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fsm_fourier.pyx
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fsm_fourier.pyx
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r"""
Fourier Coefficients
Introduction
============
Let `T` be a complete deterministic transducer reading the `q`-ary
digit expansion of a random integer `n` with `0\le n<N`.
Then the expected sum of the output labels of the transducer is
`e_T \log_q N + \Phi(\log_q N) + o(N)` for some periodic fluctuation
`\Phi` and some constant `e_T`.
This module computes the Fourier Coefficients of `\Phi`, following
[HKP2014]_.
Contents
========
:class:`FSMFourier`
-------------------
The class :class:`FSMFourier` is the main class of this module, it
precomputes values needed for all Fourier coefficients and then offers
the method :meth:`~FSMFourier.FourierCoefficient`.
.. csv-table::
:class: contentstable
:widths: 30, 70
:delim: |
:meth:`~FSMFourier.FourierCoefficient` | Compute a Fourier Coefficient
:meth:`~FSMFourier.plot_fluctuation` | Generate a plot of the fluctuation
:meth:`~FSMFourier.plot_fluctuation_asymptote` | Generate an asymptote plot of the fluctuation
:meth:`~FSMFourier.b0` | Compute the final output word
:class:`FSMFourierComponent`
------------------------------
Data corresponding to a final component of the transducer is stored in
:class:`FSMFourierComponent`. This mainly comprises eigenvectors.
.. csv-table::
:class: contentstable
:widths: 30, 70
:delim: |
:meth:`~FSMFourierComponent.mu_prime` | Derivative `\mu_{j0}'(0)`
:meth:`~FSMFourierComponent.a` | Constant `a_j`
:meth:`~FSMFourierComponent.coefficient_lambda` | Constant `\lambda_j`
:meth:`~FSMFourierComponent.right_eigenvectors` | Right eigenvectors `\mathbf{v}_{jk}(0)`
:meth:`~FSMFourierComponent.v_prime` | Derivative of the right eigenvector `\mathbf{v}'_{jk}(0)`
:meth:`~FSMFourierComponent.left_eigenvectors` | Left eigenvectors
:meth:`~FSMFourierComponent.w` | Scaled left eigenvectors `\mathbf{w}_{jk}(0)`
:meth:`~FSMFourierComponent.w_prime` | Derivative of the left eigenvector `\mathbf{w}'_{jk}(0)`
:meth:`~FSMFourierComponent.w_ell` | Left eigenvectors to given eigenvalue.
:class:`FSMFourierCache`
------------------------
This is a mostly internal class speeding up some of the computation.
.. csv-table::
:class: contentstable
:widths: 30, 70
:delim: |
:meth:`~FSMFourierCache.b` | Compute `\mathbf{b}(r)`.
:meth:`~FSMFourierCache.fluctuation_empirical` | Compute the fluctuation empirically.
:meth:`~FSMFourierCache.fluctuation_fourier` | Compute the fluctuation via an the Fourier series.
Example
=======
::
sage: function('f')
f
sage: var('n')
n
sage: from sage.combinat.fsm_fourier import FSMFourier # optional - arb
sage: T = transducers.Recursion([
....: f(2*n + 1) == f(n) + 1,
....: f(2*n) == f(n),
....: f(0) == 0],
....: f, n, 2)
sage: sage.combinat.finite_state_machine.FSMOldProcessOutput = False
sage: F = FSMFourier(T) # optional - arb
sage: F.FourierCoefficient(0) # optional - arb
-0.1455994557084? + 0.?e-16*I
sage: F.FourierCoefficient(10) # optional - arb
0.0016818573864? + 0.0003201978624?*I
AUTHORS:
- Clemens Heuberger (2014-08-14--2014-10-13): initial version
- Sara Kropf (2014-10-13--2014-10-21): corrections and 0th Fourier Coefficient
- Clemens Heuberger (2014-10-23--2014-10-26): Optimization and Preparation for Sage
ACKNOWLEDGEMENT:
- Clemens Heuberger and Sara Kropf are supported by the
Austrian Science Fund (FWF): P 24644-N26.
.. TODO::
The following private and special methods are included now for
proofreading, they should be excluded. Furthermore, this module cannot
be shown in the documentation as it depends on arb.
Classes and Methods
===================
.. autofunction:: _hurwitz_zeta_
.. autofunction:: _reduce_resolution_
.. automethod:: FSMFourierComponent.__init__
.. automethod:: FSMFourierComponent._eigenvectors_
.. automethod:: FSMFourierComponent._mask_
.. automethod:: FSMFourierCache.__init__
.. automethod:: FSMFourier.__init__
.. automethod:: FSMFourier._H_m_rhs_
.. automethod:: FSMFourier._w_H_Res_
.. automethod:: FSMFourier._H_m_
"""
#*****************************************************************************
# Copyright (C) 2014 Clemens Heuberger <clemens.heuberger@aau.at>
# 2014 Sara Kropf <sara.kropf@aau.at>
#
# Distributed under the terms of the GNU General Public License (GPL)
# as published by the Free Software Foundation; either version 2 of
# the License, or (at your option) any later version.
# http://www.gnu.org/licenses/
#*****************************************************************************
from libc.stdlib cimport malloc, realloc, free
import itertools
from sage.combinat.finite_state_machine import Transducer
from sage.functions.log import log
from sage.functions.other import floor
from sage.libs.arb.acb cimport *
from sage.libs.arb.acb_mat cimport *
from sage.matrix.constructor import matrix
from sage.matrix.matrix_acb_dense cimport (
matrix_to_acb_mat, acb_mat_to_matrix)
from sage.matrix.matrix_space import MatrixSpace
from sage.misc.cachefunc import cached_method
from sage.misc.misc import srange, verbose
from sage.modules.free_module_element import vector
from sage.modules.free_module_element cimport FreeModuleElement_generic_dense
import sage.rings.arith
from sage.rings.complex_interval cimport ComplexIntervalFieldElement
from sage.rings.complex_interval_acb cimport ComplexIntervalFieldElement_to_acb
from sage.rings.complex_interval_field import ComplexIntervalField
from sage.functions.other import ceil
from sage.rings.infinity import infinity
from sage.rings.integer_ring import ZZ
from sage.rings.rational_field import QQ
from sage.rings.real_arb cimport arb_to_mpfi
from sage.rings.real_mpfi import min_RIF, max_RIF, RealIntervalField
from sage.rings.real_mpfi cimport RealIntervalFieldElement
from sage.structure.sage_object cimport SageObject
def infinity_vector_norm(v):
"""
Compute the infinity norm of a vector of
:class:`~sage.rings.real_mpfi.RealIntervalFieldElement`.
INPUT:
- ``v`` -- a vector of
:class:`~sage.rings.real_mpfi.RealIntervalFieldElement`.
OUTPUT:
A :class:`~sage.rings.real_mpfi.RealIntervalFieldElement` element.
EXAMPLES::
sage: from sage.combinat.fsm_fourier import infinity_vector_norm # optional - arb
sage: a = RIF(1, 4)
sage: b = RIF(-3, -2)
sage: infinity_vector_norm(vector([a, b])).endpoints() # optional - arb
(2.00000000000000, 4.00000000000000)
sage: infinity_vector_norm(vector([b, a])).endpoints() # optional - arb
(2.00000000000000, 4.00000000000000)
Note that the
:meth:`sage.modules.free_module_element.FreeModuleElement.norm`
method is inadequate, as it uses the Python max instead of
:func:`~sage.rings.real_mpfi.max_RIF`::
sage: vector([a, b]).norm(infinity).endpoints()
(1.00000000000000, 4.00000000000000)
sage: vector([b, a]).norm(infinity).endpoints()
(2.00000000000000, 3.00000000000000)
"""
return max_RIF(abs(r) for r in v)
def infinity_matrix_norm(A):
"""
Compute the infinity norm of a matrix A, in the same ring as
the original matrix A.
INPUT:
- ``A`` -- a matrix.
OUTPUT:
An entry.
In contrast to :meth:`sage.matrix.matrix2.Matrix.norm`,
sparse zero matrices are allowed and the result is not
necessarily a
:class:`~sage.rings.real_double.RealDoubleElement`.
EXAMPLES::
sage: from sage.combinat.fsm_fourier import infinity_matrix_norm # optional - arb
sage: M = matrix([[1, 2], [-5, -1]])
sage: infinity_matrix_norm(M) # optional - arb
6
sage: M.norm(infinity)
6.0
sage: M = matrix([[0]], sparse=True)
sage: infinity_matrix_norm(M) # optional - arb
0
sage: M.norm(infinity)
Traceback (most recent call last):
...
TypeError: base_ring (=Category of objects) must be a ring
"""
if A.is_zero():
return 0
return max(sum(r) for r in A.apply_map(abs).rows())
def _hurwitz_zeta_(s, alpha, m=0, max_approximation_error=0):
r"""
Compute the truncated Hurwitz zeta function `\sum_{k\ge m} (k+\alpha)^{-s}`.
INPUT:
- ``s`` -- a :class:`ComplexIntervalFieldElement`
- ``alpha`` -- a :class:`~sage.rings.real_mpfi.RealIntervalFieldElement`
- ``m`` -- a positive integer
- ``max_approximation_error`` -- a non-negative number; an
approximation error less than ``max_approximation_error`` is
accepted even if it is not small with respect to the result.
OUTPUT:
A :class:`ComplexIntervalFieldElement` in the same field as ``s``.
EXAMPLES:
- Simple example::
sage: from sage.combinat.fsm_fourier import _hurwitz_zeta_ # optional - arb
sage: _hurwitz_zeta_(CIF(2), RIF(3/4), 10) # optional - arb
0.097483848201852? + 0.?e-17*I
- Compare with well-known value `\zeta(2)=\zeta(2, 1)=\pi^2/6`::
sage: _hurwitz_zeta_(CIF(2), 1) # optional - arb
1.64493406684823? + 0.?e-17*I
sage: (_hurwitz_zeta_(CIF(2), 1) - CIF(pi)^2/6).abs()<10^(-13) # optional - arb
True
sage: _hurwitz_zeta_(CIF(2*pi*I/log(2)), 1) # optional - arb
1.598734526809? + 0.278338669639?*I
- There is a singularity at `s=1`. ::
sage: _hurwitz_zeta_(CIF(RIF(0.9, 1.1), (-0.1, 0.1)), 1) # optional - arb
Traceback (most recent call last):
...
ZeroDivisionError: zeta is singular at 1.
- Checking the function for non-positive integers::
sage: zeta(-1) in _hurwitz_zeta_(CIF(-1), 1) # optional - arb
True
sage: zeta(0) in _hurwitz_zeta_(CIF(0), 1) # optional - arb
True
- Debugging output can be enabled using
:func:`~sage.misc.misc.set_verbose`. ::
sage: set_verbose(2)
sage: _hurwitz_zeta_(CIF(1+100/log(2)*I), 1) # optional - arb
verbose 1 (...) _hurwitz_zeta_(1 + 144.2695040888963?*I, 1, 0): M = 172
verbose 2 (...) N = 2, error = 0.0352354068797?, acceptable_error = 3.41199291042927e-13, result = 2.125571548789? + 0.511221280470?*I
verbose 2 (...) N = 4, error = 0.000310532577681?, acceptable_error = 3.41254802194158e-13, result = 2.125575595864? + 0.51121897538?*I
verbose 2 (...) N = 6, error = 3.65215306101?e-6, acceptable_error = 3.41351946708813e-13, result = 2.125575660430? + 0.511218933060?*I
verbose 2 (...) N = 8, error = 4.83820940904?e-8, acceptable_error = 3.41407457860044e-13, result = 2.125575661484? + 0.511218932225?*I
verbose 2 (...) N = 10, error = 6.84787778802?e-10, acceptable_error = 3.41504602374699e-13, result = 2.125575661501? + 0.511218932208?*I
verbose 2 (...) N = 12, error = 1.011644165731?e-11, acceptable_error = 3.41560113525930e-13, result = 2.125575661501? + 0.51121893221?*I
verbose 2 (...) N = 14, error = 1.54088223831?e-13, acceptable_error = 3.41615624677161e-13, result = 2.125575661501? + 0.51121893221?*I
verbose 1 (...) N = 14, error = 1.54088223831?e-13, acceptable_error = 3.41615624677161e-13, result = 2.125575661501? + 0.51121893221?*I
2.125575661501? + 0.51121893221?*I
sage: set_verbose(0)
- The current implementation does not work well with negative real
values, all precision is lost::
sage: _hurwitz_zeta_(CIF(-15+I), 1) # optional - arb
0.?e12 + 0.?e12*I
sage: hurwitz_zeta(ComplexField(200)(-15 + I), 1) # optional - arb
0.66621329305522618549073659441004805750538410627754288912677
- 0.84614995218731390314834131849322502368334938802936485299779*I
A work-around is to start with higher precision; however, we have
to clear the cache first::
sage: _hurwitz_zeta_(ComplexIntervalField(200)(-15 + I), 1) # optional - arb
0.66621329305522618549073660?
- 0.8461499521873139031483414?*I
"""
from sage.rings.arith import bernoulli, falling_factorial
CIF = s.parent()
RIF = s.real().parent()
if ZZ(1) in s:
raise ZeroDivisionError("zeta is singular at 1.")
# We rely on (2pi)^-N for convergence of the error term.
# As a conservative estimate, 2pi is approximately 2^2,
# so we will finally need N ~ s.prec()/2 to achieve an error which
# is less than the resolution of s.
# In order to have the falling factorial (-s)^\underline{N}
# smaller than (M+a)^N, we choose M>|s|+N
M = max(m, (s.prec()/ZZ(2)).ceil() + ZZ(s.abs().upper().ceil()))
verbose("_hurwitz_zeta_(%s, %s, %s): M = %d" % (s, alpha, m, M),
level=1)
sigma = s.real()
result = sum((r + alpha)**(-s) for r in reversed(srange(m, M)))
result += (M + alpha)**(1-s) / (s-1)
factor = (M + alpha)**(-s)
result += factor/2
N = 0
error_factor = RIF(4)
N_factorial = ZZ(1)
while True:
N += 2
factor *= (-s - N + 2)/(M + alpha)
#assert factor.overlaps(falling_factorial(-s, N - 1)/(M + alpha)**(s + N - 1))
N_factorial *= (N - 1)*N
#assert ZZ(N).factorial() == N_factorial
result -= bernoulli(N)/N_factorial * factor
factor *= (-s - N + 1)
#assert factor.overlaps(falling_factorial(-s, N)/(M + alpha)**(s + N - 1))
error_factor /= 4*RIF.pi()**2
#assert error_factor.overlaps(4/(2*RIF.pi())**N)
error_bound = error_factor / (sigma + N - 1) * factor.abs()
if result.abs().upper().is_zero():
error_acceptable = 0
else:
rounding_error = result.real().upper() + result.imag().upper() \
- result.real().lower() - result.imag().lower()
error_acceptable = max(max_approximation_error, rounding_error/8)
verbose(" N = %d, error = %s, acceptable_error = %s, result = %s" %
(N, error_bound, error_acceptable, result), level=2)
if error_bound.abs() < error_acceptable:
error_real = RIF(-error_bound, error_bound)
error = CIF(error_real, error_real)
verbose(" N = %d, error = %s, acceptable_error = %s, result = %s" %
(N, error_bound, error_acceptable, result), level=1)
return result + error
factor /= (M + alpha)
#assert factor.overlaps(falling_factorial(-s, N)/(M + alpha)**(s + N))
def _reduce_resolution_(data, x_min, x_max, resolution):
r"""
Reduce a list of pairs `(x, y)` to a list of triples `(x,
y_\mathrm{min}, y_\mathrm{max})` corresponding to
``resolution`` equidistant `x` values.
INPUT:
- ``data`` -- a list (or iterable) of pairs of doubles.
- ``x_min`` -- a double, start of the interval.
- ``x_max`` -- a double, end of the interval.
- ``resolution`` -- a positive integer, the number of points
in `x` direction.
OUTPUT:
A list of triples of doubles.
Each `(x, y)` is mapped to some
`(x', y_\mathrm{min}, y_\mathrm{max})`
such that `y_\mathrm{min}\le y\le y_\mathrm{max}` and
`0\le x-x'< (x_\mathrm{max}-x_\mathrm{min})/\mathit{resolution}`.
A list plot of the original list thus corresponds to the line
segments defined by the output.
EXAMPLE::
sage: from sage.combinat.fsm_fourier import _reduce_resolution_ # optional - arb
sage: _reduce_resolution_(((i/10, i) for i in range(10)), # optional - arb
....: 0, 1, 2)
[(0.0, 0, 4), (0.5, 5, 9)]
TESTS::
sage: _reduce_resolution_([(1, 10)], # optional - arb
....: 0, 1, 2)
Traceback (most recent call last):
...
ValueError: x values must be >= x_min and < x_max.
sage: _reduce_resolution_([(-1, 10)], # optional - arb
....: 0, 1, 2)
Traceback (most recent call last):
...
ValueError: x values must be >= x_min and < x_max.
"""
result = [None for _ in range(resolution)]
f = (<double> resolution)/(x_max-x_min)
for (x, y) in data:
if x < x_min and x > x_min - 1/f:
# Allow undershooting x_min by one pixel in order to be
# more robust against numerical noise.
x = x_min
if x < x_min or x >= x_max:
raise ValueError(
"x values must be >= x_min and < x_max.")
i = floor((x - x_min)*f)
current = result[i]
if current is None:
result[i] = (y, y)
else:
result[i] = (min(y, current[0]), max(y, current[0]))
return [(x_min + (<double> i)/f, y[0], y[1])
for i, y in enumerate(result)
if y is not None]
class FSMFourierComponent(SageObject):
r"""
Final component of a
:class:`~sage.combinat.finite_state_machine.Transducer` and its
associated data for computing the Fourier coefficients of the
fluctuations of the sum of output.
INPUT:
- ``fsm`` -- the final component as a
:class:`~sage.combinat.finite_state_machine.Transducer`.
- ``parent`` -- an instance of
:class:`~sage.combinat.fsm_fourier.FSMFourier` holding all
relevant data for the full transducer.
EXAMPLES::
sage: sage.combinat.finite_state_machine.FSMOldProcessOutput = False
sage: function('f')
f
sage: var('n')
n
sage: from sage.combinat.fsm_fourier import FSMFourier # optional - arb
sage: F = FSMFourier(transducers.Recursion([ # indirect doctest; optional - arb
....: f(2*n + 1) == f(n) + 1,
....: f(2*n) == f(n),
....: f(0) == 0],
....: f, n, 2))
sage: F.components[0] # optional - arb
<class 'sage.combinat.fsm_fourier.FSMFourierComponent'>
sage: F.components[0].period # optional - arb
1
"""
def __init__(self, fsm, parent):
r"""
Initialize the :class:`FSMFourierComponent`.
INPUT:
- ``fsm`` -- the final component as a
:class:`~sage.combinat.finite_state_machine.Transducer`.
- ``parent`` -- an instance of :class:`FSMFourier` holding all
relevant data for the full transducer.
EXAMPLE::
sage: sage.combinat.finite_state_machine.FSMOldProcessOutput = False
sage: function('f')
f
sage: var('n')
n
sage: from sage.combinat.fsm_fourier import FSMFourier # optional - arb
sage: F = FSMFourier(transducers.Recursion([ # indirect doctest; optional - arb
....: f(2*n + 1) == f(n) + 1,
....: f(2*n) == f(n),
....: f(0) == 0],
....: f, n, 2))
sage: F.components[0] # optional - arb
<class 'sage.combinat.fsm_fourier.FSMFourierComponent'>
sage: F.components[0].period # optional - arb
1
"""
self.fsm = fsm
self.period = fsm.graph().period()
self.n_states = len(self.fsm.states())
self.parent = parent
def _mask_(self):
r"""
Return a matrix whose rows consist of the incidence
vectors of the states contained in the other final components
of ``self.parent`` than self.
INPUT:
None.
OUTPUT:
A matrix.
EXAMPLES::
sage: from sage.combinat.fsm_fourier import FSMFourier # optional - arb
sage: T = Transducer([(0, 1, 0, 0), (0, 2, 1, 0),
....: (1, 1, 0, 0), (1, 1, 1, 0),
....: (2, 2, 0, 0), (2, 2, 1, 0)],
....: initial_states=[0],
....: final_states=[0, 1, 2]) # optional - arb
sage: FC = FSMFourier(T).components[0] # optional - arb
sage: FC._mask_() # optional - arb
[0 0 1]
"""
n = self.parent.M.nrows()
nrows = sum(c.n_states
for c in self.parent.components if c != self)
_mask_ = matrix(
nrows, n,
[self.parent.standard_basis[
self.parent.positions[state.label()]]
for other in self.parent.components
if other != self
for state in other.fsm.iter_states()])
return _mask_
def _eigenvectors_(self, M):
r"""
Determine the dominant eigenvectors of the given full
adjacency matrix where all coordinates corresponding to final
components other than self vanish.
INPUT:
- ``M`` -- a matrix.
OUTPUT:
A list of vectors, the `k`-th entry is the eigenvector to
the eigenvalue `q \exp(2\pi i k/p_j)` where `p_j` is
the period of this component.
The first eigenvector (to the positive eigenvalue) is
normalized such that its entries sum up to one.
This method is called by :meth:`.right_eigenvectors` with
``M = self.parent.M`` and by :meth:`.left_eigenvectors`
with ``M = self.parent.M.transpose()``.
EXAMPLES::
sage: from sage.combinat.fsm_fourier import FSMFourier # optional - arb
sage: T = Transducer([(0, 1, 0, 0), (0, 2, 1, 0),
....: (1, 1, 0, 0), (1, 1, 1, 0),
....: (2, 2, 0, 0), (2, 2, 1, 0)],
....: initial_states=[0],
....: final_states=[0, 1, 2]) # optional - arb
sage: F = FSMFourier(T) # optional - arb
sage: FC = F.components[0] # optional - arb
sage: FC._eigenvectors_(F.M) # optional - arb
[(1/3, 2/3, 0)]
.. SEEALSO::
:meth:`.right_eigenvectors`, :meth:`.left_eigenvectors`
"""
mask = self._mask_()
def eigenvector(j):
eigenvalue = self.parent.q * self.parent.alpha**(
j * self.parent.common_period / self.period)
S = matrix.block(
[[M - eigenvalue*matrix.identity(M.nrows())],
[mask]],
subdivide=False)
kernel = S.right_kernel_matrix()
assert kernel.nrows() == 1
if j == 0:
#normalize for positive eigenvector
return kernel.row(0) / sum(kernel.row(0))
else:
return kernel.row(0)
return [eigenvector(j) for j in range(self.period)]
@cached_method()
def right_eigenvectors(self):
r"""
Determine the dominant right eigenvectors of the full
adjacency matrix where all coordinates corresponding to final
components other than self vanish.
INPUT:
None.
OUTPUT:
A list of vectors, the `k`-th entry is the eigenvector to
the eigenvalue `q \exp(2\pi i k/p_j)` where `p_j` is
the period of this component.
The first eigenvector (to the positive eigenvalue) is
normalized such that its entries sum up to one.
EXAMPLES::
sage: from sage.combinat.fsm_fourier import FSMFourier # optional - arb
sage: T = Transducer([(0, 1, 0, 0), (0, 2, 1, 0),
....: (1, 1, 0, 0), (1, 1, 1, 0),
....: (2, 2, 0, 0), (2, 2, 1, 0)],
....: initial_states=[0],
....: final_states=[0, 1, 2]) # optional - arb
sage: F = FSMFourier(T) # optional - arb
sage: FC = F.components[0] # optional - arb
sage: FC.right_eigenvectors() # optional - arb
[(1/3, 2/3, 0)]
.. SEEALSO::
:meth:`.left_eigenvectors`
"""
return self._eigenvectors_(self.parent.M)
@cached_method()
def left_eigenvectors(self):
r"""
Determine the dominant left eigenvectors of the full
adjacency matrix where all coordinates corresponding to final
components other than self vanish.
INPUT:
None.
OUTPUT:
A list of vectors, the `k`-th entry is the eigenvector to
the eigenvalue `q \exp(2\pi i k/p_j)` where `p_j` is
the period of this component.
The eigenvectors are normalized such that their product
with the corresponding right eigenvectors computed by
:meth:`.right_eigenvectors` is `1`.
EXAMPLES::
sage: from sage.combinat.fsm_fourier import FSMFourier # optional - arb
sage: T = Transducer([(0, 1, 0, 0), (0, 2, 1, 0),
....: (1, 1, 0, 0), (1, 1, 1, 0),
....: (2, 2, 0, 0), (2, 2, 1, 0)],
....: initial_states=[0],
....: final_states=[0, 1, 2]) # optional - arb
sage: F = FSMFourier(T) # optional - arb
sage: FC = F.components[0] # optional - arb
sage: FC.left_eigenvectors() # optional - arb
[(0, 3/2, 0)]
.. SEEALSO::
:meth:`.right_eigenvectors`
"""
left_eigenvectors = self._eigenvectors_(self.parent.M.transpose())
return [w/(v*w) for v, w
in itertools.izip(self.right_eigenvectors(),
left_eigenvectors)]
@cached_method()
def w(self):
r"""
Compute `w_{jk}` for `0\le k< p_j`.
INPUT:
None.
OUTPUT:
A list of vectors.
EXAMPLES::
sage: from sage.combinat.fsm_fourier import FSMFourier # optional - arb
sage: T = Transducer([(0, 1, 0, 0), (0, 2, 1, 0),
....: (1, 1, 0, 0), (1, 1, 1, 0),
....: (2, 2, 0, 0), (2, 2, 1, 0)],
....: initial_states=[0],
....: final_states=[0, 1, 2]) # optional - arb
sage: F = FSMFourier(T) # optional - arb
sage: FC = F.components[0] # optional - arb
sage: FC.w() # optional - arb
[(0, 1/2, 0)]
.. SEEALSO::
:meth:`.left_eigenvectors`, :meth:`.right_eigenvectors`
"""
return [(self.parent.initial_vector*v)*w for v, w
in itertools.izip(self.right_eigenvectors(),
self.left_eigenvectors())]
@cached_method()
def coefficient_lambda(self):
r"""
Compute `\lambda_j`.
INPUT:
None.
OUTPUT:
An element of a cyclotomic field representing the value
`\lambda_j`.
EXAMPLES::
sage: from sage.combinat.fsm_fourier import FSMFourier # optional - arb
sage: T = Transducer([(0, 1, 0, 0), (0, 2, 1, 0),
....: (1, 1, 0, 0), (1, 1, 1, 0),
....: (2, 2, 0, 0), (2, 2, 1, 0)],
....: initial_states=[0],
....: final_states=[0, 1, 2]) # optional - arb
sage: F = FSMFourier(T) # optional - arb
sage: FC = F.components[0] # optional - arb
sage: FC.coefficient_lambda() # optional - arb
1/2
"""
products = [w*self.parent.ones for w in self.w()]
assert all(e.is_zero() for e in products[1:])
return products[0]
@cached_method()
def mu_prime(self):
r"""
Compute `\mu_{j0}'(0)`, the derivative of the dominant positive
eigenvector of `M(t)` corresponding to ``self`` at `t=0`.
INPUT:
None.
OUTPUT:
An element of a cyclotomic field representing the value
`\mu'_{j0}(0)`.
EXAMPLES::
sage: from sage.combinat.fsm_fourier import FSMFourier # optional - arb
sage: T = Transducer([(0, 1, 0, 0), (0, 2, 1, 0),
....: (1, 1, 0, 0), (1, 1, 1, 0),
....: (2, 2, 0, 0), (2, 2, 1, 0)],
....: initial_states=[0],
....: final_states=[0, 1, 2]) # optional - arb
sage: F = FSMFourier(T) # optional - arb
sage: FC = F.components[0] # optional - arb
sage: FC.mu_prime() # optional - arb
0
"""
Y = self.parent.Y
p = self.fsm.adjacency_matrix(
entry=lambda t:Y**sum(t.word_out)).charpoly('Z')
Z = p.parent().gen()
assert p(Y=1, Z=self.parent.q) == 0
mu_prime_Z = (- p.derivative(Y)/p.derivative(Z))(
Y=1, Z=self.parent.q)
return self.parent.I*mu_prime_Z
@cached_method()
def a(self):
r"""
Compute the constant `a_{j}`.
INPUT:
None.
OUTPUT:
An element of a cyclotomic field representing the value
`a_j`.
EXAMPLES::
sage: from sage.combinat.fsm_fourier import FSMFourier # optional - arb
sage: T = Transducer([(0, 1, 0, 0), (0, 2, 1, 0),
....: (1, 1, 0, 0), (1, 1, 1, 0),
....: (2, 2, 0, 0), (2, 2, 1, 0)],
....: initial_states=[0],
....: final_states=[0, 1, 2]) # optional - arb
sage: F = FSMFourier(T) # optional - arb
sage: FC = F.components[0] # optional - arb
sage: FC.a() # optional - arb
0
"""
return QQ(-self.parent.I * self.mu_prime()/self.parent.q)
def w_ell(self, ell):
r"""
Compute the left eigenvector to the eigenvalue
`\exp(2\pi i \ell/p)` to `M(0)` if the former is an eigenvalue
of `M(0)`. Otherwise, return the zero vector.
INPUT:
- ``ell`` -- an integer.
OUTPUT:
A vector over a complex interval field or the zero vector.
EXAMPLES::
sage: from sage.combinat.fsm_fourier import FSMFourier # optional - arb
sage: T = Transducer([(0, 1, 0, 0), (0, 2, 1, 0),
....: (1, 1, 0, 0), (1, 1, 1, 0),
....: (2, 3, 0, 0), (2, 3, 1, 0),
....: (3, 2, 0, 0), (3, 2, 1, 0)],
....: initial_states=[0],
....: final_states=[0, 1, 2, 3]) # optional - arb
sage: F = FSMFourier(T) # optional - arb
sage: FC = F.components[0] # optional - arb
sage: FC.w_ell(0) # optional - arb
(0, 0.50000000000000000?, 0, 0)
sage: FC.w_ell(1) # optional - arb
(0, 0, 0, 0)
"""
if self.parent.common_period.divides(ell*self.period):
k = self.period*ell/self.parent.common_period % self.period
return vector(self.parent.field_to_CIF(c) for c in self.w()[k])
else:
return vector(0 for _ in self.parent.ones)
def v_prime(self, k):
r"""
Compute the derivative `v_{jk}'(0)`, the right
eigenvector to the `k`-th dominant eigenvalue.
INPUT:
- ``k`` -- a non-negative integer.
OUTPUT:
A vector.
EXAMPLES::
sage: from sage.combinat.fsm_fourier import FSMFourier # optional - arb
sage: T = Transducer([(0, 1, 0, 0), (0, 2, 1, 0),
....: (1, 1, 0, 0), (1, 1, 1, 0),
....: (2, 3, 0, 0), (2, 3, 1, 0),
....: (3, 2, 0, 0), (3, 2, 1, 0)],
....: initial_states=[0],
....: final_states=[0, 1, 2, 3]) # optional - arb
sage: F = FSMFourier(T) # optional - arb
sage: FC = F.components[0] # optional - arb
sage: FC.v_prime(0) # optional - arb
(0, 0, 0, 0)
"""
M = self.parent.M
mask = self._mask_()
eigenvalue = self.parent.q * self.parent.alpha**(
k * self.parent.common_period / self.period)
S = matrix.block(
[[M - eigenvalue*matrix.identity(M.nrows())],
[matrix(self.parent.ones)],
[mask]],
subdivide=False)
eigenvector_right = vector(self.right_eigenvectors()[k])
M_prime = self.parent.I*self.parent.Delta
right_side = - matrix.block(
[[M_prime - self.mu_prime()*matrix.identity(M.nrows())],
[0*matrix(self.parent.ones)],
[0*mask]],
subdivide=False) * eigenvector_right
v_prime = S.solve_right(right_side)
return v_prime
def w_prime(self, k):
r"""
Compute the derivative `w_{jk}'(0)`, the left
eigenvector to the `k`-th dominant eigenvalue.
INPUT:
- ``k`` -- a non-negative integer.
OUTPUT:
A vector.
EXAMPLES::
sage: from sage.combinat.fsm_fourier import FSMFourier # optional - arb
sage: T = Transducer([(0, 1, 0, 0), (0, 2, 1, 0),
....: (1, 1, 0, 0), (1, 1, 1, 0),
....: (2, 3, 0, 0), (2, 3, 1, 0),
....: (3, 2, 0, 0), (3, 2, 1, 0)],
....: initial_states=[0],
....: final_states=[0, 1, 2, 3]) # optional - arb
sage: F = FSMFourier(T) # optional - arb
sage: FC = F.components[0] # optional - arb
sage: FC.w_prime(0) # optional - arb
(0, 0, 0, 0)
"""
M = self.parent.M
mask = self._mask_()
eigenvalue = self.parent.q * self.parent.alpha**(
k * self.parent.common_period / self.period)
eigenvector_right = vector(self.right_eigenvectors()[k])
eigenvector_left = vector(self.left_eigenvectors()[k])
M_prime = self.parent.I*self.parent.Delta
S = matrix.block(
[[M.transpose() - eigenvalue*matrix.identity(M.nrows())],
[matrix(eigenvector_right)],
[mask]],
subdivide=False)
right_side = - matrix.block(
[[M_prime.transpose() - self.mu_prime()*matrix.identity(M.nrows())],
[matrix(self.v_prime(k))],
[0*mask]],
subdivide=False) * eigenvector_left
left_prime = S.solve_right(right_side)
w_prime = self.parent.initial_vector*self.v_prime(k)*eigenvector_left \
+ self.parent.initial_vector*eigenvector_right*left_prime
return w_prime
cdef class FSMFourierCache(SageObject):
r"""
Compute and cache `\mathbf{b}(r)` for increased performance
and compute partial sums of the Dirichlet series `\mathbf{H}(s)`.
INPUT:
- ``parent`` -- a :class:`FSMFourier` instance.
OUTPUT:
None.
EXAMPLES::
sage: sage.combinat.finite_state_machine.FSMOldProcessOutput = False
sage: from sage.combinat.fsm_fourier import FSMFourier # optional - arb
sage: T = Transducer([(0, 0, 0, 0), (0, 0, 1, 1)],
....: initial_states=[0],
....: final_states=[0])
sage: FSMFourier(T).cache # optional - arb
<type 'sage.combinat.fsm_fourier.FSMFourierCache'>
"""
cdef acb_mat_t *bb
cdef unsigned long bb_computed
cdef unsigned long bb_allocated
cdef acb_mat_t *Delta_epsilon_ones
cdef acb_mat_t *M_epsilon
cdef unsigned long q
cdef unsigned long precision
cdef unsigned long n
cdef object parent
def __init__(self, parent):
"""