,1+a-b,-1/x)``. This is not the same as Kummer's
`M`-hypergeometric function, denoted sometimes as
``_1F_1(alpha,beta,x)``, though it satisfies the same DE that
`U` does.
.. warning::
In the literature, both are called "Kummer confluent
hypergeometric" functions.
EXAMPLES::
sage: hypergeometric_U(1,1,1,"scipy")
0.596347362323...
sage: hypergeometric_U(1,1,1)
0.59634736232319...
sage: hypergeometric_U(1,1,1,"pari",70)
0.59634736232319407434...
"""
if algorithm == "scipy":
if prec != 53:
raise ValueError(
"for the scipy algorithm the precision must be 53")
import scipy.special
ans = str(scipy.special.hyperu(float(alpha), float(beta), float(x)))
ans = ans.replace("(", "")
ans = ans.replace(")", "")
ans = ans.replace("j", "*I")
return sage_eval(ans)
elif algorithm == 'pari':
from sage.libs.pari.all import pari
R = RealField(prec)
return R(pari(R(alpha)).hyperu(R(beta), R(x), precision=prec))
else:
raise ValueError("unknown algorithm '%s'" % algorithm)
def best_known_covering_design_www(v, k, t, verbose=False):
r"""
Gives the best known (v,k,t) covering design, using database at
\url{http://www.ccrwest.org/}.
INPUT:
v -- integer, the size of the point set for the design
k -- integer, the number of points per block
t -- integer, the size of sets covered by the blocks
verbose -- bool (default=False), print verbose message
OUTPUT:
CoveringDesign -- (v,k,t) covering design with smallest number of blocks
EXAMPLES:
sage: C = best_known_covering_design_www(7, 3, 2) # optional - internet
sage: print C # optional - internet
C(7,3,2) = 7
Method: lex covering
Submitted on: 1996-12-01 00:00:00
0 1 2
0 3 4
0 5 6
1 3 5
1 4 6
2 3 6
2 4 5
This function raises a ValueError if the (v,k,t) parameters are
not found in the database.
"""
v = int(v)
k = int(k)
t = int(t)
param = ("?v=%s&k=%s&t=%s"%(v,k,t))
url = "http://www.ccrwest.org/cover/get_cover.php"+param
if verbose:
print "Looking up the bounds at %s"%url
f = urllib.urlopen(url)
s = f.read()
f.close()
if 'covering not in database' in s: #not found
str = "no (%d,%d,%d) covering design in database\n"%(v,k,t)
raise ValueError, str
return sage_eval(s)
def __init__(self, collection, name):
dct = {'collection': collection, 'name': name}
smple = find_sample(dct)
assert smple, '%s: sample does not exist' % dct
self.__collection = collection
self.__name = name
weight = smple.get('weight')
field = smple.get('field')
fcs = smple.get('Fourier_coefficients')
evs = smple.get('eigenvalues')
self.__weight = Integer(weight) if weight else weight
R = PolynomialRing(IntegerRing(), name='x')
self.__field = sage_eval(field,
locals=R.gens_dict()) if field else field
loc_f = self.__field.gens_dict()
P = PolynomialRing(self.__field, names='x,y')
loc = P.gens_dict()
loc.update(loc_f)
self.__fcs = dict((tuple(eval(f)), sage_eval(fcs[f], locals=loc))
for f in fcs) if fcs else fcs
loc = self.field().gens_dict()
self.__evs = dict( (eval(l), sage_eval( evs[l], locals=loc_f)) for l in evs)\
if evs else evs
self.__explicit_formula = smple.get('explicit_formula')
def __init__(self, collection, name):
dct = {'collection': collection, 'name': name}
smple = find_sample(dct)
assert smple, '%s: sample does not exist' % dct
self.__collection = collection
self.__name = name
weight = smple.get('weight')
field = smple.get('field')
fcs = smple.get('Fourier_coefficients')
evs = smple.get('eigenvalues')
self.__weight = Integer(weight) if weight else weight
R = PolynomialRing(IntegerRing(), name='x')
self.__field = sage_eval(field, locals=R.gens_dict()) if field else field
loc_f = self.__field.gens_dict()
P = PolynomialRing(self.__field, names='x,y')
loc = P.gens_dict()
loc.update(loc_f)
self.__fcs = dict((tuple(eval(f)),
sage_eval(fcs[f], locals=loc))
for f in fcs) if fcs else fcs
loc = self.field().gens_dict()
self.__evs = dict( (eval(l), sage_eval( evs[l], locals=loc_f)) for l in evs)\
if evs else evs
self.__explicit_formula = smple.get('explicit_formula')
def chebyshev_U(n, x):
"""
Returns the Chebyshev function of the second kind for integers `n>-1`.
REFERENCE:
- AS, 22.8.3 page 783 and AS 6.1.22 page 256.
EXAMPLES::
sage: x = PolynomialRing(QQ, 'x').gen()
sage: chebyshev_U(2,x)
4*x^2 - 1
"""
_init()
return sage_eval(maxima.eval("chebyshev_u(%s,x)" % ZZ(n)), locals={"x": x})
def hermite(n,x):
"""
Returns the Hermite polynomial for integers `n > -1`.
REFERENCE:
- [ASHandbook]_ 22.5.40 and 22.5.41, page 779.
EXAMPLES::
sage: x = PolynomialRing(QQ, 'x').gen()
sage: hermite(2,x)
4*x^2 - 2
sage: hermite(3,x)
8*x^3 - 12*x
sage: hermite(3,2)
40
sage: S.<y> = PolynomialRing(RR)
sage: hermite(3,y)
8.00000000000000*y^3 - 12.0000000000000*y
sage: R.<x,y> = QQ[]
sage: hermite(3,y^2)
8*y^6 - 12*y^2
sage: w = var('w')
sage: hermite(3,2*w)
8*(8*w^2 - 3)*w
Check that :trac:`17192` is fixed::
sage: x = PolynomialRing(QQ, 'x').gen()
sage: hermite(0,x)
1
sage: hermite(-1,x)
Traceback (most recent call last):
...
ValueError: n must be greater than -1, got n = -1
sage: hermite(-7,x)
Traceback (most recent call last):
...
ValueError: n must be greater than -1, got n = -7
"""
if not (n > -1):
raise ValueError("n must be greater than -1, got n = {0}".format(n))
_init()
return sage_eval(maxima.eval('hermite(%s,x)'%ZZ(n)), locals={'x':x})
def __call__(self, x, im=None):
"""
Construct an element.
EXAMPLES::
sage: CIF(2) # indirect doctest
2
sage: CIF(CIF.0)
1*I
sage: CIF('1+I')
1 + 1*I
sage: CIF(2,3)
2 + 3*I
sage: CIF(pi, e)
3.141592653589794? + 2.718281828459046?*I
sage: ComplexIntervalField(100)(CIF(RIF(2,3)))
3.?
"""
if im is None:
if isinstance(x, complex_interval.ComplexIntervalFieldElement):
if x.parent() is self:
return x
else:
return complex_interval.ComplexIntervalFieldElement(self, x)
elif isinstance(x, complex_double.ComplexDoubleElement):
return complex_interval.ComplexIntervalFieldElement(self, x.real(), x.imag())
elif isinstance(x, str):
# TODO: this is probably not the best and most
# efficient way to do this. -- Martin Albrecht
return complex_interval.ComplexIntervalFieldElement(self,
sage_eval(x.replace(' ',''), locals={"I":self.gen(),"i":self.gen()}))
late_import()
if isinstance(x, NumberFieldElement_quadratic) and list(x.parent().polynomial()) == [1, 0, 1]:
(re, im) = list(x)
return complex_interval.ComplexIntervalFieldElement(self, re, im)
try:
return x._complex_mpfi_( self )
except AttributeError:
pass
try:
return x._complex_mpfr_field_( self )
except AttributeError:
pass
return complex_interval.ComplexIntervalFieldElement(self, x, im)
def _element_constructor_(self, f, prec=infinity, check=True):
"""
Coerce object to this power series ring.
Returns a new instance unless the parent of f is self, in which
case f is returned (since f is immutable).
INPUT:
- ``f`` - object, e.g., a power series ring element
- ``prec`` - (default: infinity); truncation precision
for coercion
- ``check`` - bool (default: True), whether to verify
that the coefficients, etc., coerce in correctly.
EXAMPLES::
sage: R.<t> = PowerSeriesRing(ZZ)
sage: R(t+O(t^5)) # indirect doctest
t + O(t^5)
sage: R(13)
13
sage: R(2/3)
Traceback (most recent call last):
...
TypeError: no conversion of this rational to integer
sage: R([1,2,3])
1 + 2*t + 3*t^2
sage: S.<w> = PowerSeriesRing(QQ)
sage: R(w + 3*w^2 + O(w^3))
t + 3*t^2 + O(t^3)
sage: x = polygen(QQ,'x')
sage: R(x + x^2 + x^3 + x^5, 3)
t + t^2 + O(t^3)
"""
if isinstance(f, power_series_ring_element.PowerSeries) and f.parent() is self:
if prec >= f.prec():
return f
f = f.truncate(prec)
elif isinstance(f, MagmaElement) and str(f.Type()) == 'RngSerPowElt':
v = sage_eval(f.Eltseq())
return self(v) * (self.gen(0)**f.Valuation())
return self.element_class(self, f, prec, check=check)
def chebyshev_T(n, x):
"""
Returns the Chebyshev function of the first kind for integers
`n>-1`.
REFERENCE:
- AS 22.5.31 page 778 and AS 6.1.22 page 256.
EXAMPLES::
sage: x = PolynomialRing(QQ, 'x').gen()
sage: chebyshev_T(2,x)
2*x^2 - 1
"""
_init()
return sage_eval(maxima.eval("chebyshev_t(%s,x)" % ZZ(n)), locals={"x": x})
def eval(self, s, locals=None):
r"""
Evaluate embedded <sage> tags
INPUT:
- ``s`` -- string.
- ``globals`` -- dictionary. The global variables when
evaluating ``s``. Default: the current global variables.
OUTPUT:
A :class:`HtmlFragment` instance.
EXAMPLES::
sage: a = 123
sage: html.eval('<sage>a</sage>')
<script type="math/tex">123</script>
sage: html.eval('<sage>a</sage>', locals={'a': 456})
<script type="math/tex">456</script>
"""
if hasattr(s, '_html_'):
deprecation(18292, 'html.eval() is for strings, use html() for sage objects')
return s._html_()
if locals is None:
from sage.repl.user_globals import get_globals
locals = get_globals()
s = str(s)
s = math_parse(s)
t = ''
while len(s) > 0:
i = s.find('<sage>')
if i == -1:
t += s
break
j = s.find('</sage>')
if j == -1:
t += s
break
t += s[:i] + '<script type="math/tex">%s</script>'%\
latex(sage_eval(s[6+i:j], locals=locals))
s = s[j+7:]
return HtmlFragment(t)
def ultraspherical(n,a,x):
"""
Returns the ultraspherical (or Gegenbauer) polynomial for integers
`n > -1`.
Computed using Maxima.
REFERENCE:
- [ASHandbook]_ 22.5.27
EXAMPLES::
sage: x = PolynomialRing(QQ, 'x').gen()
sage: ultraspherical(2,3/2,x)
15/2*x^2 - 3/2
sage: ultraspherical(2,1/2,x)
3/2*x^2 - 1/2
sage: ultraspherical(1,1,x)
2*x
sage: t = PolynomialRing(RationalField(),"t").gen()
sage: gegenbauer(3,2,t)
32*t^3 - 12*t
Check that :trac:`17192` is fixed::
sage: x = PolynomialRing(QQ, 'x').gen()
sage: ultraspherical(0,1,x)
1
sage: ultraspherical(-1,1,x)
Traceback (most recent call last):
...
ValueError: n must be greater than -1, got n = -1
sage: ultraspherical(-7,1,x)
Traceback (most recent call last):
...
ValueError: n must be greater than -1, got n = -7
"""
if not (n > -1):
raise ValueError("n must be greater than -1, got n = {0}".format(n))
_init()
return sage_eval(maxima.eval('ultraspherical(%s,%s,x)'%(ZZ(n),a)), locals={'x':x})
