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 _evalf_(self, x, **kwargs):
"""
EXAMPLES::
sage: from sage.functions.airy import airy_ai_simple
sage: airy_ai_simple(0.0)
0.355028053887817
sage: airy_ai_simple(1.0 * I)
0.331493305432141 - 0.317449858968444*I
We can use several methods for numerical evaluation::
sage: airy_ai_simple(3).n(algorithm='mpmath')
0.00659113935746072
sage: airy_ai_simple(3).n(algorithm='mpmath', prec=100)
0.0065911393574607191442574484080
sage: airy_ai_simple(3).n(algorithm='scipy') # rel tol 1e-10
0.006591139357460719
sage: airy_ai_simple(I).n(algorithm='scipy') # rel tol 1e-10
0.33149330543214117 - 0.3174498589684438*I
TESTS::
sage: parent(airy_ai_simple(3).n(algorithm='scipy'))
Real Field with 53 bits of precision
sage: airy_ai_simple(3).n(algorithm='scipy', prec=200)
Traceback (most recent call last):
...
NotImplementedError: airy_ai not implemented for precision > 53
"""
algorithm = kwargs.get('algorithm', 'mpmath') or 'mpmath'
parent = kwargs.get('parent')
if algorithm == 'scipy':
if hasattr(parent, 'prec') and parent.prec() > 53:
raise NotImplementedError(
"%s not implemented for precision > 53" % self.name())
from sage.rings.all import RR, CC
from sage.functions.other import real, imag
from scipy.special import airy as airy
if x in RR:
y = airy(real(x))[0]
if parent is None:
return RR(y)
else:
y = airy(complex(real(x), imag(x)))[0]
if parent is None:
return CC(y)
return parent(y)
elif algorithm == 'mpmath':
import mpmath
from sage.libs.mpmath import utils as mpmath_utils
return mpmath_utils.call(mpmath.airyai, x, parent=parent)
else:
raise ValueError("unknown algorithm '%s'" % algorithm)
def _evalf_(self, x, **kwargs):
"""
EXAMPLES::
sage: from sage.functions.airy import airy_bi_simple
sage: airy_bi_simple(0.0)
0.614926627446001
sage: airy_bi_simple(1.0 * I)
0.648858208330395 + 0.344958634768048*I
We can use several methods for numerical evaluation::
sage: airy_bi_simple(3).n(algorithm='mpmath')
14.0373289637302
sage: airy_bi_simple(3).n(algorithm='mpmath', prec=100)
14.037328963730232031740267314
sage: airy_bi_simple(3).n(algorithm='scipy') # rel tol 1e-10
14.037328963730136
sage: airy_bi_simple(I).n(algorithm='scipy') # rel tol 1e-10
0.648858208330395 + 0.34495863476804844*I
TESTS::
sage: parent(airy_bi_simple(3).n(algorithm='scipy'))
Real Field with 53 bits of precision
sage: airy_bi_simple(3).n(algorithm='scipy', prec=200)
Traceback (most recent call last):
...
NotImplementedError: airy_bi not implemented for precision > 53
"""
algorithm = kwargs.get('algorithm', 'mpmath') or 'mpmath'
parent = kwargs.get('parent', None)
if algorithm == 'scipy':
if hasattr(parent, 'prec') and parent.prec() > 53:
raise NotImplementedError(
"%s not implemented for precision > 53" % self.name())
from sage.rings.all import RR, CC
from sage.functions.other import real, imag
from scipy.special import airy as airy
if x in RR:
y = airy(real(x))[2]
if parent is None:
return RR(y)
else:
y = airy(complex(real(x), imag(x)))[2]
if parent is None:
return CC(y)
return parent(y)
elif algorithm == 'mpmath':
import mpmath
from sage.libs.mpmath import utils as mpmath_utils
return mpmath_utils.call(mpmath.airybi, x, parent=parent)
else:
raise ValueError("unknown algorithm '%s'" % algorithm)
def _evalf_(self, x, **kwargs):
"""
EXAMPLES::
sage: airy_ai_prime(0.0)
-0.258819403792807
We can use several methods for numerical evaluation::
sage: airy_ai_prime(4).n(algorithm='mpmath')
-0.00195864095020418
sage: airy_ai_prime(4).n(algorithm='mpmath', prec=100)
-0.0019586409502041789001381409184
sage: airy_ai_prime(4).n(algorithm='scipy') # rel tol 1e-10
-0.00195864095020418
sage: airy_ai_prime(I).n(algorithm='scipy') # rel tol 1e-10
-0.43249265984180707 + 0.09804785622924324*I
TESTS::
sage: parent(airy_ai_prime(3).n(algorithm='scipy'))
Real Field with 53 bits of precision
sage: airy_ai_prime(3).n(algorithm='scipy', prec=200)
Traceback (most recent call last):
...
NotImplementedError: airy_ai_prime not implemented for precision > 53
"""
algorithm = kwargs.get('algorithm', 'mpmath') or 'mpmath'
parent = kwargs.get('parent', None)
if algorithm == 'scipy':
if hasattr(parent, 'prec') and parent.prec() > 53:
raise NotImplementedError(
"%s not implemented for precision > 53" % self.name())
from sage.rings.all import RR, CC
from sage.functions.other import real, imag
from scipy.special import airy as airy
if x in RR:
y = airy(real(x))[1]
if parent is None:
return RR(y)
else:
y = airy(complex(real(x), imag(x)))[1]
if parent is None:
return CC(y)
return parent(y)
elif algorithm == 'mpmath':
import mpmath
from sage.libs.mpmath import utils as mpmath_utils
return mpmath_utils.call(mpmath.airyai,
x,
derivative=1,
parent=parent)
else:
raise ValueError("unknown algorithm '%s'" % algorithm)
def _evalf_(self, x, **kwargs):
"""
EXAMPLES::
sage: airy_bi_prime(0.0)
0.448288357353826
We can use several methods for numerical evaluation::
sage: airy_bi_prime(4).n(algorithm='mpmath')
161.926683504613
sage: airy_bi_prime(4).n(algorithm='mpmath', prec=100)
161.92668350461340184309492429
sage: airy_bi_prime(4).n(algorithm='scipy') # rel tol 1e-10
161.92668350461398
sage: airy_bi_prime(I).n(algorithm='scipy') # rel tol 1e-10
0.135026646710819 - 0.1288373867812549*I
TESTS::
sage: parent(airy_bi_prime(3).n(algorithm='scipy'))
Real Field with 53 bits of precision
sage: airy_bi_prime(3).n(algorithm='scipy', prec=200)
Traceback (most recent call last):
...
NotImplementedError: airy_bi_prime not implemented for precision > 53
"""
algorithm = kwargs.get('algorithm', 'mpmath') or 'mpmath'
parent = kwargs.get('parent', None)
if algorithm == 'scipy':
if hasattr(parent, 'prec') and parent.prec() > 53:
raise NotImplementedError(
"%s not implemented for precision > 53" % self.name())
from sage.rings.all import RR, CC
from sage.functions.other import real, imag
from scipy.special import airy as airy
if x in RR:
y = airy(real(x))[3]
if parent is None:
return RR(y)
else:
y = airy(complex(real(x), imag(x)))[3]
if parent is None:
return CC(y)
return parent(y)
elif algorithm == 'mpmath':
import mpmath
from sage.libs.mpmath import utils as mpmath_utils
return mpmath_utils.call(mpmath.airybi,
x,
derivative=1,
parent=parent)
else:
raise ValueError("unknown algorithm '%s'" % algorithm)
def _evalf_(self, x, **kwargs):
"""
EXAMPLES::
sage: from sage.functions.airy import airy_ai_simple
sage: airy_ai_simple(0.0)
0.355028053887817
sage: airy_ai_simple(1.0 * I)
0.331493305432141 - 0.317449858968444*I
We can use several methods for numerical evaluation::
sage: airy_ai_simple(3).n(algorithm='mpmath')
0.00659113935746072
sage: airy_ai_simple(3).n(algorithm='mpmath', prec=100)
0.0065911393574607191442574484080
sage: airy_ai_simple(3).n(algorithm='scipy') # rel tol 1e-10
0.006591139357460719
sage: airy_ai_simple(I).n(algorithm='scipy') # rel tol 1e-10
0.33149330543214117 - 0.3174498589684438*I
TESTS::
sage: parent(airy_ai_simple(3).n(algorithm='scipy'))
Real Field with 53 bits of precision
sage: airy_ai_simple(3).n(algorithm='scipy', prec=200)
Traceback (most recent call last):
...
NotImplementedError: airy_ai not implemented for precision > 53
"""
algorithm = kwargs.get('algorithm', 'mpmath') or 'mpmath'
parent = kwargs.get('parent')
if algorithm == 'scipy':
if hasattr(parent, 'prec') and parent.prec() > 53:
raise NotImplementedError("%s not implemented for precision > 53"%self.name())
from sage.rings.all import RR, CC
from sage.functions.other import real,imag
from scipy.special import airy as airy
if x in RR:
y = airy(real(x))[0]
if parent is None:
return RR(y)
else:
y = airy(complex(real(x),imag(x)))[0]
if parent is None:
return CC(y)
return parent(y)
elif algorithm == 'mpmath':
import mpmath
from sage.libs.mpmath import utils as mpmath_utils
return mpmath_utils.call(mpmath.airyai, x, parent=parent)
else:
raise ValueError("unknown algorithm '%s'" % algorithm)
def _evalf_(self, x, **kwargs):
"""
EXAMPLES::
sage: from sage.functions.airy import airy_bi_simple
sage: airy_bi_simple(0.0)
0.614926627446001
sage: airy_bi_simple(1.0 * I)
0.648858208330395 + 0.344958634768048*I
We can use several methods for numerical evaluation::
sage: airy_bi_simple(3).n(algorithm='mpmath')
14.0373289637302
sage: airy_bi_simple(3).n(algorithm='mpmath', prec=100)
14.037328963730232031740267314
sage: airy_bi_simple(3).n(algorithm='scipy') # rel tol 1e-10
14.037328963730136
sage: airy_bi_simple(I).n(algorithm='scipy') # rel tol 1e-10
0.648858208330395 + 0.34495863476804844*I
TESTS::
sage: parent(airy_bi_simple(3).n(algorithm='scipy'))
Real Field with 53 bits of precision
sage: airy_bi_simple(3).n(algorithm='scipy', prec=200)
Traceback (most recent call last):
...
NotImplementedError: airy_bi not implemented for precision > 53
"""
algorithm = kwargs.get('algorithm', 'mpmath') or 'mpmath'
parent = kwargs.get('parent', None)
if algorithm == 'scipy':
if hasattr(parent, 'prec') and parent.prec() > 53:
raise NotImplementedError("%s not implemented for precision > 53"%self.name())
from sage.rings.all import RR, CC
from sage.functions.other import real,imag
from scipy.special import airy as airy
if x in RR:
y = airy(real(x))[2]
if parent is None:
return RR(y)
else:
y = airy(complex(real(x),imag(x)))[2]
if parent is None:
return CC(y)
return parent(y)
elif algorithm == 'mpmath':
import mpmath
from sage.libs.mpmath import utils as mpmath_utils
return mpmath_utils.call(mpmath.airybi, x, parent=parent)
else:
raise ValueError("unknown algorithm '%s'" % algorithm)
def isometry_from_fixed_points(self, repel, attract):
r"""
Given two fixed points ``repel`` and ``attract`` as complex
numbers return a hyperbolic isometry with ``repel`` as repelling
fixed point and ``attract`` as attracting fixed point.
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: UHP.isometry_from_fixed_points(2 + I, 3 + I)
Traceback (most recent call last):
...
ValueError: fixed points of hyperbolic elements must be ideal
sage: UHP.isometry_from_fixed_points(2, 0)
Isometry in UHP
[ -1 0]
[-1/3 -1/3]
TESTS::
sage: UHP = HyperbolicPlane().UHP()
sage: UHP.isometry_from_fixed_points(0, 4)
Isometry in UHP
[ -1 0]
[-1/5 -1/5]
sage: UHP.isometry_from_fixed_points(UHP.get_point(0), UHP.get_point(4))
Isometry in UHP
[ -1 0]
[-1/5 -1/5]
"""
if isinstance(repel, HyperbolicPoint):
repel = repel._coordinates
if isinstance(attract, HyperbolicPoint):
attract = attract._coordinates
if imag(repel) + imag(attract) > EPSILON:
raise ValueError(
"fixed points of hyperbolic elements must be ideal")
repel = real(repel)
attract = real(attract)
if repel == infinity:
A = self._moebius_sending([infinity, attract, attract + 1],
[infinity, attract, attract + 2])
elif attract == infinity:
A = self._moebius_sending([repel, infinity, repel + 1],
[repel, infinity, repel + 2])
else:
A = self._moebius_sending(
[repel, attract, infinity],
[repel, attract, max(repel, attract) + 1])
return self.get_isometry(A)
def sexp_0(self,t):
#convergence radius 1
t = self._in_prec(t)
sexp = self.sexp_0
b = self.b
#development point -1 convergence radius 1
if real(t)>1:
return b**(sexp(t-1))
if real(t)<0:
#sage bug, log(z,b) does not work for complex z
return log(sexp(t+1))/log(b)
return self.sexp_0_raw(t)
def show(self, boundary=True, **options):
r"""
Plot ``self``.
EXAMPLES:
First some lines::
sage: PD = HyperbolicPlane().PD()
sage: PD.get_geodesic(0, 1).show()
Graphics object consisting of 2 graphics primitives
sage: PD.get_geodesic(0, 0.3+0.8*I).show()
Graphics object consisting of 2 graphics primitives
Then some generic geodesics::
sage: PD.get_geodesic(-0.5, 0.3+0.4*I).show()
Graphics object consisting of 2 graphics primitives
sage: PD.get_geodesic(-1, exp(3*I*pi/7)).show(linestyle="dashed", color="red")
Graphics object consisting of 2 graphics primitives
sage: PD.get_geodesic(exp(2*I*pi/11), exp(1*I*pi/11)).show(thickness=6, color="orange")
Graphics object consisting of 2 graphics primitives
"""
opts = {'axes': False, 'aspect_ratio': 1}
opts.update(self.graphics_options())
opts.update(options)
end_1, end_2 = [CC(k.coordinates()) for k in self.endpoints()]
bd_1, bd_2 = [CC(k.coordinates()) for k in self.ideal_endpoints()]
# Check to see if it's a line
if abs(bd_1 + bd_2) < EPSILON:
pic = line([end_1, end_2], **opts)
else:
# If we are here, we know it's not a line
# So we compute the center and radius of the circle
invdet = RR.one() / (real(bd_1)*imag(bd_2) - real(bd_2)*imag(bd_1))
centerx = (imag(bd_2) - imag(bd_1)) * invdet
centery = (real(bd_1) - real(bd_2)) * invdet
center = centerx + I * centery
radius = RR(abs(bd_1 - center))
# Now we calculate the angles for the arc
theta1 = CC(end_1 - center).arg()
theta2 = CC(end_2 - center).arg()
theta1, theta2 = sorted([theta1, theta2])
# Make sure the sector is inside the disk
if theta2 - theta1 > pi:
theta1 += 2 * pi
pic = arc((centerx, centery), radius,
sector=(theta1, theta2), **opts)
if boundary:
pic += self._model.get_background_graphic()
return pic
def _evalf_(self, x, **kwargs):
"""
EXAMPLES::
sage: airy_ai_prime(0.0)
-0.258819403792807
We can use several methods for numerical evaluation::
sage: airy_ai_prime(4).n(algorithm='mpmath')
-0.00195864095020418
sage: airy_ai_prime(4).n(algorithm='mpmath', prec=100)
-0.0019586409502041789001381409184
sage: airy_ai_prime(4).n(algorithm='scipy') # rel tol 1e-10
-0.00195864095020418
sage: airy_ai_prime(I).n(algorithm='scipy') # rel tol 1e-10
-0.43249265984180707 + 0.09804785622924324*I
TESTS::
sage: parent(airy_ai_prime(3).n(algorithm='scipy'))
Real Field with 53 bits of precision
sage: airy_ai_prime(3).n(algorithm='scipy', prec=200)
Traceback (most recent call last):
...
NotImplementedError: airy_ai_prime not implemented for precision > 53
"""
algorithm = kwargs.get('algorithm', 'mpmath') or 'mpmath'
parent = kwargs.get('parent', None)
if algorithm == 'scipy':
if hasattr(parent, 'prec') and parent.prec() > 53:
raise NotImplementedError("%s not implemented for precision > 53"%self.name())
from sage.rings.all import RR, CC
from sage.functions.other import real,imag
from scipy.special import airy as airy
if x in RR:
y = airy(real(x))[1]
if parent is None:
return RR(y)
else:
y = airy(complex(real(x),imag(x)))[1]
if parent is None:
return CC(y)
return parent(y)
elif algorithm == 'mpmath':
import mpmath
from sage.libs.mpmath import utils as mpmath_utils
return mpmath_utils.call(mpmath.airyai, x, derivative=1,
parent=parent)
else:
raise ValueError("unknown algorithm '%s'" % algorithm)
def show(self, boundary=True, **options):
r"""
Plot ``self``.
EXAMPLES:
First some lines::
sage: PD = HyperbolicPlane().PD()
sage: PD.get_geodesic(0, 1).show()
Graphics object consisting of 2 graphics primitives
sage: PD.get_geodesic(0, 0.3+0.8*I).show()
Graphics object consisting of 2 graphics primitives
Then some generic geodesics::
sage: PD.get_geodesic(-0.5, 0.3+0.4*I).show()
Graphics object consisting of 2 graphics primitives
sage: PD.get_geodesic(-1, exp(3*I*pi/7)).show(linestyle="dashed", color="red")
Graphics object consisting of 2 graphics primitives
sage: PD.get_geodesic(exp(2*I*pi/11), exp(1*I*pi/11)).show(thickness=6, color="orange")
Graphics object consisting of 2 graphics primitives
"""
opts = {'axes': False, 'aspect_ratio': 1}
opts.update(self.graphics_options())
opts.update(options)
end_1, end_2 = [CC(k.coordinates()) for k in self.endpoints()]
bd_1, bd_2 = [CC(k.coordinates()) for k in self.ideal_endpoints()]
# Check to see if it's a line
if abs(bd_1 + bd_2) < EPSILON:
pic = line([end_1, end_2], **opts)
else:
# If we are here, we know it's not a line
# So we compute the center and radius of the circle
invdet = RR.one() / (real(bd_1)*imag(bd_2) - real(bd_2)*imag(bd_1))
centerx = (imag(bd_2) - imag(bd_1)) * invdet
centery = (real(bd_1) - real(bd_2)) * invdet
center = centerx + I * centery
radius = RR(abs(bd_1 - center))
# Now we calculate the angles for the arc
theta1 = CC(end_1 - center).arg()
theta2 = CC(end_2 - center).arg()
theta1, theta2 = sorted([theta1, theta2])
# Make sure the sector is inside the disk
if theta2 - theta1 > pi:
theta1 += 2 * pi
pic = arc((centerx, centery), radius,
sector=(theta1, theta2), **opts)
if boundary:
pic += self._model.get_background_graphic()
return pic
def _evalf_(self, x, **kwargs):
"""
EXAMPLES::
sage: airy_bi_prime(0.0)
0.448288357353826
We can use several methods for numerical evaluation::
sage: airy_bi_prime(4).n(algorithm='mpmath')
161.926683504613
sage: airy_bi_prime(4).n(algorithm='mpmath', prec=100)
161.92668350461340184309492429
sage: airy_bi_prime(4).n(algorithm='scipy') # rel tol 1e-10
161.92668350461398
sage: airy_bi_prime(I).n(algorithm='scipy') # rel tol 1e-10
0.135026646710819 - 0.1288373867812549*I
TESTS::
sage: parent(airy_bi_prime(3).n(algorithm='scipy'))
Real Field with 53 bits of precision
sage: airy_bi_prime(3).n(algorithm='scipy', prec=200)
Traceback (most recent call last):
...
NotImplementedError: airy_bi_prime not implemented for precision > 53
"""
algorithm = kwargs.get('algorithm', 'mpmath') or 'mpmath'
parent = kwargs.get('parent', None)
if algorithm == 'scipy':
if hasattr(parent, 'prec') and parent.prec() > 53:
raise NotImplementedError("%s not implemented for precision > 53"%self.name())
from sage.rings.all import RR, CC
from sage.functions.other import real,imag
from scipy.special import airy as airy
if x in RR:
y = airy(real(x))[3]
if parent is None:
return RR(y)
else:
y = airy(complex(real(x),imag(x)))[3]
if parent is None:
return CC(y)
return parent(y)
elif algorithm == 'mpmath':
import mpmath
from sage.libs.mpmath import utils as mpmath_utils
return mpmath_utils.call(mpmath.airybi, x, derivative=1,
parent=parent)
else:
raise ValueError("unknown algorithm '%s'" % algorithm)
def sexp_1(self,t):
t = self._in_prec(t)
sexp = self.sexp_1
b = self.b
IM = self.IM
N = self.N
#development point 0 convergence radius 2
if real(t)>1:
return b**(sexp(t-1))
if real(t)<0:
#sage bug, log(z,b) does not work for complex z
return log(sexp(t+1))/log(b)
return self.sexp_1_raw(t)
def isometry_from_fixed_points(self, repel, attract):
r"""
Given two fixed points ``repel`` and ``attract`` as complex
numbers return a hyperbolic isometry with ``repel`` as repelling
fixed point and ``attract`` as attracting fixed point.
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: UHP.isometry_from_fixed_points(2 + I, 3 + I)
Traceback (most recent call last):
...
ValueError: fixed points of hyperbolic elements must be ideal
sage: UHP.isometry_from_fixed_points(2, 0)
Isometry in UHP
[ -1 0]
[-1/3 -1/3]
TESTS::
sage: UHP = HyperbolicPlane().UHP()
sage: UHP.isometry_from_fixed_points(0, 4)
Isometry in UHP
[ -1 0]
[-1/5 -1/5]
sage: UHP.isometry_from_fixed_points(UHP.get_point(0), UHP.get_point(4))
Isometry in UHP
[ -1 0]
[-1/5 -1/5]
"""
if isinstance(repel, HyperbolicPoint):
repel = repel._coordinates
if isinstance(attract, HyperbolicPoint):
attract = attract._coordinates
if imag(repel) + imag(attract) > EPSILON:
raise ValueError("fixed points of hyperbolic elements must be ideal")
repel = real(repel)
attract = real(attract)
if repel == infinity:
A = self._moebius_sending([infinity, attract, attract + 1],
[infinity, attract, attract + 2])
elif attract == infinity:
A = self._moebius_sending([repel, infinity, repel + 1],
[repel, infinity, repel + 2])
else:
A = self._moebius_sending([repel, attract, infinity],
[repel, attract, max(repel, attract) + 1])
return self.get_isometry(A)
def show(self, boundary=True, **options):
r"""
Plot ``self``.
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: UHP.get_geodesic(0, 1).show()
Graphics object consisting of 2 graphics primitives
sage: UHP.get_geodesic(I, 3+4*I).show(linestyle="dashed", color="red")
Graphics object consisting of 2 graphics primitives
"""
opts = {'axes': False, 'aspect_ratio': 1}
opts.update(self.graphics_options())
opts.update(options)
end_1, end_2 = [CC(k.coordinates()) for k in self.endpoints()]
bd_1, bd_2 = [CC(k.coordinates()) for k in self.ideal_endpoints()]
if (abs(real(end_1) - real(end_2)) < EPSILON) \
or CC(infinity) in [end_1, end_2]: # on same vertical line
# If one of the endpoints is infinity, we replace it with a
# large finite point
if end_1 == CC(infinity):
end_1 = (real(end_2), (imag(end_2) + 10))
end_2 = (real(end_2), imag(end_2))
elif end_2 == CC(infinity):
end_2 = (real(end_1), (imag(end_1) + 10))
end_1 = (real(end_1), imag(end_1))
pic = line((end_1, end_2), **opts)
if boundary:
cent = min(bd_1, bd_2)
bd_dict = {'bd_min': cent - 3, 'bd_max': cent + 3}
bd_pic = self._model.get_background_graphic(**bd_dict)
pic = bd_pic + pic
return pic
else:
center = (bd_1 + bd_2) / 2 # Circle center
radius = abs(bd_1 - bd_2) / 2
theta1 = CC(end_1 - center).arg()
theta2 = CC(end_2 - center).arg()
if abs(theta1 - theta2) < EPSILON:
theta2 += pi
pic = arc((real(center), imag(center)),
radius,
sector=(theta1, theta2),
**opts)
if boundary:
# We want to draw a segment of the real line. The
# computations below compute the projection of the
# geodesic to the real line, and then draw a little
# to the left and right of the projection.
shadow_1, shadow_2 = [real(k) for k in [end_1, end_2]]
midpoint = (shadow_1 + shadow_2) / 2
length = abs(shadow_1 - shadow_2)
bd_dict = {
'bd_min': midpoint - length,
'bd_max': midpoint + length
}
bd_pic = self._model.get_background_graphic(**bd_dict)
pic = bd_pic + pic
return pic
def show(self, boundary=True, **options):
r"""
Plot ``self``.
EXAMPLES::
sage: HyperbolicPlane().UHP().get_geodesic(0, 1).show()
Graphics object consisting of 2 graphics primitives
"""
opts = {'axes': False, 'aspect_ratio': 1}
opts.update(self.graphics_options())
opts.update(options)
end_1, end_2 = [CC(k.coordinates()) for k in self.endpoints()]
bd_1, bd_2 = [CC(k.coordinates()) for k in self.ideal_endpoints()]
if (abs(real(end_1) - real(end_2)) < EPSILON) \
or CC(infinity) in [end_1, end_2]: #on same vertical line
# If one of the endpoints is infinity, we replace it with a
# large finite point
if end_1 == CC(infinity):
end_1 = (real(end_2), (imag(end_2) + 10))
end_2 = (real(end_2), imag(end_2))
elif end_2 == CC(infinity):
end_2 = (real(end_1), (imag(end_1) + 10))
end_1 = (real(end_1), imag(end_1))
from sage.plot.line import line
pic = line((end_1, end_2), **opts)
if boundary:
cent = min(bd_1, bd_2)
bd_dict = {'bd_min': cent - 3, 'bd_max': cent + 3}
bd_pic = self._model.get_background_graphic(**bd_dict)
pic = bd_pic + pic
return pic
else:
center = (bd_1 + bd_2)/2 # Circle center
radius = abs(bd_1 - bd_2)/2
theta1 = CC(end_1 - center).arg()
theta2 = CC(end_2 - center).arg()
if abs(theta1 - theta2) < EPSILON:
theta2 += pi
[theta1, theta2] = sorted([theta1, theta2])
from sage.calculus.var import var
from sage.plot.plot import parametric_plot
x = var('x')
pic = parametric_plot((radius*cos(x) + real(center),
radius*sin(x) + imag(center)),
(x, theta1, theta2), **opts)
if boundary:
# We want to draw a segment of the real line. The
# computations below compute the projection of the
# geodesic to the real line, and then draw a little
# to the left and right of the projection.
shadow_1, shadow_2 = [real(k) for k in [end_1, end_2]]
midpoint = (shadow_1 + shadow_2)/2
length = abs(shadow_1 - shadow_2)
bd_dict = {'bd_min': midpoint - length, 'bd_max': midpoint +
length}
bd_pic = self._model.get_background_graphic(**bd_dict)
pic = bd_pic + pic
return pic
def image_coordinates(self, x):
"""
Return the image of the coordinates of the hyperbolic point ``x``
under ``self``.
EXAMPLES::
sage: PD = HyperbolicPlane().PD()
sage: KM = HyperbolicPlane().KM()
sage: phi = KM.coerce_map_from(PD)
sage: phi.image_coordinates(0.5+0.5*I)
(0.666666666666667, 0.666666666666667)
"""
return (2*real(x)/(Integer(1) + real(x)**2 + imag(x)**2),
2*imag(x)/(Integer(1) + real(x)**2 + imag(x)**2))
def image_coordinates(self, x):
"""
Return the image of the coordinates of the hyperbolic point ``x``
under ``self``.
EXAMPLES::
sage: PD = HyperbolicPlane().PD()
sage: KM = HyperbolicPlane().KM()
sage: phi = KM.coerce_map_from(PD)
sage: phi.image_coordinates(0.5+0.5*I)
(0.666666666666667, 0.666666666666667)
"""
return (2 * real(x) / (Integer(1) + real(x)**2 + imag(x)**2),
2 * imag(x) / (Integer(1) + real(x)**2 + imag(x)**2))
def image_coordinates(self, x):
"""
Return the image of the coordinates of the hyperbolic point ``x``
under ``self``.
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: HM = HyperbolicPlane().HM()
sage: phi = HM.coerce_map_from(UHP)
sage: phi.image_coordinates(3 + I)
(3, 9/2, 11/2)
"""
return vector((real(x)/imag(x),
(real(x)**2 + imag(x)**2 - 1)/(2*imag(x)),
(real(x)**2 + imag(x)**2 + 1)/(2*imag(x))))
def image_coordinates(self, x):
"""
Return the image of the coordinates of the hyperbolic point ``x``
under ``self``.
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: HM = HyperbolicPlane().HM()
sage: phi = HM.coerce_map_from(UHP)
sage: phi.image_coordinates(3 + I)
(3, 9/2, 11/2)
"""
return vector(
(real(x) / imag(x), (real(x)**2 + imag(x)**2 - 1) / (2 * imag(x)),
(real(x)**2 + imag(x)**2 + 1) / (2 * imag(x))))
def reflection_involution(self):
r"""
Return the isometry of the involution fixing the geodesic ``self``.
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: g1 = UHP.get_geodesic(0, 1)
sage: g1.reflection_involution()
Isometry in UHP
[ 1 0]
[ 2 -1]
sage: UHP.get_geodesic(I, 2*I).reflection_involution()
Isometry in UHP
[ 1 0]
[ 0 -1]
"""
x, y = [real(k.coordinates()) for k in self.ideal_endpoints()]
if x == infinity:
M = matrix([[1, -2 * y], [0, -1]])
elif y == infinity:
M = matrix([[1, -2 * x], [0, -1]])
else:
M = matrix([[(x + y) / (y - x), -2 * x * y / (y - x)],
[2 / (y - x), -(x + y) / (y - x)]])
return self._model.get_isometry(M)
def reflection_involution(self):
r"""
Return the isometry of the involution fixing the geodesic ``self``.
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: g1 = UHP.get_geodesic(0, 1)
sage: g1.reflection_involution()
Isometry in UHP
[ 1 0]
[ 2 -1]
sage: UHP.get_geodesic(I, 2*I).reflection_involution()
Isometry in UHP
[ 1 0]
[ 0 -1]
"""
x, y = [real(k.coordinates()) for k in self.ideal_endpoints()]
if x == infinity:
M = matrix([[1, -2*y], [0, -1]])
elif y == infinity:
M = matrix([[1, -2*x], [0, -1]])
else:
M = matrix([[(x+y)/(y-x), -2*x*y/(y-x)], [2/(y-x), -(x+y)/(y-x)]])
return self._model.get_isometry(M)
def image_coordinates(self, x):
"""
Return the image of the coordinates of the hyperbolic point ``x``
under ``self``.
EXAMPLES::
sage: PD = HyperbolicPlane().PD()
sage: HM = HyperbolicPlane().HM()
sage: phi = HM.coerce_map_from(PD)
sage: phi.image_coordinates(0.5+0.5*I)
(2.00000000000000, 2.00000000000000, 3.00000000000000)
"""
return vector(
(2 * real(x) / (1 - real(x)**2 - imag(x)**2),
2 * imag(x) / (1 - real(x)**2 - imag(x)**2),
(real(x)**2 + imag(x)**2 + 1) / (1 - real(x)**2 - imag(x)**2)))
def image_coordinates(self, x):
"""
Return the image of the coordinates of the hyperbolic point ``x``
under ``self``.
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: KM = HyperbolicPlane().KM()
sage: phi = KM.coerce_map_from(UHP)
sage: phi.image_coordinates(3 + I)
(6/11, 9/11)
"""
if x == infinity:
return (0, 1)
return ((2 * real(x)) / (real(x)**2 + imag(x)**2 + 1),
(real(x)**2 + imag(x)**2 - 1) / (real(x)**2 + imag(x)**2 + 1))
def image_coordinates(self, x):
"""
Return the image of the coordinates of the hyperbolic point ``x``
under ``self``.
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: KM = HyperbolicPlane().KM()
sage: phi = KM.coerce_map_from(UHP)
sage: phi.image_coordinates(3 + I)
(6/11, 9/11)
"""
if x == infinity:
return (0, 1)
return ((2*real(x))/(real(x)**2 + imag(x)**2 + 1),
(real(x)**2 + imag(x)**2 - 1)/(real(x)**2 + imag(x)**2 + 1))
def image_coordinates(self, x):
"""
Return the image of the coordinates of the hyperbolic point ``x``
under ``self``.
EXAMPLES::
sage: PD = HyperbolicPlane().PD()
sage: HM = HyperbolicPlane().HM()
sage: phi = HM.coerce_map_from(PD)
sage: phi.image_coordinates(0.5+0.5*I)
(2.00000000000000, 2.00000000000000, 3.00000000000000)
"""
return vector((2*real(x)/(1 - real(x)**2 - imag(x)**2),
2*imag(x)/(1 - real(x)**2 - imag(x)**2),
(real(x)**2 + imag(x)**2 + 1) /
(1 - real(x)**2 - imag(x)**2)))
def super(self,x):
"""
Development point is x0-1
"""
if isinstance(x,float) and self.iprec != None:
x = RealField(self.iprec)(x)
super = self.super
super_raw = self.super_raw
b = self.b
c = self.c
xt = x - c
if real(xt)<-0.5:
return log(super(x+1))/log(b)
if real(xt)>0.5:
return b**(super(x-1))
return super_raw(x)
def _dist_points(self, p1, p2):
r"""
Compute the distance between two points in the Upper Half Plane
using the hyperbolic metric.
INPUT:
- ``p1``, ``p2`` -- the coordinates of the points
EXAMPLES::
sage: HyperbolicPlane().UHP()._dist_points(4.0*I, I)
1.38629436111989
"""
num = (real(p2) - real(p1))**2 + (imag(p2) - imag(p1))**2
denom = 2 * imag(p1) * imag(p2)
if denom == 0:
return infinity
return arccosh(1 + num / denom)
def _dist_points(self, p1, p2):
r"""
Compute the distance between two points in the Upper Half Plane
using the hyperbolic metric.
INPUT:
- ``p1``, ``p2`` -- the coordinates of the points
EXAMPLES::
sage: HyperbolicPlane().UHP()._dist_points(4.0*I, I)
1.38629436111989
"""
num = (real(p2) - real(p1))**2 + (imag(p2) - imag(p1))**2
denom = 2 * imag(p1) * imag(p2)
if denom == 0:
return infinity
return arccosh(1 + num/denom)
def midpoint(self): # UHP
r"""
Return the (hyperbolic) midpoint of ``self`` if it exists.
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: g = UHP.random_geodesic()
sage: m = g.midpoint()
sage: d1 = UHP.dist(m, g.start())
sage: d2 = UHP.dist(m, g.end())
sage: bool(abs(d1 - d2) < 10**-9)
True
Infinite geodesics have no midpoint::
sage: UHP.get_geodesic(0, 2).midpoint()
Traceback (most recent call last):
...
ValueError: the length must be finite
"""
if self.length() == infinity:
raise ValueError("the length must be finite")
start = self._start.coordinates()
end = self._end.coordinates()
d = self._model._dist_points(start, end) / 2
S = self.complete()._to_std_geod(start)
T = matrix([[exp(d), 0], [0, 1]])
M = S.inverse() * T * S
if ((real(start - end) < EPSILON)
or (abs(real(start - end)) < EPSILON
and imag(start - end) < EPSILON)):
end_p = start
else:
end_p = end
return self._model.get_point(mobius_transform(M, end_p))
def midpoint(self): # UHP
r"""
Return the (hyperbolic) midpoint of ``self`` if it exists.
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: g = UHP.random_geodesic()
sage: m = g.midpoint()
sage: d1 = UHP.dist(m, g.start())
sage: d2 = UHP.dist(m, g.end())
sage: bool(abs(d1 - d2) < 10**-9)
True
Infinite geodesics have no midpoint::
sage: UHP.get_geodesic(0, 2).midpoint()
Traceback (most recent call last):
...
ValueError: the length must be finite
"""
if self.length() == infinity:
raise ValueError("the length must be finite")
start = self._start.coordinates()
end = self._end.coordinates()
d = self._model._dist_points(start, end) / 2
S = self.complete()._to_std_geod(start)
T = matrix([[exp(d), 0], [0, 1]])
M = S.inverse() * T * S
if ((real(start - end) < EPSILON)
or (abs(real(start - end)) < EPSILON
and imag(start - end) < EPSILON)):
end_p = start
else:
end_p = end
return self._model.get_point(mobius_transform(M, end_p))
def symmetry_involution(self):
r"""
Return the involutory isometry fixing the given point.
EXAMPLES::
sage: HyperbolicPlane().UHP().get_point(3 + 2*I).symmetry_involution()
Isometry in UHP
[ 3/2 -13/2]
[ 1/2 -3/2]
"""
p = self._coordinates
x, y = real(p), imag(p)
if y > 0:
M = matrix([[x/y, -(x**2/y) - y], [1/y, -(x/y)]])
return self.parent().get_isometry(M)
raise ValueError("cannot determine the isometry of a boundary point")
def symmetry_involution(self):
r"""
Return the involutory isometry fixing the given point.
EXAMPLES::
sage: HyperbolicPlane().UHP().get_point(3 + 2*I).symmetry_involution()
Isometry in UHP
[ 3/2 -13/2]
[ 1/2 -3/2]
"""
p = self._coordinates
x, y = real(p), imag(p)
if y > 0:
M = matrix([[x / y, -(x**2 / y) - y], [1 / y, -(x / y)]])
return self.parent().get_isometry(M)
raise ValueError("cannot determine the isometry of a boundary point")
def SL2R_to_SO21(A):
r"""
Given a matrix in `SL(2, \RR)` return its irreducible representation in
`O(2,1)`.
Note that this is not the only homomorphism, but it is the only one
that works in the context of the implemented 2D hyperbolic geometry
models.
EXAMPLES::
sage: from sage.geometry.hyperbolic_space.hyperbolic_coercion import SL2R_to_SO21
sage: A = SL2R_to_SO21(identity_matrix(2))
sage: J = matrix([[1,0,0],[0,1,0],[0,0,-1]]) #Lorentzian Gram matrix
sage: norm(A.transpose()*J*A - J) < 10**-4
True
"""
a, b, c, d = (A/A.det().sqrt()).list()
# Kill ~0 imaginary parts
components = [
a*d + b*c, a*c - b*d, a*c + b*d, a*b - c*d,
Integer(1)/Integer(2)*a**2 - Integer(1)/Integer(2)*b**2 -
Integer(1)/Integer(2)*c**2 + Integer(1)/Integer(2)*d**2,
Integer(1)/Integer(2)*a**2 + Integer(1)/Integer(2)*b**2 -
Integer(1)/Integer(2)*c**2 - Integer(1)/Integer(2)*d**2,
a*b + c*d, Integer(1)/Integer(2)*a**2 -
Integer(1)/Integer(2)*b**2 + Integer(1)/Integer(2)*c**2 -
Integer(1)/Integer(2)*d**2, Integer(1)/Integer(2)*a**2 +
Integer(1)/Integer(2)*b**2 + Integer(1)/Integer(2)*c**2 +
Integer(1)/Integer(2)*d**2
]
B = matrix(3, [real(c) for c in components])
#B = B.apply_map(attrcall('real'))
if A.det() > 0:
return B
else:
# Orientation-reversing isometries swap the nappes of
# the lightcone. This fixes that issue.
return -B
def SL2R_to_SO21(A):
r"""
Given a matrix in `SL(2, \RR)` return its irreducible representation in
`O(2,1)`.
Note that this is not the only homomorphism, but it is the only one
that works in the context of the implemented 2D hyperbolic geometry
models.
EXAMPLES::
sage: from sage.geometry.hyperbolic_space.hyperbolic_coercion import SL2R_to_SO21
sage: A = SL2R_to_SO21(identity_matrix(2))
sage: J = matrix([[1,0,0],[0,1,0],[0,0,-1]]) #Lorentzian Gram matrix
sage: norm(A.transpose()*J*A - J) < 10**-4
True
"""
a, b, c, d = (A / A.det().sqrt()).list()
# Kill ~0 imaginary parts
components = [
a * d + b * c, a * c - b * d, a * c + b * d, a * b - c * d,
Integer(1) / Integer(2) * a**2 - Integer(1) / Integer(2) * b**2 -
Integer(1) / Integer(2) * c**2 + Integer(1) / Integer(2) * d**2,
Integer(1) / Integer(2) * a**2 + Integer(1) / Integer(2) * b**2 -
Integer(1) / Integer(2) * c**2 - Integer(1) / Integer(2) * d**2,
a * b + c * d,
Integer(1) / Integer(2) * a**2 - Integer(1) / Integer(2) * b**2 +
Integer(1) / Integer(2) * c**2 - Integer(1) / Integer(2) * d**2,
Integer(1) / Integer(2) * a**2 + Integer(1) / Integer(2) * b**2 +
Integer(1) / Integer(2) * c**2 + Integer(1) / Integer(2) * d**2
]
B = matrix(3, [real(comp) for comp in components])
#B = B.apply_map(attrcall('real'))
if A.det() > 0:
return B
else:
# Orientation-reversing isometries swap the nappes of
# the lightcone. This fixes that issue.
return -B
def show(self, boundary=True, **options):
r"""
Plot ``self``.
EXAMPLES::
sage: HyperbolicPlane().UHP().get_point(I).show()
Graphics object consisting of 2 graphics primitives
sage: HyperbolicPlane().UHP().get_point(0).show()
Graphics object consisting of 2 graphics primitives
sage: HyperbolicPlane().UHP().get_point(infinity).show()
Traceback (most recent call last):
...
NotImplementedError: can't draw the point infinity
"""
p = self.coordinates()
if p == infinity:
raise NotImplementedError("can't draw the point infinity")
opts = {'axes': False, 'aspect_ratio': 1}
opts.update(self.graphics_options())
opts.update(options)
from sage.misc.functional import numerical_approx
p = numerical_approx(p + 0 * I)
from sage.plot.point import point
if self._bdry:
pic = point((p, 0), **opts)
if boundary:
bd_pic = self.parent().get_background_graphic(bd_min=p - 1,
bd_max=p + 1)
pic = bd_pic + pic
else:
pic = point(p, **opts)
if boundary:
cent = real(p)
bd_pic = self.parent().get_background_graphic(bd_min=cent - 1,
bd_max=cent + 1)
pic = bd_pic + pic
return pic
def ideal_endpoints(self):
r"""
Determine the ideal (boundary) endpoints of the complete
hyperbolic geodesic corresponding to ``self``.
OUTPUT:
- a list of 2 boundary points
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: UHP.get_geodesic(I, 2*I).ideal_endpoints()
[Boundary point in UHP 0,
Boundary point in UHP +Infinity]
sage: UHP.get_geodesic(1 + I, 2 + 4*I).ideal_endpoints()
[Boundary point in UHP -sqrt(65) + 9,
Boundary point in UHP sqrt(65) + 9]
"""
start = self._start.coordinates()
end = self._end.coordinates()
[x1, x2] = [real(k) for k in [start, end]]
[y1, y2] = [imag(k) for k in [start, end]]
M = self._model
# infinity is the first endpoint, so the other ideal endpoint
# is just the real part of the second coordinate
if start == infinity:
return [M.get_point(start), M.get_point(x2)]
# Same idea as above
if end == infinity:
return [M.get_point(x1), M.get_point(end)]
# We could also have a vertical line with two interior points
if x1 == x2:
return [M.get_point(x1), M.get_point(infinity)]
# Otherwise, we have a semicircular arc in the UHP
c = ((x1 + x2) * (x2 - x1) + (y1 + y2) * (y2 - y1)) / (2 * (x2 - x1))
r = sqrt((c - x1)**2 + y1**2)
return [M.get_point(c - r), M.get_point(c + r)]
def show(self, boundary=True, **options):
r"""
Plot ``self``.
EXAMPLES::
sage: HyperbolicPlane().UHP().get_point(I).show()
Graphics object consisting of 2 graphics primitives
sage: HyperbolicPlane().UHP().get_point(0).show()
Graphics object consisting of 2 graphics primitives
sage: HyperbolicPlane().UHP().get_point(infinity).show()
Traceback (most recent call last):
...
NotImplementedError: can't draw the point infinity
"""
p = self.coordinates()
if p == infinity:
raise NotImplementedError("can't draw the point infinity")
opts = {'axes': False, 'aspect_ratio': 1}
opts.update(self.graphics_options())
opts.update(options)
from sage.misc.functional import numerical_approx
p = numerical_approx(p + 0 * I)
from sage.plot.point import point
if self._bdry:
pic = point((p, 0), **opts)
if boundary:
bd_pic = self.parent().get_background_graphic(bd_min=p - 1,
bd_max=p + 1)
pic = bd_pic + pic
else:
pic = point(p, **opts)
if boundary:
cent = real(p)
bd_pic = self.parent().get_background_graphic(bd_min=cent - 1,
bd_max=cent + 1)
pic = bd_pic + pic
return pic
def ideal_endpoints(self):
r"""
Determine the ideal (boundary) endpoints of the complete
hyperbolic geodesic corresponding to ``self``.
OUTPUT:
- a list of 2 boundary points
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: UHP.get_geodesic(I, 2*I).ideal_endpoints()
[Boundary point in UHP 0,
Boundary point in UHP +Infinity]
sage: UHP.get_geodesic(1 + I, 2 + 4*I).ideal_endpoints()
[Boundary point in UHP -sqrt(65) + 9,
Boundary point in UHP sqrt(65) + 9]
"""
start = self._start.coordinates()
end = self._end.coordinates()
[x1, x2] = [real(k) for k in [start, end]]
[y1, y2] = [imag(k) for k in [start, end]]
M = self._model
# infinity is the first endpoint, so the other ideal endpoint
# is just the real part of the second coordinate
if start == infinity:
return [M.get_point(start), M.get_point(x2)]
# Same idea as above
if end == infinity:
return [M.get_point(x1), M.get_point(end)]
# We could also have a vertical line with two interior points
if x1 == x2:
return [M.get_point(x1), M.get_point(infinity)]
# Otherwise, we have a semicircular arc in the UHP
c = ((x1+x2)*(x2-x1) + (y1+y2)*(y2-y1)) / (2*(x2-x1))
r = sqrt((c - x1)**2 + y1**2)
return [M.get_point(c - r), M.get_point(c + r)]
def cmp_ir(self,z):
"""
returns -1 for left, 0 for in, and 1 for right from initial region
cut line is on the north ray from L.
"""
L = self.L
x0 = self.x0
if x0 > 0.5:
if real(z) > real(L) and abs(z) < abs(L):
return 0
if real(z) < real(L):
return -1
if real(z) > real(L):
return 1
else:
if imag(z) > imag(L):
if real(z) > real(L):
return 1
if real(z) < real(L):
return -1
if real(z) < real(L) and real(z) > log(real(L)) + log(sqrt(1+tan(imag(z))**2)):
return 0
if real(z) > real(L):
return 1
if real(z) < real(L):
return -1
def cmp_ir(self,z):
"""
returns -1 for left, 0 for in, and 1 for right from initial region
cut line is on the north ray from pfp.
Works only for real x0.
"""
pfp = self.pfp
x0 = self.x0
if x0 > 0.5:
print z,abs(z)
if real(z) >= real(pfp) and abs(z) < abs(pfp):
return 0
if real(z) < real(pfp):
return -1
if real(z) > real(pfp):
return 1
else:
if imag(z) > imag(pfp):
if real(z) > real(pfp):
return 1
if real(z) < real(pfp):
return -1
if real(z) < real(pfp) and real(z) > log(real(pfp)) + log(sqrt(1+tan(imag(z))**2)):
return 0
if real(z) > real(pfp):
return 1
if real(z) < real(pfp):
return -1
def as_tuple_b(self, b): b2 = QQ(real(b)) * 2 b1 = QQ(b2) * (-self.D) / 2 - QQ(imag(b) * self.field(-self.D).sqrt()) assert self.from_tuple_b(b1, b2) == b return (b1, b2)
def bessel_K(nu,z,algorithm="pari",prec=53):
r"""
Implements the "K-Bessel function", or "modified Bessel function,
2nd kind", with index (or "order") nu and argument z. Defn::
pi*(bessel_I(-nu, z) - bessel_I(nu, z))
----------------------------------------
2*sin(pi*nu)
if nu is not an integer and by taking a limit otherwise.
Sometimes bessel_K(nu,z) is denoted K_nu(z) in the literature. In
PARI, nu can be complex and z must be real and positive.
EXAMPLES::
sage: bessel_K(3,2,"scipy")
0.64738539094...
sage: bessel_K(3,2)
0.64738539094...
sage: bessel_K(1,1)
0.60190723019...
sage: bessel_K(1,1,"pari",10)
0.60
sage: bessel_K(1,1,"pari",100)
0.60190723019723457473754000154
TESTS::
sage: bessel_K(2,1.1, algorithm="maxima")
Traceback (most recent call last):
...
NotImplementedError: The K-Bessel function is only implemented for the pari and scipy algorithms
Check whether the return value is real whenever the argument is real (#10251)::
sage: bessel_K(5, 1.5, algorithm='scipy') in RR
True
"""
if algorithm=="scipy":
if prec != 53:
raise ValueError, "for the scipy algorithm the precision must be 53"
import scipy.special
ans = str(scipy.special.kv(float(nu),float(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 == 'pari':
from sage.libs.pari.all import pari
try:
R = RealField(prec)
nu = R(nu)
z = R(z)
except TypeError:
C = ComplexField(prec)
nu = C(nu)
z = C(z)
K = C
K = z.parent()
return K(pari(nu).besselk(z, precision=prec))
elif algorithm == 'maxima':
raise NotImplementedError, "The K-Bessel function is only implemented for the pari and scipy algorithms"
else:
raise ValueError, "unknown algorithm '%s'"%algorithm
def bessel_J(nu,z,algorithm="pari",prec=53):
r"""
Return value of the "J-Bessel function", or "Bessel function, 1st
kind", with index (or "order") nu and argument z.
::
Defn:
Maxima:
inf
==== - nu - 2 k nu + 2 k
\ (-1)^k 2 z
> -------------------------
/ k! Gamma(nu + k + 1)
====
k = 0
PARI:
inf
==== - 2k 2k
\ (-1)^k 2 z Gamma(nu + 1)
> ----------------------------
/ k! Gamma(nu + k + 1)
====
k = 0
Sometimes bessel_J(nu,z) is denoted J_nu(z) in the literature.
.. warning::
Inaccurate for small values of z.
EXAMPLES::
sage: bessel_J(2,1.1)
0.136564153956658
sage: bessel_J(0,1.1)
0.719622018527511
sage: bessel_J(0,1)
0.765197686557967
sage: bessel_J(0,0)
1.00000000000000
sage: bessel_J(0.1,0.1)
0.777264368097005
We check consistency of PARI and Maxima::
sage: n(bessel_J(3,10,"maxima"))
0.0583793793051...
sage: n(bessel_J(3,10,"pari"))
0.0583793793051868
sage: bessel_J(3,10,"scipy")
0.0583793793052...
Check whether the return value is real whenever the argument is real (#10251)::
sage: bessel_J(5, 1.5, algorithm='scipy') in RR
True
"""
if algorithm=="pari":
from sage.libs.pari.all import pari
try:
R = RealField(prec)
nu = R(nu)
z = R(z)
except TypeError:
C = ComplexField(prec)
nu = C(nu)
z = C(z)
K = C
if nu == 0:
nu = ZZ(0)
K = z.parent()
return K(pari(nu).besselj(z, precision=prec))
elif algorithm=="scipy":
if prec != 53:
raise ValueError, "for the scipy algorithm the precision must be 53"
import scipy.special
ans = str(scipy.special.jv(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 maxima_function("bessel_j")(nu, z)
else:
raise ValueError, "unknown algorithm '%s'"%algorithm
def bessel_I(nu,z,algorithm = "pari",prec=53):
r"""
Implements the "I-Bessel function", or "modified Bessel function,
1st kind", with index (or "order") nu and argument z.
INPUT:
- ``nu`` - a real (or complex, for pari) number
- ``z`` - a real (positive) algorithm - "pari" or
"maxima" or "scipy" prec - real precision (for PARI only)
DEFINITION::
Maxima:
inf
==== - nu - 2 k nu + 2 k
\ 2 z
> -------------------
/ k! Gamma(nu + k + 1)
====
k = 0
PARI:
inf
==== - 2 k 2 k
\ 2 z Gamma(nu + 1)
> -----------------------
/ k! Gamma(nu + k + 1)
====
k = 0
Sometimes ``bessel_I(nu,z)`` is denoted
``I_nu(z)`` in the literature.
.. warning::
In Maxima (the manual says) i0 is deprecated but
``bessel_i(0,*)`` is broken. (Was fixed in recent CVS patch
though.)
EXAMPLES::
sage: bessel_I(1,1,"pari",500)
0.565159103992485027207696027609863307328899621621092009480294489479255640964371134092664997766814410064677886055526302676857637684917179812041131208121
sage: bessel_I(1,1)
0.565159103992485
sage: bessel_I(2,1.1,"maxima")
0.16708949925104...
sage: bessel_I(0,1.1,"maxima")
1.32616018371265...
sage: bessel_I(0,1,"maxima")
1.2660658777520...
sage: bessel_I(1,1,"scipy")
0.565159103992...
Check whether the return value is real whenever the argument is real (#10251)::
sage: bessel_I(5, 1.5, algorithm='scipy') in RR
True
"""
if algorithm=="pari":
from sage.libs.pari.all import pari
try:
R = RealField(prec)
nu = R(nu)
z = R(z)
except TypeError:
C = ComplexField(prec)
nu = C(nu)
z = C(z)
K = C
K = z.parent()
return K(pari(nu).besseli(z, precision=prec))
elif algorithm=="scipy":
if prec != 53:
raise ValueError, "for the scipy algorithm the precision must be 53"
import scipy.special
ans = str(scipy.special.iv(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 # Return real value when arg is real
elif algorithm == "maxima":
if prec != 53:
raise ValueError, "for the maxima algorithm the precision must be 53"
return sage_eval(maxima.eval("bessel_i(%s,%s)"%(float(nu),float(z))))
else:
raise ValueError, "unknown algorithm '%s'"%algorithm
def angle(self, other): # UHP
r"""
Return the angle between any two given completed geodesics if
they intersect.
INPUT:
- ``other`` -- a hyperbolic geodesic in the UHP model
OUTPUT:
- the angle in radians between the two given geodesics
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: g = UHP.get_geodesic(2, 4)
sage: h = UHP.get_geodesic(3, 3 + I)
sage: g.angle(h)
1/2*pi
sage: numerical_approx(g.angle(h))
1.57079632679490
If the geodesics are identical, return angle 0::
sage: g.angle(g)
0
It is an error to ask for the angle of two geodesics that do not
intersect::
sage: g = UHP.get_geodesic(2, 4)
sage: h = UHP.get_geodesic(5, 7)
sage: g.angle(h)
Traceback (most recent call last):
...
ValueError: geodesics do not intersect
"""
if self.is_parallel(other):
raise ValueError("geodesics do not intersect")
# Make sure the segments are complete or intersect
if (not(self.is_complete() and other.is_complete())
and not self.intersection(other)):
print("Warning: Geodesic segments do not intersect. "
"The angle between them is not defined.\n"
"Returning the angle between their completions.")
# Make sure both are in the same model
if other._model is not self._model:
other = other.to_model(self._model)
(p1, p2) = sorted([k.coordinates()
for k in self.ideal_endpoints()], key=str)
(q1, q2) = sorted([k.coordinates()
for k in other.ideal_endpoints()], key=str)
# if the geodesics are equal, the angle between them is 0
if (abs(p1 - q1) < EPSILON
and abs(p2 - q2) < EPSILON):
return 0
elif p2 != infinity: # geodesic not a straight line
# So we send it to the geodesic with endpoints [0, oo]
T = HyperbolicGeodesicUHP._crossratio_matrix(p1, (p1 + p2) / 2, p2)
else:
# geodesic is a straight line, so we send it to the geodesic
# with endpoints [0,oo]
T = HyperbolicGeodesicUHP._crossratio_matrix(p1, p1 + 1, p2)
# b1 and b2 are the endpoints of the image of other
b1, b2 = [mobius_transform(T, k) for k in [q1, q2]]
# If other is now a straight line...
if (b1 == infinity or b2 == infinity):
# then since they intersect, they are equal
return 0
return real(arccos((b1 + b2) / abs(b2 - b1)))
def bessel_I(nu, z, algorithm="pari", prec=53):
r"""
Implements the "I-Bessel function", or "modified Bessel function,
1st kind", with index (or "order") nu and argument z.
INPUT:
- ``nu`` - a real (or complex, for pari) number
- ``z`` - a real (positive) algorithm - "pari" or
"maxima" or "scipy" prec - real precision (for PARI only)
DEFINITION::
Maxima:
inf
==== - nu - 2 k nu + 2 k
\ 2 z
> -------------------
/ k! Gamma(nu + k + 1)
====
k = 0
PARI:
inf
==== - 2 k 2 k
\ 2 z Gamma(nu + 1)
> -----------------------
/ k! Gamma(nu + k + 1)
====
k = 0
Sometimes ``bessel_I(nu,z)`` is denoted
``I_nu(z)`` in the literature.
.. warning::
In Maxima (the manual says) i0 is deprecated but
``bessel_i(0,*)`` is broken. (Was fixed in recent CVS patch
though.)
EXAMPLES::
sage: bessel_I(1,1,"pari",500)
0.565159103992485027207696027609863307328899621621092009480294489479255640964371134092664997766814410064677886055526302676857637684917179812041131208121
sage: bessel_I(1,1)
0.565159103992485
sage: bessel_I(2,1.1,"maxima")
0.16708949925104...
sage: bessel_I(0,1.1,"maxima")
1.32616018371265...
sage: bessel_I(0,1,"maxima")
1.2660658777520...
sage: bessel_I(1,1,"scipy")
0.565159103992...
Check whether the return value is real whenever the argument is real (#10251)::
sage: bessel_I(5, 1.5, algorithm='scipy') in RR
True
"""
if algorithm == "pari":
from sage.libs.pari.all import pari
try:
R = RealField(prec)
nu = R(nu)
z = R(z)
except TypeError:
C = ComplexField(prec)
nu = C(nu)
z = C(z)
K = C
K = z.parent()
return K(pari(nu).besseli(z, precision=prec))
elif algorithm == "scipy":
if prec != 53:
raise ValueError, "for the scipy algorithm the precision must be 53"
import scipy.special
ans = str(scipy.special.iv(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 # Return real value when arg is real
elif algorithm == "maxima":
if prec != 53:
raise ValueError, "for the maxima algorithm the precision must be 53"
return sage_eval(maxima.eval("bessel_i(%s,%s)" %
(float(nu), float(z))))
else:
raise ValueError, "unknown algorithm '%s'" % algorithm
def angle(self, other): # UHP
r"""
Return the angle between any two given completed geodesics if
they intersect.
INPUT:
- ``other`` -- a hyperbolic geodesic in the UHP model
OUTPUT:
- the angle in radians between the two given geodesics
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: g = UHP.get_geodesic(2, 4)
sage: h = UHP.get_geodesic(3, 3 + I)
sage: g.angle(h)
1/2*pi
sage: numerical_approx(g.angle(h))
1.57079632679490
If the geodesics are identical, return angle 0::
sage: g.angle(g)
0
It is an error to ask for the angle of two geodesics that do not
intersect::
sage: g = UHP.get_geodesic(2, 4)
sage: h = UHP.get_geodesic(5, 7)
sage: g.angle(h)
Traceback (most recent call last):
...
ValueError: geodesics do not intersect
"""
if self.is_parallel(other):
raise ValueError("geodesics do not intersect")
# Make sure the segments are complete or intersect
if (not (self.is_complete() and other.is_complete())
and not self.intersection(other)):
print("Warning: Geodesic segments do not intersect. "
"The angle between them is not defined.\n"
"Returning the angle between their completions.")
# Make sure both are in the same model
if other._model is not self._model:
other = other.to_model(self._model)
(p1, p2) = sorted([k.coordinates() for k in self.ideal_endpoints()],
key=str)
(q1, q2) = sorted([k.coordinates() for k in other.ideal_endpoints()],
key=str)
# if the geodesics are equal, the angle between them is 0
if (abs(p1 - q1) < EPSILON and abs(p2 - q2) < EPSILON):
return 0
elif p2 != infinity: # geodesic not a straight line
# So we send it to the geodesic with endpoints [0, oo]
T = HyperbolicGeodesicUHP._crossratio_matrix(p1, (p1 + p2) / 2, p2)
else:
# geodesic is a straight line, so we send it to the geodesic
# with endpoints [0,oo]
T = HyperbolicGeodesicUHP._crossratio_matrix(p1, p1 + 1, p2)
# b1 and b2 are the endpoints of the image of other
b1, b2 = [mobius_transform(T, k) for k in [q1, q2]]
# If other is now a straight line...
if (b1 == infinity or b2 == infinity):
# then since they intersect, they are equal
return 0
return real(arccos((b1 + b2) / abs(b2 - b1)))
def bessel_J(nu, z, algorithm="pari", prec=53):
r"""
Return value of the "J-Bessel function", or "Bessel function, 1st
kind", with index (or "order") nu and argument z.
::
Defn:
Maxima:
inf
==== - nu - 2 k nu + 2 k
\ (-1)^k 2 z
> -------------------------
/ k! Gamma(nu + k + 1)
====
k = 0
PARI:
inf
==== - 2k 2k
\ (-1)^k 2 z Gamma(nu + 1)
> ----------------------------
/ k! Gamma(nu + k + 1)
====
k = 0
Sometimes bessel_J(nu,z) is denoted J_nu(z) in the literature.
.. warning::
Inaccurate for small values of z.
EXAMPLES::
sage: bessel_J(2,1.1)
0.136564153956658
sage: bessel_J(0,1.1)
0.719622018527511
sage: bessel_J(0,1)
0.765197686557967
sage: bessel_J(0,0)
1.00000000000000
sage: bessel_J(0.1,0.1)
0.777264368097005
We check consistency of PARI and Maxima::
sage: n(bessel_J(3,10,"maxima"))
0.0583793793051...
sage: n(bessel_J(3,10,"pari"))
0.0583793793051868
sage: bessel_J(3,10,"scipy")
0.0583793793052...
Check whether the return value is real whenever the argument is real (#10251)::
sage: bessel_J(5, 1.5, algorithm='scipy') in RR
True
"""
if algorithm == "pari":
from sage.libs.pari.all import pari
try:
R = RealField(prec)
nu = R(nu)
z = R(z)
except TypeError:
C = ComplexField(prec)
nu = C(nu)
z = C(z)
K = C
if nu == 0:
nu = ZZ(0)
K = z.parent()
return K(pari(nu).besselj(z, precision=prec))
elif algorithm == "scipy":
if prec != 53:
raise ValueError, "for the scipy algorithm the precision must be 53"
import scipy.special
ans = str(scipy.special.jv(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 maxima_function("bessel_j")(nu, z)
else:
raise ValueError, "unknown algorithm '%s'" % algorithm
def bessel_K(nu, z, algorithm="pari", prec=53):
r"""
Implements the "K-Bessel function", or "modified Bessel function,
2nd kind", with index (or "order") nu and argument z. Defn::
pi*(bessel_I(-nu, z) - bessel_I(nu, z))
----------------------------------------
2*sin(pi*nu)
if nu is not an integer and by taking a limit otherwise.
Sometimes bessel_K(nu,z) is denoted K_nu(z) in the literature. In
PARI, nu can be complex and z must be real and positive.
EXAMPLES::
sage: bessel_K(3,2,"scipy")
0.64738539094...
sage: bessel_K(3,2)
0.64738539094...
sage: bessel_K(1,1)
0.60190723019...
sage: bessel_K(1,1,"pari",10)
0.60
sage: bessel_K(1,1,"pari",100)
0.60190723019723457473754000154
TESTS::
sage: bessel_K(2,1.1, algorithm="maxima")
Traceback (most recent call last):
...
NotImplementedError: The K-Bessel function is only implemented for the pari and scipy algorithms
Check whether the return value is real whenever the argument is real (#10251)::
sage: bessel_K(5, 1.5, algorithm='scipy') in RR
True
"""
if algorithm == "scipy":
if prec != 53:
raise ValueError, "for the scipy algorithm the precision must be 53"
import scipy.special
ans = str(scipy.special.kv(float(nu), float(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 == 'pari':
from sage.libs.pari.all import pari
try:
R = RealField(prec)
nu = R(nu)
z = R(z)
except TypeError:
C = ComplexField(prec)
nu = C(nu)
z = C(z)
K = C
K = z.parent()
return K(pari(nu).besselk(z, precision=prec))
elif algorithm == 'maxima':
raise NotImplementedError, "The K-Bessel function is only implemented for the pari and scipy algorithms"
else:
raise ValueError, "unknown algorithm '%s'" % algorithm
def __init__(self,b,N,iprec,x0=0):
self.bsym = b
self.N = N
self.iprec = iprec
self.x0sym = x0
self.prec = None
bname = repr(b).strip('0').replace('.',',')
if b == sqrt(2):
bname = "sqrt2"
if b == e**(1/e):
bname = "eta"
x0name = repr(x0)
if x0name.find('.') > -1:
if x0.is_real():
x0name = repr(float(real(x0))).strip('0').replace('.',',')
else:
x0name = repr(complex(x0)).strip('0').replace('.',',')
# by some reason save does not work with additional . inside the path
self.path = "savings/msexp_%s"%bname + "_N%04d"%N + "_iprec%05d"%iprec + "_a%s"%x0name
b = b.n(iprec)
self.b = b
x0 = x0.n(iprec)
self.x0 = x0
if isinstance(x0,RealNumber):
R = RealField(iprec)
else:
R = ComplexField(iprec)
#symbolic carleman matrix
if x0 == 0:
#C = Matrix([[ m**n*log(b)**n/factorial(n) for n in range(N)] for m in range(N)])
coeffs = [log(b)**n/factorial(n) for n in xrange(N)]
else:
#too slow
#c = b**x0
#C = Matrix([ [log(b)**n/factorial(n)*sum([binomial(m,k)*k**n*c**k*(-x0)**(m-k) for k in range(m+1)]) for n in range(N)] for m in range(N)])
coeffs = [b**x0-x0]+[b**x0*log(b)**n/factorial(n) for n in xrange(1,N)]
def psmul(A,B):
N = len(B)
return [sum([A[k]*B[n-k] for k in xrange(n+1)]) for n in xrange(N)]
C = Matrix(R,N)
row = vector(R,[1]+(N-1)*[0])
C[0] = row
for m in xrange(1,N):
row = psmul(row,coeffs)
C[m] = row
print "Carleman matrix created."
#numeric matrix and eigenvalues
#self.CM = C.n(iprec) #n seems to reduce to a RealField
self.CM = C
self.eigenvalues = self.CM.eigenvalues()
print "Eigenvalues computed."
self.IM = None
self.calc_IM()
print "Iteration matrix computed."
self.coeffs_1 = self.IM * vector([1]+[(1-x0)**n for n in range(1,N)])
if x0 == 0:
self.coeffs_0 = self.IM.column(0)
else:
self.coeffs_0 = self.IM * vector([1]+[(-x0)**n for n in range(1,N)])
#there would also be a method to only coeffs_0 with x0,
#this would require to make the offset -1-slog(0)
self.L = None