def _sollya_annotate(self, center, rad, polys):
import sagesollya as sollya
logger = logging.getLogger(__name__ + ".sollya")
logger.info("calling annotatefunction() on %s derivatives", len(polys))
center = QQ(center)
sollya_fun = self._sollya_object
# sollya keeps all annotations, let's not bother with selecting
# the best one
for ord, pol0 in enumerate(polys):
pol = ZZ(ord).factorial() * pol0
sollya_pol = sum(
[c.center() * sollya.x**k for k, c in enumerate(pol)])
dom = RIF(center - rad,
center + rad) # XXX: dangerous when inexact
err_pol = pol.map_coefficients(lambda c: c - c.squash())
err = RIF(err_pol(RBF.zero().add_error(rad)))
with sollya.settings(display=sollya.dyadic):
logger = logging.getLogger(__name__ + ".sollya.annotate")
logger.debug("annotatefunction(%s, %s, %s, %s, %s);",
sollya_fun, sollya_pol, sollya.SollyaObject(dom),
sollya.SollyaObject(err),
sollya.SollyaObject(center))
sollya.annotatefunction(sollya_fun, sollya_pol, dom, err, center)
sollya_fun = sollya.diff(sollya_fun)
logger.info("...done")
def _update_approx(self, center, rad, prec, derivatives):
ini, path = self._path_to(center, prec)
eps = RBF.one() >> prec
# keep="all" won't do anything until _path_to returns better paths
ctx = ancont.Context(self.dop, path, eps, keep="all")
pairs = ancont.analytic_continuation(ctx, ini=ini)
for (vert, val) in pairs:
known = self._inivecs.get(vert)
if known is None or known[0].accuracy() < val[0][0].accuracy():
self._inivecs[vert] = [c[0] for c in val]
logger.info(
"computing new polynomial approximations: "
"ini=%s, path=%s, rad=%s, eps=%s, ord=%s", ini, path, rad, eps,
derivatives)
polys = polapprox.doit(self.dop,
ini=ini,
path=path,
rad=rad,
eps=eps,
derivatives=derivatives,
x_is_real=True,
economization=polapprox.chebyshev_economization)
logger.info("...done")
approx = self._polys.get(center, [])
new_approx = []
for ord, pol in enumerate(polys):
if ord >= len(approx) or approx[ord].prec < prec:
new_approx.append(RealPolApprox(pol, prec))
else:
new_approx.append(approx[ord])
self._update_approx_hook(center, rad, polys)
self._polys[center] = new_approx
return polys
def accuracy(self):
infinity = RBF.maximal_accuracy()
if self.universe.is_exact():
return infinity
elif isinstance(self.universe, (RealBallField, ComplexBallField)):
return min(
infinity,
*(x.accuracy() for val in self.shift.values() for x in val))
else:
raise ValueError
def wrapper(pt, ord, prec):
if RBF(pt.diameter()) >= self.max_rad / 4:
return self._known_bound(RBF(pt), post_transform=Dx**ord)
try:
val = self.approx(pt, prec, post_transform=Dx**ord)
except Exception:
logger.info("pt=%s, ord=%s, prec=%s, error",
pt,
ord,
prec,
exc_info=(pt.absolute_diameter() < .5))
return RIF('nan')
logger.debug("pt=%s, ord=%s, prec=%s, val=%s", pt, ord, prec, val)
if not pt.overlaps(self._sollya_domain):
backtrace = sollya.getbacktrace()
logger.debug("%s not in %s", pt.str(style='brackets'),
self._sollya_domain.str(style='brackets'))
logger.debug("sollya backtrace: %s", [
sollya.objectname(t.struct.called_proc) for t in backtrace
])
return val
def _disk(self, pt):
assert pt.is_real()
# Since approximation disks satisfy 2·rad ≤ dist(center, sing), any
# approximation disk containing pt must have rad ≤ dist(pt, sing)
max_rad = pt.dist_to_sing().min(self.max_rad)
# What we want is the largest such disk containing pt
expo = ZZ(max_rad.log(2).upper().ceil()) - 1 # rad = 2^expo
logger.log(logging.DEBUG - 2, "max_rad = %s, expo = %s", max_rad, expo)
while True:
approx_pt = pt.approx_abs_real(-expo)
mantissa = (approx_pt.squash() >> expo).floor()
if ZZ(mantissa) % 2 == 0:
mantissa += 1
center = mantissa << expo
dist = Point(center, pt.dop).dist_to_sing()
rad = RBF.one() << expo
logger.log(
logging.DEBUG - 2,
"candidate disk: approx_pt = %s, mantissa = %s, "
"center = %s, dist = %s, rad = %s", approx_pt, mantissa,
center, dist, rad)
if safe_ge(dist >> 1, rad):
break
expo -= 1
logger.debug("disk for %s: center=%s, rad=%s", pt, center, rad)
# pt may be a ball with nonzero radius: check that it is contained in
# our candidate disk
log = RBF.zero() if 0 in approx_pt else approx_pt.abs().log(2)
F = RealBallField(ZZ((expo - log).max(0).upper().ceil()) + 10)
dist_to_center = (F(approx_pt) - F(center)).abs()
if not safe_le(dist_to_center, rad):
assert not safe_gt((approx_pt.squash() - center).squash(), rad)
logger.info("check that |%s - %s| < %s failed", approx_pt, center,
rad)
return None, None
# exactify center so that subsequent computations are not limited by the
# precision of its parent
center = QQ(center)
return center, rad
def __init__(self,
dop,
ini,
name="dfinitefun",
max_prec=256,
max_rad=RBF('inf')):
self.dop = dop = DifferentialOperator(dop)
if not isinstance(ini, dict):
ini = {0: ini}
if len(ini) != 1:
# In the future, we should support specifying several vectors of
# initial values.
raise NotImplementedError
self.ini = ini
self.name = name
# Global maximum width for the approximation intervals. In the case of
# equations with no finite singular point, we try to avoid cancellation
# and interval blowup issues by taking polynomial approximations on
# intervals on which the general term of the series doesn't grow too
# large. The expected order of magnitude of the “top of the hump” is
# about exp(κ·|αx|^(1/κ)) and doesn't depend on the base point. We also
# let the user impose a maximum width, even in other cases.
self.max_rad = RBF(max_rad)
if dop.leading_coefficient().is_constant():
kappa, alpha = _growth_parameters(dop)
self.max_rad = self.max_rad.min(1 / (alpha * RBF(kappa)**kappa))
self.max_prec = max_prec
self._inivecs = {}
self._polys = {}
self._sollya_object = None
self._sollya_domain = RIF('-inf', 'inf')
self._max_derivatives = dop.order()
self._update_approx_hook = (lambda *args: None)
def monodromy_matrices(dop, base, eps=1e-16, algorithm="connect"):
r"""
Compute generators of the monodromy group of ``dop`` with base point
``base``.
OUTPUT:
A list of matrices, each encoding the analytic continuation of solutions
along a simple positive loop based in ``base`` around a singular point
of ``dop`` (with no other singular point inside the loop). Identity matrices
may be omitted. The precise choice of elements of the fundamental group
corresponding to each matrix (position with respect to the other
singular points, order) are unspecified.
EXAMPLES::
sage: from ore_algebra import *
sage: from ore_algebra.analytic.monodromy import monodromy_matrices
sage: Dops, x, Dx = DifferentialOperators()
sage: monodromy_matrices(Dx*x*Dx, 1)
[
[ 1.0000... [6.2831853071795...]*I]
[ 0 1.0000...]
]
sage: monodromy_matrices(Dx*x*Dx, 1, algorithm="loop")
[
[ 1.0000... [+/- ...] + [6.283185307179...]*I]
[ 0 [1.000000000000...] + [+/- ...]*I]
]
sage: monodromy_matrices(Dx*x*Dx, 1/2)
[
[ 1.0000... [+/- ...] + [3.1415926535897...]*I]
[ 0 [1.0000000000000...]]
]
"""
dop = DifferentialOperator(dop)
base = path.Point(base, dop)
if not base.is_regular():
raise ValueError("base point must be regular")
eps = RBF(eps)
if not (algorithm == "connect" or algorithm == "loop"):
raise ValueError("unknown algorithm")
id_mat = _identity_matrix(base, eps)
def matprod(elts):
return prod(reversed(elts), id_mat)
# TODO: filter out the factors of the leading coefficient that correspond to
# apparent singularities (may require improvements to the analytic
# continuation code)
tree = _sing_tree(dop, base)
polygon_base, local_monodromy_base = _local_monodromy(dop, base, eps, algorithm)
result = [] if base.is_ordinary() else local_monodromy_base
def dfs(x, path, path_mat, polygon_x, local_monodromy_x):
x.seen = True
for y in [z for z in tree.neighbors(x) if not z.seen]:
logger.info("Computing local monodromy around %s via %s", y, path)
polygon_y, local_monodromy_y = _local_monodromy(dop, y, eps, algorithm)
anchor_index_x, anchor_x = _closest_unsafe(polygon_x, y)
anchor_index_y, anchor_y = _closest_unsafe(polygon_y, x)
bypass_mat_x = matprod(local_monodromy_x[:anchor_index_x])
if anchor_index_y > 0:
bypass_mat_y = matprod(local_monodromy_y[anchor_index_y:])
else:
bypass_mat_y = id_mat
edge_mat = dop.numerical_transition_matrix([anchor_x, anchor_y], eps, assume_analytic=True)
new_path_mat = bypass_mat_y*edge_mat*bypass_mat_x*path_mat
assert isinstance(new_path_mat, Matrix_complex_ball_dense)
local_mat = matprod(local_monodromy_y)
based_mat = (~new_path_mat)*local_mat*new_path_mat
result.append(based_mat)
dfs(y, path + [y], new_path_mat, polygon_y, local_monodromy_y)
for x in tree:
x.seen = False
dfs(base, [base], id_mat, polygon_base, local_monodromy_base)
return result
def _test_monodromy_matrices():
r"""
TESTS::
sage: from ore_algebra.analytic.monodromy import _test_monodromy_matrices
sage: _test_monodromy_matrices()
"""
from sage.all import matrix
from ore_algebra import DifferentialOperators
Dops, x, Dx = DifferentialOperators()
h = QQ(1) / 2
i = QQi.gen()
def norm(m):
return sum(c.abs()**2 for c in m.list()).sqrtpos()
mon = monodromy_matrices((x**2 + 1) * Dx - 1, QQ(1000000))
assert norm(mon[0] - CBF(pi).exp()) < RBF(1e-10)
assert norm(mon[1] - CBF(-pi).exp()) < RBF(1e-10)
mon = monodromy_matrices((x**2 - 1) * Dx - 1, QQ(0))
assert all(m == -1 for m in mon)
dop = (x**2 + 1) * Dx**2 + 2 * x * Dx
mon = monodromy_matrices(dop, QQbar(i + 1)) # mon[0] <--> i
assert norm(mon[0] - matrix(CBF, [[1, pi *
(1 + 2 * i)], [0, 1]])) < RBF(1e-10)
assert norm(mon[1] - matrix(CBF, [[1, -pi *
(1 + 2 * i)], [0, 1]])) < RBF(1e-10)
mon = monodromy_matrices(dop, QQbar(-i + 1)) # mon[0] <--> -i
assert norm(mon[0] - matrix(CBF, [[1, pi *
(-1 + 2 * i)], [0, 1]])) < RBF(1e-10)
assert norm(mon[1] - matrix(CBF, [[1, pi *
(1 - 2 * i)], [0, 1]])) < RBF(1e-10)
mon = monodromy_matrices(dop, QQbar(i)) # mon[0] <--> i
assert norm(mon[0] - matrix(CBF, [[1, 0], [2 * pi * i, 1]])) < RBF(1e-10)
assert norm(mon[1] - matrix(CBF, [[1, 0], [-2 * pi * i, 1]])) < RBF(1e-10)
mon = monodromy_matrices(dop, QQbar(i), sing=[QQbar(i)])
assert len(mon) == 1
assert norm(mon[0] - matrix(CBF, [[1, 0], [2 * pi * i, 1]])) < RBF(1e-10)
mon = monodromy_matrices(dop, QQbar(i), sing=[QQbar(-i)])
assert len(mon) == 1
assert norm(mon[0] - matrix(CBF, [[1, 0], [-2 * pi * i, 1]])) < RBF(1e-10)
mon = monodromy_matrices(dop, QQbar(-i), sing=[QQbar(i)])
assert len(mon) == 1
assert norm(mon[0] - matrix(CBF, [[1, 0], [-2 * pi * i, 1]])) < RBF(1e-10)
mon = monodromy_matrices(dop, QQbar(i), sing=[])
assert mon == []
dop = (x**2 + 1) * (x**2 - 1) * Dx**2 + 1
mon = monodromy_matrices(dop, QQ(0), sing=[QQ(1), QQbar(i)])
m0 = dop.numerical_transition_matrix([0, i + 1, 2 * i, i - 1, 0])
assert norm(m0 - mon[0]) < RBF(1e-10)
m1 = dop.numerical_transition_matrix([0, 1 - i, 2, 1 + i, 0])
assert norm(m1 - mon[1]) < RBF(1e-10)
dop = x * (x - 3) * (x - 4) * (x**2 - 6 * x + 10) * Dx**2 - 1
mon = monodromy_matrices(dop, QQ(-1))
m0 = dop.numerical_transition_matrix([-1, -i, 1, i, -1])
assert norm(m0 - mon[0]) < RBF(1e-10)
m1 = dop.numerical_transition_matrix(
[-1, i / 2, 3 - i / 2, 3 + h, 3 + i / 2, i / 2, -1])
assert norm(m1 - mon[1]) < RBF(1e-10)
m2 = dop.numerical_transition_matrix(
[-1, i / 2, 3 + i / 2, 4 - i / 2, 4 + h, 4 + i / 2, i / 2, -1])
assert norm(m2 - mon[2]) < RBF(1e-10)
m3 = dop.numerical_transition_matrix([-1, 3 + i + h, 3 + 2 * i, -1])
assert norm(m3 - mon[3]) < RBF(1e-10)
m4 = dop.numerical_transition_matrix(
[-1, 3 - 2 * i, 3 - i + h, 3 - i / 2, -1])
assert norm(m4 - mon[4]) < RBF(1e-10)
dop = (x - i)**2 * (x + i) * Dx - 1
mon = monodromy_matrices(dop, 0)
assert norm(mon[0] + i) < RBF(1e-10)
assert norm(mon[1] - i) < RBF(1e-10)
dop = (x - i)**2 * (x + i) * Dx**2 - 1
mon = monodromy_matrices(dop, 0)
m0 = dop.numerical_transition_matrix([0, i + 1, 2 * i, i - 1, 0])
assert norm(m0 - mon[0]) < RBF(1e-10)
m1 = dop.numerical_transition_matrix([0, -i - 1, -2 * i, -i + 1, 0])
assert norm(m1 - mon[1]) < RBF(1e-10)
def _monodromy_matrices(dop, base, eps=1e-16, sing=None):
r"""
EXAMPLES::
sage: from ore_algebra import *
sage: from ore_algebra.analytic.monodromy import _monodromy_matrices
sage: Dops, x, Dx = DifferentialOperators()
sage: rat = 1/(x^2-1)
sage: dop = (rat*Dx - rat.derivative()).lclm(Dx*x*Dx)
sage: [rec.point for rec in _monodromy_matrices(dop, 0) if not rec.is_scalar]
[0]
TESTS::
sage: from ore_algebra.examples import fcc
sage: mon = list(_monodromy_matrices(fcc.dop5, -1, 2**(-2**7))) # long time (2.3 s)
sage: [rec.monodromy[0][0] for rec in mon if rec.point == -5/3] # long time
[[1.01088578589319884254557667137848...]]
Thanks to Alexandre Goyer for this example::
sage: L1 = ((x^5 - x^4 + x^3)*Dx^3 + (27/8*x^4 - 25/9*x^3 + 8*x^2)*Dx^2
....: + (37/24*x^3 - 25/9*x^2 + 14*x)*Dx - 2*x^2 - 3/4*x + 4)
sage: L2 = ((x^5 - 9/4*x^4 + x^3)*Dx^3 + (11/6*x^4 - 31/4*x^3 + 7*x^2)*Dx^2
....: + (7/30*x^3 - 101/20*x^2 + 10*x)*Dx + 4/5*x^2 + 5/6*x + 2)
sage: L = L1*L2
sage: L = L.parent()(L.annihilator_of_composition(x+1))
sage: mon = list(_monodromy_matrices(L, 0, eps=1e-30)) # long time (1.3-1.7 s)
sage: mon[-1][0], mon[-1][1][0][0] # long time
(0.6403882032022075?,
[1.15462187280628880820271...] + [-0.018967673022432256251718...]*I)
"""
dop = DifferentialOperator(dop)
base = QQbar.coerce(base)
eps = RBF(eps)
if sing is None:
sing = dop._singularities(QQbar)
else:
sing = [QQbar.coerce(s) for s in sing]
todo = {x: TodoItem(x, dop, want_self=True, want_conj=False) for x in sing}
base = todo.setdefault(base, TodoItem(base, dop))
if not base.point().is_regular():
raise ValueError("irregular singular base point")
# If the coefficients are rational, reduce to handling singularities in the
# same half-plane as the base point, and share some computations between
# Galois conjugates.
need_conjugates = False
crit_cache = None
if all(c in QQ for pol in dop for c in pol):
need_conjugates = _merge_conjugate_singularities(dop, sing, base, todo)
# TODO: do something like that even over number fields?
# XXX this is actually a bit costly: do it only after checking that the
# monodromy is not scalar?
# XXX keep the cache from one run to the next when increasing prec?
crit_cache = {}
Scalars = ComplexBallField(utilities.prec_from_eps(eps))
id_mat = matrix.identity_matrix(Scalars, dop.order())
def matprod(elts):
return prod(reversed(elts), id_mat)
for key, todoitem in list(todo.items()):
point = todoitem.point()
# We could call _local_monodromy_loop() if point is irregular, but
# delaying it may allow us to start returning results earlier.
if point.is_regular():
if crit_cache is None or point.algdeg() == 1:
crit = _critical_monomials(dop.shift(point))
emb = point.value.parent().hom(Scalars)
else:
mpol = point.value.minpoly()
try:
NF, crit = crit_cache[mpol]
except KeyError:
NF = point.value.parent()
crit = _critical_monomials(dop.shift(point))
# Only store the critical monomials for reusing when all
# local exponents are rational. We need to restrict to this
# case because we do not have the technology in place to
# follow algebraic exponents along the embedding of NF in ℂ.
# (They are represented as elements of "new" number fields
# given by as_embedded_number_field_element(), even when
# they actually lie in NF itself as opposed to a further
# algebraic extension. XXX: Ideally, LocalBasisMapper should
# give us access to the tower of extensions in which the
# exponents "naturally" live.)
if all(sol.leftmost.parent() is QQ for sol in crit):
crit_cache[mpol] = NF, crit
emb = NF.hom([Scalars(point.value.parent().gen())],
check=False)
mon, scalar = _formal_monodromy_from_critical_monomials(crit, emb)
if scalar:
# No need to compute the connection matrices then!
# XXX When we do need them, though, it would be better to get
# the formal monodromy as a byproduct of their computation.
if todoitem.want_self:
yield LocalMonodromyData(key, mon, True)
if todoitem.want_conj:
conj = key.conjugate()
logger.info(
"Computing local monodromy around %s by "
"complex conjugation", conj)
conj_mat = ~mon.conjugate()
yield LocalMonodromyData(conj, conj_mat, True)
if todoitem is not base:
del todo[key]
continue
todoitem.local_monodromy = [mon]
todoitem.polygon = [point]
if need_conjugates:
base_conj_mat = dop.numerical_transition_matrix(
[base.point(), base.point().conjugate()],
eps,
assume_analytic=True)
def conjugate_monodromy(mat):
return ~base_conj_mat * ~mat.conjugate() * base_conj_mat
tree = _spanning_tree(base, todo.values())
def dfs(x, path, path_mat):
logger.info("Computing local monodromy around %s via %s", x, path)
local_mat = matprod(x.local_monodromy)
based_mat = (~path_mat) * local_mat * path_mat
if x.want_self:
yield LocalMonodromyData(x.alg, based_mat, False)
if x.want_conj:
conj = x.alg.conjugate()
logger.info(
"Computing local monodromy around %s by complex "
"conjugation", conj)
conj_mat = conjugate_monodromy(based_mat)
yield LocalMonodromyData(conj, conj_mat, False)
x.done = True
for y in tree.neighbors(x):
if y.done:
continue
if y.local_monodromy is None:
y.polygon, y.local_monodromy = _local_monodromy_loop(
dop, y.point(), eps)
new_path_mat = _extend_path_mat(dop, path_mat, x, y, eps, matprod)
yield from dfs(y, path + [y], new_path_mat)
yield from dfs(base, [base], id_mat)
def _monodromy_matrices(dop, base, eps=1e-16, sing=None):
r"""
EXAMPLES::
sage: from ore_algebra import *
sage: from ore_algebra.analytic.monodromy import _monodromy_matrices
sage: Dops, x, Dx = DifferentialOperators()
sage: rat = 1/(x^2-1)
sage: dop = (rat*Dx - rat.derivative()).lclm(Dx*x*Dx)
sage: [rec.point for rec in _monodromy_matrices(dop, 0) if not rec.is_scalar]
[0]
TESTS::
sage: from ore_algebra.examples import fcc
sage: mon = list(_monodromy_matrices(fcc.dop5, -1, 2**(-2**7))) # long time (2.3 s)
sage: [rec.monodromy[0][0] for rec in mon if rec.point == -5/3] # long time
[[1.01088578589319884254557667137848...]]
"""
dop = DifferentialOperator(dop)
base = QQbar.coerce(base)
eps = RBF(eps)
if sing is None:
sing = dop._singularities(QQbar)
else:
sing = [QQbar.coerce(s) for s in sing]
todo = {x: TodoItem(x, dop, want_self=True, want_conj=False) for x in sing}
base = todo.setdefault(base, TodoItem(base, dop))
if not base.point().is_regular():
raise ValueError("irregular singular base point")
# If the coefficients are rational, reduce to handling singularities in the
# same half-plane as the base point, and share some computations between
# Galois conjugates.
need_conjugates = False
crit_cache = None
if all(c in QQ for pol in dop for c in pol):
need_conjugates = _merge_conjugate_singularities(dop, sing, base, todo)
# TODO: do something like that even over number fields?
# XXX this is actually a bit costly: do it only after checking that the
# monodromy is not scalar?
# XXX keep the cache from one run to the next when increasing prec?
crit_cache = {}
Scalars = ComplexBallField(utilities.prec_from_eps(eps))
id_mat = matrix.identity_matrix(Scalars, dop.order())
def matprod(elts):
return prod(reversed(elts), id_mat)
for key, todoitem in list(todo.items()):
point = todoitem.point()
# We could call _local_monodromy_loop() if point is irregular, but
# delaying it may allow us to start returning results earlier.
if point.is_regular():
if crit_cache is None or point.algdeg() == 1:
crit = _critical_monomials(dop.shift(point))
emb = point.value.parent().hom(Scalars)
else:
mpol = point.value.minpoly()
try:
NF, crit = crit_cache[mpol]
except KeyError:
NF = point.value.parent()
crit = _critical_monomials(dop.shift(point))
crit_cache[mpol] = NF, crit
emb = NF.hom([Scalars(point.value.parent().gen())],
check=False)
mon, scalar = _formal_monodromy_from_critical_monomials(crit, emb)
if scalar:
# No need to compute the connection matrices then!
# XXX When we do need them, though, it would be better to get
# the formal monodromy as a byproduct of their computation.
if todoitem.want_self:
yield LocalMonodromyData(key, mon, True)
if todoitem.want_conj:
conj = key.conjugate()
logger.info(
"Computing local monodromy around %s by "
"complex conjugation", conj)
conj_mat = ~mon.conjugate()
yield LocalMonodromyData(conj, conj_mat, True)
if todoitem is not base:
del todo[key]
continue
todoitem.local_monodromy = [mon]
todoitem.polygon = [point]
if need_conjugates:
base_conj_mat = dop.numerical_transition_matrix(
[base.point(), base.point().conjugate()],
eps,
assume_analytic=True)
def conjugate_monodromy(mat):
return ~base_conj_mat * ~mat.conjugate() * base_conj_mat
tree = _spanning_tree(base, todo.values())
def dfs(x, path, path_mat):
logger.info("Computing local monodromy around %s via %s", x, path)
local_mat = matprod(x.local_monodromy)
based_mat = (~path_mat) * local_mat * path_mat
if x.want_self:
yield LocalMonodromyData(x.alg, based_mat, False)
if x.want_conj:
conj = x.alg.conjugate()
logger.info(
"Computing local monodromy around %s by complex "
"conjugation", conj)
conj_mat = conjugate_monodromy(based_mat)
yield LocalMonodromyData(conj, conj_mat, False)
x.done = True
for y in tree.neighbors(x):
if y.done:
continue
if y.local_monodromy is None:
y.polygon, y.local_monodromy = _local_monodromy_loop(
dop, y.point(), eps)
new_path_mat = _extend_path_mat(dop, path_mat, x, y, eps, matprod)
yield from dfs(y, path + [y], new_path_mat)
yield from dfs(base, [base], id_mat)
class DFiniteFunction(object):
r"""
At the moment, this class just provides a simple caching mechanism for
repeated evaluations of a D-Finite function on the real line. It may
evolve to support evaluations on the complex plane, branch cuts, ring
operations on D-Finite functions, and more. Do not expect any API stability.
TESTS::
sage: from ore_algebra import *
sage: from ore_algebra.analytic.function import DFiniteFunction
sage: from ore_algebra.analytic.path import Point
sage: from sage.rings.infinity import AnInfinity
sage: Dops, x, Dx = DifferentialOperators()
sage: f = DFiniteFunction(Dx - 1, [1], max_rad=7/8)
sage: f._disk(Point(pi, dop=Dx-1))
(7/2, 0.5000000000000000)
sage: f = DFiniteFunction(Dx^2 - x,
....: [1/(gamma(2/3)*3^(2/3)), -1/(gamma(1/3)*3^(1/3))],
....: name='my_Ai')
sage: f.plot((-5,5))
Graphics object consisting of 1 graphics primitive
sage: f._known_bound(RBF(RIF(1/3, 2/3)), post_transform=Dops.one())
[0.2...]
sage: f._known_bound(RBF(RIF(-1, 5.99)), post_transform=Dops.one())
[+/- ...]
sage: f._known_bound(RBF(RIF(-1, 6.01)), Dops.one())
[+/- inf]
sage: f = DFiniteFunction((x^2 + 1)*Dx^2 + 2*x*Dx, [0, 1])
sage: f.plot((0, 4))
Graphics object consisting of 1 graphics primitive
sage: f._known_bound(RBF(RIF(1,4)), Dops.one())
[1e+0 +/- 0.3...]
sage: f.approx(1, post_transform=Dx), f.approx(2, post_transform=Dx)
([0.5000000000000...], [0.2000000000000...])
sage: f._known_bound(RBF(RIF(1.1, 2.9)), post_transform=Dx)
[+/- 0.4...]
sage: f._known_bound(RBF(AnInfinity()), Dops.one())
[+/- inf]
sage: f = DFiniteFunction((x + 1)*Dx^2 + 2*x*Dx, [0, 1])
sage: f._known_bound(RBF(RIF(1,10)),Dops.one())
nan
sage: f._known_bound(RBF(AnInfinity()), Dops.one())
nan
sage: f = DFiniteFunction(Dx^2 + 2*x*Dx, [1, -2/sqrt(pi)], name='my_erfc')
sage: f._known_bound(RBF(RIF(-1/2,1/2)), post_transform=Dx^2)
[+/- inf]
sage: _ = f.approx(1/2, post_transform=Dx^2)
sage: _ = f.approx(-1/2, post_transform=Dx^2)
sage: f._known_bound(RBF(RIF(-1/2,1/2)), post_transform=Dx^2)
[+/- 1.5...]
"""
# Stupid, but simple and deterministic caching strategy:
#
# - To any center c = m·2^k with k ∈ ℤ and m *odd*, we associate the disk of
# radius 2^k. Any two disks with the same k have at most one point in
# common.
#
# - Thus, in the case of an equation with at least one finite singular
# point, there is a unique largest disk of the previous collection that
# contains any given ordinary x ∈ ℝ \ ℤ·2^(-∞) while staying “far enough”
# from the singularities.
#
# - When asked to evaluate f at x, we actually compute and store a
# polynomial approximation on (the real trace of) the corresponding disk
# and/or a vector of initial conditions at its center, which can then be
# reused for subsequent evaluations.
#
# - We may additionally want to allow disks (of any radius?) centered at
# real regular singular points, and perhaps, as a special case, at 0.
# These would be used when possible, and one would revert to the other
# family otherwise.
def __init__(self,
dop,
ini,
name="dfinitefun",
max_prec=256,
max_rad=RBF('inf')):
self.dop = dop = DifferentialOperator(dop)
if not isinstance(ini, dict):
ini = {0: ini}
if len(ini) != 1:
# In the future, we should support specifying several vectors of
# initial values.
raise NotImplementedError
self.ini = ini
self.name = name
# Global maximum width for the approximation intervals. In the case of
# equations with no finite singular point, we try to avoid cancellation
# and interval blowup issues by taking polynomial approximations on
# intervals on which the general term of the series doesn't grow too
# large. The expected order of magnitude of the “top of the hump” is
# about exp(κ·|αx|^(1/κ)) and doesn't depend on the base point. We also
# let the user impose a maximum width, even in other cases.
self.max_rad = RBF(max_rad)
if dop.leading_coefficient().is_constant():
kappa, alpha = _growth_parameters(dop)
self.max_rad = self.max_rad.min(1 / (alpha * RBF(kappa)**kappa))
self.max_prec = max_prec
self._inivecs = {}
self._polys = {}
self._sollya_object = None
self._sollya_domain = RIF('-inf', 'inf')
self._max_derivatives = dop.order()
self._update_approx_hook = (lambda *args: None)
def __repr__(self):
return self.name
@cached_method
def _is_everywhere_defined(self):
return not any(
rt.imag().contains_zero()
for rt, mult in self.dop.leading_coefficient().roots(CIF))
def _disk(self, pt):
assert pt.is_real()
# Since approximation disks satisfy 2·rad ≤ dist(center, sing), any
# approximation disk containing pt must have rad ≤ dist(pt, sing)
max_rad = pt.dist_to_sing().min(self.max_rad)
# What we want is the largest such disk containing pt
expo = ZZ(max_rad.log(2).upper().ceil()) - 1 # rad = 2^expo
logger.log(logging.DEBUG - 2, "max_rad = %s, expo = %s", max_rad, expo)
while True:
approx_pt = pt.approx_abs_real(-expo)
mantissa = (approx_pt.squash() >> expo).floor()
if ZZ(mantissa) % 2 == 0:
mantissa += 1
center = mantissa << expo
dist = Point(center, pt.dop).dist_to_sing()
rad = RBF.one() << expo
logger.log(
logging.DEBUG - 2,
"candidate disk: approx_pt = %s, mantissa = %s, "
"center = %s, dist = %s, rad = %s", approx_pt, mantissa,
center, dist, rad)
if safe_ge(dist >> 1, rad):
break
expo -= 1
logger.debug("disk for %s: center=%s, rad=%s", pt, center, rad)
# pt may be a ball with nonzero radius: check that it is contained in
# our candidate disk
log = RBF.zero() if 0 in approx_pt else approx_pt.abs().log(2)
F = RealBallField(ZZ((expo - log).max(0).upper().ceil()) + 10)
dist_to_center = (F(approx_pt) - F(center)).abs()
if not safe_le(dist_to_center, rad):
assert not safe_gt((approx_pt.squash() - center).squash(), rad)
logger.info("check that |%s - %s| < %s failed", approx_pt, center,
rad)
return None, None
# exactify center so that subsequent computations are not limited by the
# precision of its parent
center = QQ(center)
return center, rad
def _rad(self, center):
return RBF.one() << QQ(center).valuation(2)
def _path_to(self, dest, prec=None):
r"""
Find a path from a point with known "initial" values to pt
"""
# TODO:
# - attempt to start as close as possible to the destination
# [and perhaps add logic to change for a starting point with exact
# initial values if loosing too much precision]
# - return a path passing through "interesting" points (and cache the
# associated initial vectors)
start, ini = self.ini.items()[0]
return ini, [start, dest]
# Having the update (rather than the full test-and-update) logic in a
# separate method is convenient to override it in subclasses.
def _update_approx(self, center, rad, prec, derivatives):
ini, path = self._path_to(center, prec)
eps = RBF.one() >> prec
# keep="all" won't do anything until _path_to returns better paths
ctx = ancont.Context(self.dop, path, eps, keep="all")
pairs = ancont.analytic_continuation(ctx, ini=ini)
for (vert, val) in pairs:
known = self._inivecs.get(vert)
if known is None or known[0].accuracy() < val[0][0].accuracy():
self._inivecs[vert] = [c[0] for c in val]
logger.info(
"computing new polynomial approximations: "
"ini=%s, path=%s, rad=%s, eps=%s, ord=%s", ini, path, rad, eps,
derivatives)
polys = polapprox.doit(self.dop,
ini=ini,
path=path,
rad=rad,
eps=eps,
derivatives=derivatives,
x_is_real=True,
economization=polapprox.chebyshev_economization)
logger.info("...done")
approx = self._polys.get(center, [])
new_approx = []
for ord, pol in enumerate(polys):
if ord >= len(approx) or approx[ord].prec < prec:
new_approx.append(RealPolApprox(pol, prec))
else:
new_approx.append(approx[ord])
self._update_approx_hook(center, rad, polys)
self._polys[center] = new_approx
return polys
def _sollya_annotate(self, center, rad, polys):
import sagesollya as sollya
logger = logging.getLogger(__name__ + ".sollya")
logger.info("calling annotatefunction() on %s derivatives", len(polys))
center = QQ(center)
sollya_fun = self._sollya_object
# sollya keeps all annotations, let's not bother with selecting
# the best one
for ord, pol0 in enumerate(polys):
pol = ZZ(ord).factorial() * pol0
sollya_pol = sum(
[c.center() * sollya.x**k for k, c in enumerate(pol)])
dom = RIF(center - rad,
center + rad) # XXX: dangerous when inexact
err_pol = pol.map_coefficients(lambda c: c - c.squash())
err = RIF(err_pol(RBF.zero().add_error(rad)))
with sollya.settings(display=sollya.dyadic):
logger = logging.getLogger(__name__ + ".sollya.annotate")
logger.debug("annotatefunction(%s, %s, %s, %s, %s);",
sollya_fun, sollya_pol, sollya.SollyaObject(dom),
sollya.SollyaObject(err),
sollya.SollyaObject(center))
sollya.annotatefunction(sollya_fun, sollya_pol, dom, err, center)
sollya_fun = sollya.diff(sollya_fun)
logger.info("...done")
def _known_bound(self, iv, post_transform):
post_transform = normalize_post_transform(self.dop, post_transform)
Balls = iv.parent()
Ivs = RealIntervalField(Balls.precision())
mid = [
c for c in self._polys.keys()
if Balls(c).add_error(self._rad(c)).overlaps(iv)
]
mid.sort()
rad = [self._rad(c) for c in mid]
crude_bound = (Balls(AnInfinity())
if self._is_everywhere_defined() else Balls('nan'))
if len(mid) < 1 or not mid[0] - rad[0] <= iv <= mid[-1] + rad[-1]:
return crude_bound
if not all(
safe_eq(mid[i] + rad[i], mid[i + 1] - rad[i + 1])
for i in range(len(mid) - 1)):
return crude_bound
bound = None
for c, r in zip(mid, rad):
if len(self._polys[c]) <= post_transform.order():
return crude_bound
polys = [a.pol for a in self._polys[c]]
dom = Balls(Ivs(Balls(c).add_error(r)).intersection(Ivs(iv)))
reduced_dom = dom - c
img = sum(
ZZ(j).factorial() * coeff(dom) * polys[j](reduced_dom)
for j, coeff in enumerate(post_transform))
bound = img if bound is None else bound.union(img)
return bound
def approx(self, pt, prec=None, post_transform=None):
r"""
TESTS::
sage: from ore_algebra import *
sage: from ore_algebra.analytic.function import DFiniteFunction
sage: DiffOps, x, Dx = DifferentialOperators()
sage: h = DFiniteFunction(Dx^3-1, [0, 0, 1])
sage: h.approx(0, post_transform=Dx^2)
[2.0000000000000...]
sage: f = DFiniteFunction((x^2 + 1)*Dx^2 + 2*x*Dx, [0, 1], max_prec=20)
sage: f.approx(1/3, prec=10)
[0.32...]
sage: f.approx(1/3, prec=40)
[0.321750554396...]
sage: f.approx(1/3, prec=10, post_transform=Dx)
[0.9...]
sage: f.approx(1/3, prec=40, post_transform=Dx)
[0.900000000000...]
sage: f.approx(1/3, prec=10, post_transform=Dx^2)
[-0.54...]
sage: f.approx(1/3, prec=40, post_transform=Dx^2)
[-0.540000000000...]
"""
pt = Point(pt, self.dop)
if prec is None:
prec = _guess_prec(pt)
if post_transform is None:
post_transform = self.dop.parent().one()
derivatives = min(post_transform.order() + 1, self._max_derivatives)
post_transform = normalize_post_transform(self.dop, post_transform)
if prec >= self.max_prec or not pt.is_real():
logger.info(
"performing high-prec evaluation "
"(pt=%s, prec=%s, post_transform=%s)", pt, prec,
post_transform)
ini, path = self._path_to(pt)
eps = RBF.one() >> prec
return self.dop.numerical_solution(ini,
path,
eps,
post_transform=post_transform)
center, rad = self._disk(pt)
if center is None:
# raise NotImplementedError
logger.info("falling back on generic evaluator")
ini, path = self._path_to(pt)
eps = RBF.one() >> prec
return self.dop.numerical_solution(ini,
path,
eps,
post_transform=post_transform)
approx = self._polys.get(center, [])
Balls = RealBallField(prec)
# due to the way the polynomials are recomputed, the precisions attached
# to the successive derivatives are nonincreasing
if (len(approx) < derivatives or approx[derivatives - 1].prec < prec):
polys = self._update_approx(center, rad, prec, derivatives)
else:
polys = [a.pol for a in approx]
bpt = Balls(pt.value)
reduced_pt = bpt - Balls(center)
val = sum(
ZZ(j).factorial() * coeff(bpt) * polys[j](reduced_pt)
for j, coeff in enumerate(post_transform))
return val
def __call__(self, x, prec=None):
return self.approx(x, prec=prec)
def plot(self, xrange, **options):
r"""
Plot this function.
The plot is intended to give an idea of the accuracy of the evaluations
that led to it. However, it may not be a rigorous enclosure of the graph
of the function.
EXAMPLES::
sage: from ore_algebra import *
sage: from ore_algebra.analytic.function import DFiniteFunction
sage: DiffOps, x, Dx = DifferentialOperators()
sage: f = DFiniteFunction(Dx^2 - x,
....: [1/(gamma(2/3)*3^(2/3)), -1/(gamma(1/3)*3^(1/3))])
sage: plot(f, (-10, 5), color='black')
Graphics object consisting of 1 graphics primitive
"""
mids = generate_plot_points(lambda x: self.approx(x, 20).mid(),
xrange,
plot_points=200)
ivs = [(x, self.approx(x, 20)) for x, _ in mids]
bounds = [(x, y.upper()) for x, y in ivs]
bounds += [(x, y.lower()) for x, y in reversed(ivs)]
options.setdefault('aspect_ratio', 'automatic')
g = plot.polygon(bounds, thickness=1, **options)
return g
def plot_known(self):
r"""
TESTS::
sage: from ore_algebra import *
sage: from ore_algebra.analytic.function import DFiniteFunction
sage: DiffOps, x, Dx = DifferentialOperators()
sage: f = DFiniteFunction((x^2 + 1)*Dx^2 + 2*x*Dx, [0, 1])
sage: f(-10, 100) # long time
[-1.4711276743037345918528755717...]
sage: f.approx(5, post_transform=Dx) # long time
[0.038461538461538...]
sage: f.plot_known() # long time
Graphics object consisting of ... graphics primitives
"""
g = plot.Graphics()
for center, polys in self._polys.iteritems():
center, rad = self._disk(Point(center, self.dop))
xrange = (center - rad).mid(), (center + rad).mid()
for i, a in enumerate(polys):
# color palette copied from sage.plot.plot.plot
color = plot.Color((0.66666 + i * 0.61803) % 1,
1,
0.4,
space='hsl')
Balls = a.pol.base_ring()
g += plot.plot(lambda x: a.pol(Balls(x)).mid(),
xrange,
color=color)
g += plot.text(str(a.prec), (center, a.pol(center).mid()),
color=color)
for point, ini in self._inivecs.iteritems():
g += plot.point2d((point, 0), size=50)
return g
def _sollya_(self):
r"""
EXAMPLES::
sage: import logging; logging.basicConfig(level=logging.INFO)
sage: logger = logging.getLogger("ore_algebra.analytic.function.sollya")
sage: logger.setLevel(logging.DEBUG)
sage: import ore_algebra
sage: DiffOps, x, Dx = ore_algebra.DifferentialOperators()
sage: from ore_algebra.analytic.function import DFiniteFunction
sage: f = DFiniteFunction(Dx - 1, [1])
sage: import sagesollya as sollya # optional - sollya
sage: sollya.plot(f, sollya.Interval(0, 1)) # not tested
...
"""
if self._sollya_object is not None:
return self._sollya_object
import sagesollya as sollya
logger = logging.getLogger(__name__ + ".sollya.eval")
Dx = self.dop.parent().gen()
def wrapper(pt, ord, prec):
if RBF(pt.diameter()) >= self.max_rad / 4:
return self._known_bound(RBF(pt), post_transform=Dx**ord)
try:
val = self.approx(pt, prec, post_transform=Dx**ord)
except Exception:
logger.info("pt=%s, ord=%s, prec=%s, error",
pt,
ord,
prec,
exc_info=(pt.absolute_diameter() < .5))
return RIF('nan')
logger.debug("pt=%s, ord=%s, prec=%s, val=%s", pt, ord, prec, val)
if not pt.overlaps(self._sollya_domain):
backtrace = sollya.getbacktrace()
logger.debug("%s not in %s", pt.str(style='brackets'),
self._sollya_domain.str(style='brackets'))
logger.debug("sollya backtrace: %s", [
sollya.objectname(t.struct.called_proc) for t in backtrace
])
return val
wrapper.__name__ = self.name
self._sollya_object = sollya.sagefunction(wrapper)
for pt, ini in self.ini.iteritems():
pt = RIF(pt)
if pt.is_exact():
fun = self._sollya_object
for ord, val in enumerate(ini):
try:
sollya.annotatefunction(fun, val, pt, RIF.zero())
except TypeError:
logger.info("annotation failed: D^%s(%s)(%s) = %s",
ord, self.name, pt, val)
fun = sollya.diff(fun)
self._update_approx_hook = self._sollya_annotate
return self._sollya_object
def approx(self, pt, prec=None, post_transform=None):
r"""
TESTS::
sage: from ore_algebra import *
sage: from ore_algebra.analytic.function import DFiniteFunction
sage: DiffOps, x, Dx = DifferentialOperators()
sage: h = DFiniteFunction(Dx^3-1, [0, 0, 1])
sage: h.approx(0, post_transform=Dx^2)
[2.0000000000000...]
sage: f = DFiniteFunction((x^2 + 1)*Dx^2 + 2*x*Dx, [0, 1], max_prec=20)
sage: f.approx(1/3, prec=10)
[0.32...]
sage: f.approx(1/3, prec=40)
[0.321750554396...]
sage: f.approx(1/3, prec=10, post_transform=Dx)
[0.9...]
sage: f.approx(1/3, prec=40, post_transform=Dx)
[0.900000000000...]
sage: f.approx(1/3, prec=10, post_transform=Dx^2)
[-0.54...]
sage: f.approx(1/3, prec=40, post_transform=Dx^2)
[-0.540000000000...]
"""
pt = Point(pt, self.dop)
if prec is None:
prec = _guess_prec(pt)
if post_transform is None:
post_transform = self.dop.parent().one()
derivatives = min(post_transform.order() + 1, self._max_derivatives)
post_transform = normalize_post_transform(self.dop, post_transform)
if prec >= self.max_prec or not pt.is_real():
logger.info(
"performing high-prec evaluation "
"(pt=%s, prec=%s, post_transform=%s)", pt, prec,
post_transform)
ini, path = self._path_to(pt)
eps = RBF.one() >> prec
return self.dop.numerical_solution(ini,
path,
eps,
post_transform=post_transform)
center, rad = self._disk(pt)
if center is None:
# raise NotImplementedError
logger.info("falling back on generic evaluator")
ini, path = self._path_to(pt)
eps = RBF.one() >> prec
return self.dop.numerical_solution(ini,
path,
eps,
post_transform=post_transform)
approx = self._polys.get(center, [])
Balls = RealBallField(prec)
# due to the way the polynomials are recomputed, the precisions attached
# to the successive derivatives are nonincreasing
if (len(approx) < derivatives or approx[derivatives - 1].prec < prec):
polys = self._update_approx(center, rad, prec, derivatives)
else:
polys = [a.pol for a in approx]
bpt = Balls(pt.value)
reduced_pt = bpt - Balls(center)
val = sum(
ZZ(j).factorial() * coeff(bpt) * polys[j](reduced_pt)
for j, coeff in enumerate(post_transform))
return val
def _rad(self, center):
return RBF.one() << QQ(center).valuation(2)
def _monodromy_matrices(dop, base, eps=1e-16, sing=None):
r"""
EXAMPLES::
sage: from ore_algebra import *
sage: from ore_algebra.analytic.monodromy import _monodromy_matrices
sage: Dops, x, Dx = DifferentialOperators()
sage: rat = 1/(x^2-1)
sage: dop = (rat*Dx - rat.derivative()).lclm(Dx*x*Dx)
sage: [rec.point for rec in _monodromy_matrices(dop, 0) if not rec.is_scalar]
[0]
"""
dop = DifferentialOperator(dop)
base = QQbar.coerce(base)
eps = RBF(eps)
if sing is None:
sing = dop._singularities(QQbar)
else:
sing = [QQbar.coerce(s) for s in sing]
todo = {
x: PointWithMonodromyData(x, dop, want_self=True, want_conj=False)
for x in sing
}
base = todo.setdefault(base, PointWithMonodromyData(base, dop))
if not base.is_regular():
raise ValueError("irregular singular base point")
# If the coefficients are rational, reduce to handling singularities in the
# same half-plane as the base point.
need_conjugates = _try_merge_conjugate_singularities(dop, sing, base, todo)
Scalars = ComplexBallField(utilities.prec_from_eps(eps))
id_mat = matrix.identity_matrix(Scalars, dop.order())
def matprod(elts):
return prod(reversed(elts), id_mat)
for key, point in list(todo.items()):
# We could call _local_monodromy_loop() if point is irregular, but
# delaying it may allow us to start returning results earlier.
if point.is_regular():
mon, scalar = _formal_monodromy_naive(dop.shift(point), Scalars)
if scalar:
# No need to compute the connection matrices then!
# XXX When we do need them, though, it would be better to get
# the formal monodromy as a byproduct of their computation.
if point.want_self:
yield LocalMonodromyData(QQbar(point), mon, True)
if point.want_conj:
conj = point.conjugate()
logger.info(
"Computing local monodromy around %s by "
"complex conjugation", conj)
conj_mat = ~mon.conjugate()
yield LocalMonodromyData(QQbar(conj), conj_mat, True)
if point is not base:
del todo[key]
continue
point.local_monodromy = [mon]
point.polygon = [point]
if need_conjugates:
base_conj_mat = dop.numerical_transition_matrix(
[base, base.conjugate()], eps, assume_analytic=True)
def conjugate_monodromy(mat):
return ~base_conj_mat * ~mat.conjugate() * base_conj_mat
tree = _spanning_tree(base, todo.values())
def dfs(x, path, path_mat):
logger.info("Computing local monodromy around %s via %s", x, path)
local_mat = matprod(x.local_monodromy)
based_mat = (~path_mat) * local_mat * path_mat
if x.want_self:
yield LocalMonodromyData(QQbar(x), based_mat, False)
if x.want_conj:
conj = x.conjugate()
logger.info(
"Computing local monodromy around %s by complex "
"conjugation", conj)
conj_mat = conjugate_monodromy(based_mat)
yield LocalMonodromyData(QQbar(x), conj_mat, False)
x.done = True
for y in tree.neighbors(x):
if y.done:
continue
if y.local_monodromy is None:
y.polygon, y.local_monodromy = _local_monodromy_loop(
dop, y, eps)
new_path_mat = _extend_path_mat(dop, path_mat, x, y, eps, matprod)
yield from dfs(y, path + [y], new_path_mat)
yield from dfs(base, [base], id_mat)