def signature(self):
r"""
Return the signature of the rational field, which is `(1,0)`, since
there are 1 real and no complex embeddings.
EXAMPLES::
sage: QQ.signature()
(1, 0)
"""
return (integer.Integer(1), integer.Integer(0))
def __iter__(self):
r"""
Creates an iterator that generates the rational numbers without
repetition, in order of the height.
See also :meth:`range_by_height()`.
EXAMPLES:
The first 17 rational numbers, ordered by height::
sage: import itertools
sage: lst = [a for a in itertools.islice(Rationals(),17)]
sage: lst
[0, 1, -1, 1/2, -1/2, 2, -2, 1/3, -1/3, 3, -3, 2/3, -2/3, 3/2, -3/2, 1/4, -1/4]
sage: [a.height() for a in lst]
[1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4]
"""
yield self(0)
yield self(1)
yield self(-1)
height = integer.Integer(1)
while True:
height = height + 1
for other in range(1, height):
if height.gcd(other) == 1:
yield self(other / height)
yield self(-other / height)
yield self(height / other)
yield self(-height / other)
def __iter__(self):
r"""
Creates an iterator that generates the rational numbers without
repetition, in order of the height.
See also :meth:`range_by_height()`.
EXAMPLES:
The first 17 rational numbers, ordered by height::
sage: import itertools
sage: lst = [a for a in itertools.islice(Rationals(),17)]
sage: lst
[0, 1, -1, 1/2, -1/2, 2, -2, 1/3, -1/3, 3, -3, 2/3, -2/3, 3/2, -3/2, 1/4, -1/4]
sage: [a.height() for a in lst]
[1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4]
"""
#The previous version of this function, implemented by Nils
#Bruin, used the sequence defined by $a_0=0$ and
#$a_{n+1}=\frac{1}{2\lfloor a_n\rfloor+1-a_n}$ and generated the
#sequence $$a_0,a_1,-a_1,a_2,-a_2,\ldots$$. This is [A002487]
#in Sloane's encyclopedia, attributed to [Stern]. It is not
#monotone in height, but has other interesting properties
#described in [CalkinWilf].
#REFERENCES:
# [A002487] Sloane's OLEIS,
# http://oeis.org/classic/A002487
# [CalkinWilf] N. Calkin and H.S. Wilf, Recounting the
# rationals, American Mathematical Monthly 107 (2000),
# 360--363
# [Stern] M.A. Stern, Ueber eine zahlentheoretische Funktion,
# Journal fuer die reine und angewandte Mathematik 55
# (1858), 193--220
#
# [beginning of Nils' code]
#from sage.rings.arith import integer_floor as floor
#
#n=self(0)
#yield n
#while True:
# n=1/(2*floor(n)+1-n)
# yield n
# yield -n
# [end of Nils' code]
yield self(0)
yield self(1)
yield self(-1)
height = integer.Integer(1)
while True:
height = height + 1
for other in range(1, height):
if height.gcd(other) == 1:
yield self(other / height)
yield self(-other / height)
yield self(height / other)
yield self(-height / other)
def absolute_degree(self):
"""
EXAMPLES::
sage: QQ.absolute_degree()
1
"""
return integer.Integer(1)
def ngens(self):
"""
EXAMPLES::
sage: QQ.ngens()
1
"""
return integer.Integer(1)
def degree(self):
"""
EXAMPLES::
sage: QQ.degree()
1
"""
return integer.Integer(1)
def characteristic(self):
"""
Return the characteristic of the complex (interval) field, which is 0.
EXAMPLES::
sage: CIF.characteristic()
0
"""
return integer.Integer(0)
def characteristic(self):
r"""
Return the characteristic of `\CC`, which is 0.
EXAMPLES::
sage: ComplexField().characteristic()
0
"""
return integer.Integer(0)
def ngens(self):
r"""
Return the number of generators of `\QQ` which is 1.
EXAMPLES::
sage: QQ.ngens()
1
"""
return integer.Integer(1)
def discriminant(self):
"""
Return the discriminant of the field of rational numbers, which is 1.
EXAMPLES::
sage: QQ.discriminant()
1
"""
return integer.Integer(1)
def absolute_degree(self):
r"""
Return the absolute degree of `\QQ` which is 1.
EXAMPLES::
sage: QQ.absolute_degree()
1
"""
return integer.Integer(1)
def degree(self):
r"""
Return the degree of `\QQ` which is 1.
EXAMPLES::
sage: QQ.degree()
1
"""
return integer.Integer(1)
def class_number(self):
"""
Return the class number of the field of rational numbers, which is 1.
EXAMPLES::
sage: QQ.class_number()
1
"""
return integer.Integer(1)
def characteristic(self):
r"""
Return 0 since the rational field has characteristic 0.
EXAMPLES::
sage: c = QQ.characteristic(); c
0
sage: parent(c)
Integer Ring
"""
return integer.Integer(0)
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)
sage: R(1/(1-x), prec=5)
1 + t + t^2 + t^3 + t^4 + O(t^5)
sage: R(1/x, 5)
Traceback (most recent call last):
...
TypeError: no canonical coercion from Laurent Series Ring in t over Integer Ring to Power Series Ring in t over Integer Ring
sage: PowerSeriesRing(PowerSeriesRing(QQ,'x'),'y')(x)
x
sage: PowerSeriesRing(PowerSeriesRing(QQ,'y'),'x')(x)
x
sage: PowerSeriesRing(PowerSeriesRing(QQ,'t'),'y')(x)
y
sage: PowerSeriesRing(PowerSeriesRing(QQ,'t'),'y')(1/(1+x), 5)
1 - y + y^2 - y^3 + y^4 + O(y^5)
sage: PowerSeriesRing(PowerSeriesRing(QQ,'x',5),'y')(1/(1+x))
1 - x + x^2 - x^3 + x^4 + O(x^5)
sage: PowerSeriesRing(PowerSeriesRing(QQ,'y'),'x')(1/(1+x), 5)
1 - x + x^2 - x^3 + x^4 + O(x^5)
sage: PowerSeriesRing(PowerSeriesRing(QQ,'x'),'x')(x).coefficients()
[x]
Laurent series with non-negative valuation are accepted (see
:trac:`6431`)::
sage: L.<q> = LaurentSeriesRing(QQ)
sage: P = L.power_series_ring()
sage: P(q)
q
sage: P(1/q)
Traceback (most recent call last):
...
TypeError: self is not a power series
It is checked that the precision is non-negative
(see :trac:`19409`)::
sage: PowerSeriesRing(ZZ, 'x')(1, prec=-5)
Traceback (most recent call last):
...
ValueError: prec (= -5) must be non-negative
"""
if prec is not infinity:
prec = integer.Integer(prec)
if prec < 0:
raise ValueError("prec (= %s) must be non-negative" % prec)
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, laurent_series_ring_element.LaurentSeries
) and f.parent().power_series_ring() is self:
return self(f.power_series(), prec, check=check)
elif isinstance(f, MagmaElement) and str(f.Type()) == 'RngSerPowElt':
v = sage_eval(f.Eltseq())
return self(v) * (self.gen(0)**f.Valuation())
elif isinstance(f, FractionFieldElement):
if self.base_ring().has_coerce_map_from(f.parent()):
return self.element_class(self, [f], prec, check=check)
else:
num = self.element_class(self,
f.numerator(),
prec,
check=check)
den = self.element_class(self,
f.denominator(),
prec,
check=check)
return self.coerce(num / den)
return self.element_class(self, f, prec, check=check)
def characteristic(self):
return integer.Integer(0)
else:
return x
return NearZeroKiller()(fft(wrap_intermediate(x),
sign=sign,
wrap_intermediate=wrap_intermediate))
def csr_matrix_multiply(S, x):
"""Multiplies a scipy.sparse.csr_matrix S by an object-array vector x.
"""
h, w = S.shape
import numpy
result = numpy.empty_like(x)
for i in xrange(h):
result[i] = sum(S.data[idx] * x[S.indices[idx]]
for idx in range(S.indptr[i], S.indptr[i + 1]))
return result
if __name__ == "__main__":
import integer
q = integer.Integer(14)
r = integer.Integer(22)
gcd, a, b = extended_euclidean(q, r)
print gcd, "=", a, "*", q, "+", b, "*", r
print a * q + b * r
