def followstrand(f, x0, x1, y0a):
"""
Return a piecewise linear aproximation of the homotopy continuation of the root y0a
from x0 to x1
INPUT:
- ``f`` -- a polynomial in two variables
- ``x0`` -- a complex value, where the homotopy starts
- ``x1`` -- a complex value, where the homotopy ends
- ``y0a`` -- an approximate solution of the polynomial $F(y)=f(x_0,y)$
OUTPUT:
A list of values (t, ytr, yti) such that:
- ``t`` is a real number between zero and one
- $f(t\cdot x_1+(1-t)\cdot x_0, y_{tr} + I\cdot y_{ti})$ is zero (or a good enough aproximation)
- the piecewise linear path determined by the points has a tubular
neighborhood where the actual homotopy continuation path lies, and
no other root intersects it.
EXAMPLES::
sage: R.<x,y> = QQ[]
sage: f = x^2 + y^3
sage: x0 = CC(1, 0)
sage: x1 = CC(1, 0.5)
sage: followstrand(f, x0, x1, -1.0) # optional
[(0.0, -1.0, 0.0),
(0.063125, -1.0001106593601545, -0.02104011456212618),
(0.12230468750000001, -1.0004151100003031, -0.04075695242737829),
(0.17778564453125, -1.0008762007617709, -0.059227299979315154),
(0.28181243896484376, -1.0021948141328698, -0.09380038023464156),
(0.3728358840942383, -1.0038270754728402, -0.123962953123039),
(0.4524813985824585, -1.005613540368227, -0.15026634875747985),
(0.5221712237596512, -1.0074443351354077, -0.17320066690539515),
(0.5831498207896948, -1.009246118007067, -0.1931978258913501),
(0.636506093190983, -1.0109719597443307, -0.21063630045261386),
(0.6831928315421101, -1.0125937110987449, -0.2258465242053158),
(0.7648946236565826, -1.0156754074572174, -0.2523480191006915),
(0.8261709677424369, -1.0181837235093538, -0.2721208327884435),
(0.8721282258068277, -1.0201720738092597, -0.2868892148703381),
(0.9410641129034139, -1.0233210275568283, -0.3089391941950436),
(1.0, -1.026166099551513, -0.3276894025360433)]
"""
G = f.change_ring(QQbar).change_ring(CIF)
(x, y) = G.variables()
g = G.subs({x: (1 - x) * CIF(x0) + x * CIF(x1)})
coefs = []
deg = g.total_degree()
for d in range(deg + 1):
for i in range(d + 1):
c = CIF(g.coefficient({x: d - i, y: i}))
cr = c.real()
ci = c.imag()
coefs += list(cr.endpoints())
coefs += list(ci.endpoints())
yr = CC(y0a).real()
yi = CC(y0a).imag()
points = contpath(deg, coefs, yr, yi)
return points
def __getitem__(self, arg):
"""
Extend this ring by one or several elements to create a polynomial
ring, a power series ring, or an algebraic extension.
This is a convenience method intended primarily for interactive
use.
.. SEEALSO::
:func:`~sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing`,
:func:`~sage.rings.power_series_ring.PowerSeriesRing`,
:meth:`~sage.rings.ring.Ring.extension`,
:meth:`sage.rings.integer_ring.IntegerRing_class.__getitem__`,
:meth:`sage.rings.matrix_space.MatrixSpace.__getitem__`,
:meth:`sage.structure.parent.Parent.__getitem__`
EXAMPLES:
We create several polynomial rings::
sage: ZZ['x']
Univariate Polynomial Ring in x over Integer Ring
sage: QQ['x']
Univariate Polynomial Ring in x over Rational Field
sage: GF(17)['abc']
Univariate Polynomial Ring in abc over Finite Field of size 17
sage: GF(17)['a,b,c']
Multivariate Polynomial Ring in a, b, c over Finite Field of size 17
sage: GF(17)['a']['b']
Univariate Polynomial Ring in b over Univariate Polynomial Ring in a over Finite Field of size 17
We can create skew polynomial rings::
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: k['x',Frob]
Skew Polynomial Ring in x over Finite Field in t of size 5^3 twisted by t |--> t^5
We can also create power series rings by using double brackets::
sage: QQ[['t']]
Power Series Ring in t over Rational Field
sage: ZZ[['W']]
Power Series Ring in W over Integer Ring
sage: ZZ[['x,y,z']]
Multivariate Power Series Ring in x, y, z over Integer Ring
sage: ZZ[['x','T']]
Multivariate Power Series Ring in x, T over Integer Ring
Use :func:`~sage.rings.fraction_field.Frac` or
:meth:`~sage.rings.ring.CommutativeRing.fraction_field` to obtain
the fields of rational functions and Laurent series::
sage: Frac(QQ['t'])
Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: Frac(QQ[['t']])
Laurent Series Ring in t over Rational Field
sage: QQ[['t']].fraction_field()
Laurent Series Ring in t over Rational Field
Note that the same syntax can be used to create number fields::
sage: QQ[I]
Number Field in I with defining polynomial x^2 + 1
sage: QQ[I].coerce_embedding()
Generic morphism:
From: Number Field in I with defining polynomial x^2 + 1
To: Complex Lazy Field
Defn: I -> 1*I
::
sage: QQ[sqrt(2)]
Number Field in sqrt2 with defining polynomial x^2 - 2
sage: QQ[sqrt(2)].coerce_embedding()
Generic morphism:
From: Number Field in sqrt2 with defining polynomial x^2 - 2
To: Real Lazy Field
Defn: sqrt2 -> 1.414213562373095?
::
sage: QQ[sqrt(2),sqrt(3)]
Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field
and orders in number fields::
sage: ZZ[I]
Order in Number Field in I with defining polynomial x^2 + 1
sage: ZZ[sqrt(5)]
Order in Number Field in sqrt5 with defining polynomial x^2 - 5
sage: ZZ[sqrt(2)+sqrt(3)]
Order in Number Field in a with defining polynomial x^4 - 10*x^2 + 1
Embeddings are found for simple extensions (when that makes sense)::
sage: QQi.<i> = QuadraticField(-1, 'i')
sage: QQ[i].coerce_embedding()
Generic morphism:
From: Number Field in i with defining polynomial x^2 + 1
To: Complex Lazy Field
Defn: i -> 1*I
TESTS:
A few corner cases::
sage: QQ[()]
Multivariate Polynomial Ring in no variables over Rational Field
sage: QQ[[]]
Traceback (most recent call last):
...
TypeError: power series rings must have at least one variable
These kind of expressions do not work::
sage: QQ['a,b','c']
Traceback (most recent call last):
...
ValueError: variable name 'a,b' is not alphanumeric
sage: QQ[['a,b','c']]
Traceback (most recent call last):
...
ValueError: variable name 'a,b' is not alphanumeric
sage: QQ[[['x']]]
Traceback (most recent call last):
...
TypeError: expected R[...] or R[[...]], not R[[[...]]]
Extension towers are built as follows and use distinct generator names::
sage: K = QQ[2^(1/3), 2^(1/2), 3^(1/3)]
sage: K
Number Field in a with defining polynomial x^3 - 2 over its base field
sage: K.base_field()
Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field
sage: K.base_field().base_field()
Number Field in b with defining polynomial x^3 - 3
Embeddings::
sage: QQ[I](I.pyobject())
I
sage: a = 10^100; expr = (2*a + sqrt(2))/(2*a^2-1)
sage: QQ[expr].coerce_embedding() is None
False
sage: QQ[sqrt(5)].gen() > 0
True
sage: expr = sqrt(2) + I*(cos(pi/4, hold=True) - sqrt(2)/2)
sage: QQ[expr].coerce_embedding()
Generic morphism:
From: Number Field in a with defining polynomial x^2 - 2
To: Real Lazy Field
Defn: a -> 1.414213562373095?
"""
def normalize_arg(arg):
if isinstance(arg, (tuple, list)):
# Allowing arbitrary iterables would create confusion, but we
# may want to support a few more.
return tuple(arg)
elif isinstance(arg, str):
return tuple(arg.split(','))
else:
return (arg,)
# 1. If arg is a list, try to return a power series ring.
if isinstance(arg, list):
if arg == []:
raise TypeError("power series rings must have at least one variable")
elif len(arg) == 1:
# R[["a,b"]], R[[(a,b)]]...
if isinstance(arg[0], list):
raise TypeError("expected R[...] or R[[...]], not R[[[...]]]")
elts = normalize_arg(arg[0])
else:
elts = normalize_arg(arg)
from sage.rings.power_series_ring import PowerSeriesRing
return PowerSeriesRing(self, elts)
if isinstance(arg, tuple):
from sage.categories.morphism import Morphism
if len(arg) == 2 and isinstance(arg[1], Morphism):
from sage.rings.polynomial.skew_polynomial_ring_constructor import SkewPolynomialRing
return SkewPolynomialRing(self, arg[1], names=arg[0])
# 2. Otherwise, if all specified elements are algebraic, try to
# return an algebraic extension
elts = normalize_arg(arg)
try:
minpolys = [a.minpoly() for a in elts]
except (AttributeError, NotImplementedError, ValueError, TypeError):
minpolys = None
if minpolys:
# how to pass in names?
names = tuple(_gen_names(elts))
if len(elts) == 1:
from sage.rings.all import CIF, CLF, RIF, RLF
elt = elts[0]
try:
iv = CIF(elt)
except (TypeError, ValueError):
emb = None
else:
# First try creating an ANRoot manually, because
# extension(..., embedding=CLF(expr)) (or
# ...QQbar(expr)) would normalize the expression in
# QQbar, which currently is VERY slow in many cases.
# This may fail when minpoly has close roots or elt is
# a complicated symbolic expression.
# TODO: Rewrite using #19362 and/or #17886 and/or
# #15600 once those issues are solved.
from sage.rings.qqbar import AlgebraicNumber, ANRoot
try:
elt = AlgebraicNumber(ANRoot(minpolys[0], iv))
except ValueError:
pass
# Force a real embedding when possible, to get the
# right ordered ring structure.
if (iv.imag().is_zero() or iv.imag().contains_zero()
and elt.imag().is_zero()):
emb = RLF(elt)
else:
emb = CLF(elt)
return self.extension(minpolys[0], names[0], embedding=emb)
try:
# Doing the extension all at once is best, if possible...
return self.extension(minpolys, names)
except (TypeError, ValueError):
# ...but we can also construct it iteratively
return reduce(lambda R, ext: R.extension(*ext), zip(minpolys, names), self)
# 2. Otherwise, try to return a polynomial ring
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
return PolynomialRing(self, elts)
def followstrand(f, x0, x1, y0a):
"""
Return a piecewise linear aproximation of the homotopy continuation of the root y0a
from x0 to x1
INPUT:
- ``f`` -- a polynomial in two variables
- ``x0`` -- a complex value, where the homotopy starts
- ``x1`` -- a complex value, where the homotopy ends
- ``y0a`` -- an approximate solution of the polynomial $F(y)=f(x_0,y)$
OUTPUT:
A list of values (t, ytr, yti) such that:
- ``t`` is a real number between zero and one
- $f(t\cdot x_1+(1-t)\cdot x_0, y_{tr} + I\cdot y_{ti})$ is zero (or a good enough aproximation)
- the piecewise linear path determined by the points has a tubular
neighborhood where the actual homotopy continuation path lies, and
no other root intersects it.
EXAMPLES::
sage: R.<x,y> = QQ[]
sage: f = x^2 + y^3
sage: x0 = CC(1, 0)
sage: x1 = CC(1, 0.5)
sage: followstrand(f, x0, x1, -1.0) # optional
[(0.0, -1.0, 0.0),
(0.063125, -1.0001106593601545, -0.02104011456212618),
(0.12230468750000001, -1.0004151100003031, -0.04075695242737829),
(0.17778564453125, -1.0008762007617709, -0.059227299979315154),
(0.28181243896484376, -1.0021948141328698, -0.09380038023464156),
(0.3728358840942383, -1.0038270754728402, -0.123962953123039),
(0.4524813985824585, -1.005613540368227, -0.15026634875747985),
(0.5221712237596512, -1.0074443351354077, -0.17320066690539515),
(0.5831498207896948, -1.009246118007067, -0.1931978258913501),
(0.636506093190983, -1.0109719597443307, -0.21063630045261386),
(0.6831928315421101, -1.0125937110987449, -0.2258465242053158),
(0.7648946236565826, -1.0156754074572174, -0.2523480191006915),
(0.8261709677424369, -1.0181837235093538, -0.2721208327884435),
(0.8721282258068277, -1.0201720738092597, -0.2868892148703381),
(0.9410641129034139, -1.0233210275568283, -0.3089391941950436),
(1.0, -1.026166099551513, -0.3276894025360433)]
"""
G = f.change_ring(QQbar).change_ring(CIF)
(x, y) = G.variables()
g = G.subs({x: (1-x)*CIF(x0) + x*CIF(x1)})
coefs = []
deg = g.total_degree()
for d in range(deg + 1):
for i in range(d + 1):
c = CIF(g.coefficient({x: d-i, y: i}))
cr = c.real()
ci = c.imag()
coefs += list(cr.endpoints())
coefs += list(ci.endpoints())
yr = CC(y0a).real()
yi = CC(y0a).imag()
points = contpath(deg, coefs, yr, yi)
return points