def log_norm(A, p='inf'):
r"""
Compute the logarithmic norm of a matrix.
INPUT:
- ``A`` -- a rectangular (Sage dense) matrix of order `n`. The coefficients can be either real or complex
- ``p`` -- (default: ``'inf'``). The vector norm; possible choices are ``1``, ``2``, or ``'inf'``
OUTPUT:
- ``lognorm`` -- the log-norm of `A` in the `p`-norm
"""
# parse the input matrix
if 'scipy.sparse' in str(type(A)):
# cast into numpy array (or ndarray)
A = A.toarray()
n = A.shape[0]
elif 'numpy.array' in str(type(A)) or 'numpy.ndarray' in str(type(A)):
n = A.shape[0]
else:
# assuming sage matrix
n = A.nrows()
# computation, depending on the chosen norm p
if (p == 'inf' or p == oo):
z = max(
real_part(A[i][i]) + sum(abs(A[i][j])
for j in range(n)) - abs(A[i][i])
for i in range(n))
return z
elif (p == 1):
n = A.nrows()
return max(
real_part(A[j][j]) + sum(abs(A[i][j])
for i in range(n)) - abs(A[j][j])
for j in range(n))
elif (p == 2):
if not (A.base_ring() == RR or A.base_ring() == CC):
return 1 / 2 * max((A + A.H).eigenvalues())
else:
# Alternative, always numerical
z = 1 / 2 * max(
np.linalg.eigvals(np.matrix(A + A.H, dtype=complex)))
return real_part(z) if imag_part(z) == 0 else z
else:
raise NotImplementedError(
'value of p not understood or not implemented')
def plot_y(self, plot_points=128, **kwds):
r"""Plot the y-part of the path in the complex y-plane.
Additional arguments and keywords are passed to
``matplotlib.pyplot.plot``.
Parameters
----------
N : int
The number of interpolating points used to plot.
t0 : double
Starting t-value in [0,1].
t1 : double
Ending t-value in [0,1].
Returns
-------
plt : Sage plot.
A plot of the complex y-projection of the path.
"""
s = numpy.linspace(0, 1, plot_points, dtype=double)
vals = numpy.array([self.get_y(si)[0] for si in s], dtype=complex)
pts = [(real_part(y), imag_part(y)) for y in vals]
plt = line(pts, **kwds)
return plt
def plot_y(self, plot_points=128, **kwds):
r"""Plot the y-part of the path in the complex y-plane.
Additional arguments and keywords are passed to
``matplotlib.pyplot.plot``.
Parameters
----------
N : int
The number of interpolating points used to plot.
t0 : double
Starting t-value in [0,1].
t1 : double
Ending t-value in [0,1].
Returns
-------
plt : Sage plot.
A plot of the complex y-projection of the path.
"""
s = numpy.linspace(0, 1, plot_points, dtype=double)
vals = numpy.array([self.get_y(si)[0] for si in s], dtype=complex)
pts = [(real_part(y), imag_part(y)) for y in vals]
plt = line(pts, **kwds)
return plt
def _sympysage_re(self):
"""
EXAMPLES::
sage: from sympy import Symbol, re
sage: assert real_part(x)._sympy_() == re(Symbol('x'))
sage: assert real_part(x) == re(Symbol('x'))._sage_()
"""
from sage.functions.other import real_part
return real_part(self.args[0]._sage_())
def _sympysage_re(self):
"""
EXAMPLES::
sage: from sympy import Symbol, re
sage: assert real_part(x)._sympy_() == re(Symbol('x'))
sage: assert real_part(x) == re(Symbol('x'))._sage_()
"""
from sage.functions.other import real_part
return real_part(self.args[0]._sage_())
def plot_arc(radius, p, q, **opts):
# TODO: THIS SHOULD USE THE EXISTING PLOT OF ARCS!
# plot the arc from p to q differently depending on the type of self
p = ZZ(p)
q = ZZ(q)
t = var('t')
if p - q in [1, -1]:
def f(t):
return (radius * cos(t), radius * sin(t))
(p, q) = sorted([p, q])
angle_p = vertex_to_angle(p)
angle_q = vertex_to_angle(q)
return parametric_plot(f(t), (t, angle_q, angle_p), **opts)
if self.type() == 'A':
angle_p = vertex_to_angle(p)
angle_q = vertex_to_angle(q)
if angle_p < angle_q:
angle_p += 2 * pi
internal_angle = angle_p - angle_q
if internal_angle > pi:
(angle_p, angle_q) = (angle_q + 2 * pi, angle_p)
internal_angle = angle_p - angle_q
angle_center = (angle_p + angle_q) / 2
hypotenuse = radius / cos(internal_angle / 2)
radius_arc = hypotenuse * sin(internal_angle / 2)
center = (hypotenuse * cos(angle_center),
hypotenuse * sin(angle_center))
center_angle_p = angle_p + pi / 2
center_angle_q = angle_q + 3 * pi / 2
def f(t):
return (radius_arc * cos(t) + center[0],
radius_arc * sin(t) + center[1])
return parametric_plot(f(t),
(t, center_angle_p, center_angle_q),
**opts)
elif self.type() == 'D':
if p >= q:
q += self.r()
px = -2 * pi * p / self.r() + pi / 2
qx = -2 * pi * q / self.r() + pi / 2
arc_radius = (px - qx) / 2
arc_center = qx + arc_radius
def f(t):
return exp(I * ((cos(t) + I * sin(t)) * arc_radius +
arc_center)) * radius
return parametric_plot((real_part(f(t)), imag_part(f(t))),
(t, 0, pi), **opts)
def plot_arc(radius, p, q, **opts):
# TODO: THIS SHOULD USE THE EXISTING PLOT OF ARCS!
# plot the arc from p to q differently depending on the type of self
p = ZZ(p)
q = ZZ(q)
t = var('t')
if p - q in [1, -1]:
def f(t):
return (radius * cos(t), radius * sin(t))
(p, q) = sorted([p, q])
angle_p = vertex_to_angle(p)
angle_q = vertex_to_angle(q)
return parametric_plot(f(t), (t, angle_q, angle_p), **opts)
if self.type() == 'A':
angle_p = vertex_to_angle(p)
angle_q = vertex_to_angle(q)
if angle_p < angle_q:
angle_p += 2 * pi
internal_angle = angle_p - angle_q
if internal_angle > pi:
(angle_p, angle_q) = (angle_q + 2 * pi, angle_p)
internal_angle = angle_p - angle_q
angle_center = (angle_p+angle_q) / 2
hypotenuse = radius / cos(internal_angle / 2)
radius_arc = hypotenuse * sin(internal_angle / 2)
center = (hypotenuse * cos(angle_center),
hypotenuse * sin(angle_center))
center_angle_p = angle_p + pi / 2
center_angle_q = angle_q + 3 * pi / 2
def f(t):
return (radius_arc * cos(t) + center[0],
radius_arc * sin(t) + center[1])
return parametric_plot(f(t), (t, center_angle_p,
center_angle_q), **opts)
elif self.type() == 'D':
if p >= q:
q += self.r()
px = -2 * pi * p / self.r() + pi / 2
qx = -2 * pi * q / self.r() + pi / 2
arc_radius = (px - qx) / 2
arc_center = qx + arc_radius
def f(t):
return exp(I * ((cos(t) + I * sin(t)) *
arc_radius + arc_center)) * radius
return parametric_plot((real_part(f(t)), imag_part(f(t))),
(t, 0, pi), **opts)
def generalized_series_default_iota(z, j):
if j == 0:
return z - real_part(z).floor()
else:
return z - real_part(z).ceil() + 1
def is_absolutely_convergent(cls, self, a, b, z):
r"""
Determine whether ``self`` converges absolutely as an infinite
series. ``False`` is returned if not all terms are finite.
EXAMPLES:
Degree giving infinite radius of convergence::
sage: hypergeometric([2, 3], [4, 5],
....: 6).is_absolutely_convergent()
True
sage: hypergeometric([2, 3], [-4, 5],
....: 6).is_absolutely_convergent() # undefined
False
sage: (hypergeometric([2, 3], [-4, 5], Infinity)
....: .is_absolutely_convergent()) # undefined
False
Ordinary geometric series (unit radius of convergence)::
sage: hypergeometric([1], [], 1/2).is_absolutely_convergent()
True
sage: hypergeometric([1], [], 2).is_absolutely_convergent()
False
sage: hypergeometric([1], [], 1).is_absolutely_convergent()
False
sage: hypergeometric([1], [], -1).is_absolutely_convergent()
False
sage: hypergeometric([1], [], -1).n() # Sum still exists
0.500000000000000
Degree `p = q+1` (unit radius of convergence)::
sage: hypergeometric([2, 3], [4], 6).is_absolutely_convergent()
False
sage: hypergeometric([2, 3], [4], 1).is_absolutely_convergent()
False
sage: hypergeometric([2, 3], [5], 1).is_absolutely_convergent()
False
sage: hypergeometric([2, 3], [6], 1).is_absolutely_convergent()
True
sage: hypergeometric([-2, 3], [4],
....: 5).is_absolutely_convergent()
True
sage: hypergeometric([2, -3], [4],
....: 5).is_absolutely_convergent()
True
sage: hypergeometric([2, -3], [-4],
....: 5).is_absolutely_convergent()
True
sage: hypergeometric([2, -3], [-1],
....: 5).is_absolutely_convergent()
False
Degree giving zero radius of convergence::
sage: hypergeometric([1, 2, 3], [4],
....: 2).is_absolutely_convergent()
False
sage: hypergeometric([1, 2, 3], [4],
....: 1/2).is_absolutely_convergent()
False
sage: (hypergeometric([1, 2, -3], [4], 1/2)
....: .is_absolutely_convergent()) # polynomial
True
"""
p, q = len(a), len(b)
if not self.is_termwise_finite():
return False
if p <= q:
return True
if self.is_terminating():
return True
if p == q + 1:
if abs(z) < 1:
return True
if abs(z) == 1:
if real_part(sum(b) - sum(a)) > 0:
return True
return False
def is_absolutely_convergent(cls, self, a, b, z):
r"""
Determine whether ``self`` converges absolutely as an infinite
series. ``False`` is returned if not all terms are finite.
EXAMPLES:
Degree giving infinite radius of convergence::
sage: hypergeometric([2, 3], [4, 5],
....: 6).is_absolutely_convergent()
True
sage: hypergeometric([2, 3], [-4, 5],
....: 6).is_absolutely_convergent() # undefined
False
sage: (hypergeometric([2, 3], [-4, 5], Infinity)
....: .is_absolutely_convergent()) # undefined
False
Ordinary geometric series (unit radius of convergence)::
sage: hypergeometric([1], [], 1/2).is_absolutely_convergent()
True
sage: hypergeometric([1], [], 2).is_absolutely_convergent()
False
sage: hypergeometric([1], [], 1).is_absolutely_convergent()
False
sage: hypergeometric([1], [], -1).is_absolutely_convergent()
False
sage: hypergeometric([1], [], -1).n() # Sum still exists
0.500000000000000
Degree `p = q+1` (unit radius of convergence)::
sage: hypergeometric([2, 3], [4], 6).is_absolutely_convergent()
False
sage: hypergeometric([2, 3], [4], 1).is_absolutely_convergent()
False
sage: hypergeometric([2, 3], [5], 1).is_absolutely_convergent()
False
sage: hypergeometric([2, 3], [6], 1).is_absolutely_convergent()
True
sage: hypergeometric([-2, 3], [4],
....: 5).is_absolutely_convergent()
True
sage: hypergeometric([2, -3], [4],
....: 5).is_absolutely_convergent()
True
sage: hypergeometric([2, -3], [-4],
....: 5).is_absolutely_convergent()
True
sage: hypergeometric([2, -3], [-1],
....: 5).is_absolutely_convergent()
False
Degree giving zero radius of convergence::
sage: hypergeometric([1, 2, 3], [4],
....: 2).is_absolutely_convergent()
False
sage: hypergeometric([1, 2, 3], [4],
....: 1/2).is_absolutely_convergent()
False
sage: (hypergeometric([1, 2, -3], [4], 1/2)
....: .is_absolutely_convergent()) # polynomial
True
"""
p, q = len(a), len(b)
if not self.is_termwise_finite():
return False
if p <= q:
return True
if self.is_terminating():
return True
if p == q + 1:
if abs(z) < 1:
return True
if abs(z) == 1:
if real_part(sum(b) - sum(a)) > 0:
return True
return False
