def approx_abs_real(self, prec):
r"""
Compute an approximation with absolute error about 2^(-prec).
"""
if isinstance(self.value.parent(), RealBallField):
return self.value
elif self.value.is_zero():
return RealBallField(max(2, prec)).zero()
elif self.is_real():
expo = ZZ(IR(self.value).abs().log(2).upper().ceil())
rel_prec = max(2, prec + expo + 10)
val = RealBallField(rel_prec)(self.value)
return val
else:
raise ValueError("point may not be real")
def _test_fun_approx(pol, ref, disk_rad=None, interval_rad=None,
prec=53, test_count=100):
r"""
EXAMPLES::
sage: from ore_algebra.analytic.polynomial_approximation import _test_fun_approx
sage: _test_fun_approx(lambda x: x.exp(), lambda x: x.exp() + x/1000,
....: interval_rad=1)
Traceback (most recent call last):
...
AssertionError: z = ..., ref(z) = ... not in pol(z) = ...
"""
from sage.rings.real_mpfr import RealField
from sage.rings.real_arb import RealBallField
from sage.rings.complex_arb import ComplexBallField
my_RR = RealField(prec)
my_RBF = RealBallField(prec)
my_CBF = ComplexBallField(prec)
if bool(disk_rad) == bool(interval_rad):
raise ValueError
rad = disk_rad or interval_rad
for _ in range(test_count):
rho = my_RBF(my_RR.random_element(-rad, rad))
if disk_rad:
exp_i_theta = my_CBF(my_RR.random_element(0, 1)).exppii()
z = rho*exp_i_theta
elif interval_rad:
z = rho
ref_z = ref(z)
pol_z = pol(z)
if not ref_z.overlaps(pol_z):
fmt = "z = {}, ref(z) = {} not in pol(z) = {}"
raise AssertionError(fmt.format(z, ref_z, pol_z))
def analytic_continuation(ctx, ini=None, post=None):
"""
INPUT:
- ``ini`` (constant matrix, optional) - initial values, one column per
solution
- ``post`` (matrix of polynomial/rational functions, optional) - linear
combinations of the first Taylor coefficients to take, as a function of
the evaluation point
TESTS::
sage: from ore_algebra import DifferentialOperators
sage: _, x, Dx = DifferentialOperators()
sage: (Dx^2 + 2*x*Dx).numerical_solution([0, 2/sqrt(pi)], [0,i])
[+/- ...] + [1.65042575879754...]*I
"""
logger.info("path: %s", ctx.path)
eps1 = (ctx.eps / (1 + len(ctx.path))) >> 2 # TBI, +: move to ctx?
prec = utilities.prec_from_eps(eps1)
if ini is not None:
if not isinstance(ini, Matrix): # should this be here?
try:
ini = matrix(ctx.dop.order(), 1, list(ini))
except (TypeError, ValueError):
raise ValueError("incorrect initial values: {}".format(ini))
try:
ini = ini.change_ring(RealBallField(prec))
except (TypeError, ValueError):
ini = ini.change_ring(ComplexBallField(prec))
res = []
path_mat = identity_matrix(ZZ, ctx.dop.order())
def store_value_if_wanted(point):
if point.options.get('keep_value'):
value = path_mat
if ini is not None: value = value * ini
if post is not None: value = post(point.value) * value
res.append((point.value, value))
store_value_if_wanted(ctx.path.vert[0])
for step in ctx.path:
step_mat = step_transition_matrix(step, eps1, ctx=ctx)
path_mat = step_mat * path_mat
store_value_if_wanted(step.end)
cm = sage.structure.element.get_coercion_model()
OutputIntervals = cm.common_parent(
utilities.ball_field(ctx.eps, ctx.real()),
*[mat.base_ring() for pt, mat in res])
return [(pt, mat.change_ring(OutputIntervals)) for pt, mat in res]
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, point, dop=None):
"""
TESTS::
sage: from ore_algebra import *
sage: from ore_algebra.analytic.path import Point
sage: Dops, x, Dx = DifferentialOperators()
sage: [Point(z, Dx)
....: for z in [1, 1/2, 1+I, QQbar(I), RIF(1/3), CIF(1/3), pi,
....: RDF(1), CDF(I), 0.5r, 0.5jr, 10r, QQbar(1), AA(1/3)]]
[1, 1/2, I + 1, I, [0.333333333333333...], [0.333333333333333...],
3.141592653589794?, 1.000000000000000, 1.000000000000000*I,
0.5000000000000000, 0.5000000000000000*I, 10, 1, 1/3]
sage: Point(sqrt(2), Dx).iv()
[1.414...]
"""
SageObject.__init__(self)
from sage.rings.complex_double import ComplexDoubleField_class
from sage.rings.complex_field import ComplexField_class
from sage.rings.complex_interval_field import ComplexIntervalField_class
from sage.rings.real_double import RealDoubleField_class
from sage.rings.real_mpfi import RealIntervalField_class
from sage.rings.real_mpfr import RealField_class
point = sage.structure.coerce.py_scalar_to_element(point)
try:
parent = point.parent()
except AttributeError:
raise TypeError("unexpected value for point: " + repr(point))
if isinstance(point, Point):
self.value = point.value
elif isinstance(
parent,
(number_field_base.NumberField, RealBallField, ComplexBallField)):
self.value = point
elif QQ.has_coerce_map_from(parent):
self.value = QQ.coerce(point)
# must come before QQbar, due to a bogus coerce map (#14485)
elif parent is sage.symbolic.ring.SR:
try:
return self.__init__(point.pyobject(), dop)
except TypeError:
pass
try:
return self.__init__(QQbar(point), dop)
except (TypeError, ValueError, NotImplementedError):
pass
try:
self.value = RLF(point)
except (TypeError, ValueError):
self.value = CLF(point)
elif QQbar.has_coerce_map_from(parent):
alg = QQbar.coerce(point)
NF, val, hom = alg.as_number_field_element()
if NF is QQ:
self.value = QQ.coerce(val) # parent may be ZZ
else:
embNF = number_field.NumberField(NF.polynomial(),
NF.variable_name(),
embedding=hom(NF.gen()))
self.value = val.polynomial()(embNF.gen())
elif isinstance(
parent,
(RealField_class, RealDoubleField_class, RealIntervalField_class)):
self.value = RealBallField(point.prec())(point)
elif isinstance(parent, (ComplexField_class, ComplexDoubleField_class,
ComplexIntervalField_class)):
self.value = ComplexBallField(point.prec())(point)
else:
try:
self.value = RLF.coerce(point)
except TypeError:
self.value = CLF.coerce(point)
parent = self.value.parent()
assert (isinstance(
parent,
(number_field_base.NumberField, RealBallField, ComplexBallField))
or parent is RLF or parent is CLF)
self.dop = dop or point.dop
self.keep_value = False
def analytic_continuation(dop, path, eps, ctx=dctx, ini=None, post=None,
return_local_bases=False):
"""
INPUT:
- ``ini`` (constant matrix, optional) - initial values, one column per
solution
- ``post`` (matrix of polynomial/rational functions, optional) - linear
combinations of the first Taylor coefficients to take, as a function of
the evaluation point
- ``return_local_bases`` (boolean) - if True, also compute and return the
structure of local bases at all points where we are computing values of
the solution
OUTPUT:
A list of dictionaries with information on the computed solution(s) at each
evaluation point.
TESTS::
sage: from ore_algebra import DifferentialOperators
sage: _, x, Dx = DifferentialOperators()
sage: (Dx^2 + 2*x*Dx).numerical_solution([0, 2/sqrt(pi)], [0,i])
[+/- ...] + [1.65042575879754...]*I
"""
if dop.is_zero():
raise ValueError("operator must be nonzero")
_, _, _, dop = dop._normalize_base_ring()
path = _process_path(dop, path, ctx)
logger.info("path: %s", path)
eps = bounds.IR(eps)
eps1 = (eps/(1 + len(path))) >> 2
prec = utilities.prec_from_eps(eps1)
if ini is not None:
if not isinstance(ini, Matrix): # should this be here?
try:
ini = matrix(dop.order(), 1, list(ini))
except (TypeError, ValueError):
raise ValueError("incorrect initial values: {}".format(ini))
try:
ini = ini.change_ring(RealBallField(prec))
except (TypeError, ValueError):
ini = ini.change_ring(ComplexBallField(prec))
def point_dict(point, value):
if ini is not None:
value = value*ini
if post is not None and not post.is_one():
value = post(point.value)*value
rec = {"point": point.value, "value": value}
if return_local_bases:
rec["structure"] = point.local_basis_structure()
return rec
res = []
z0 = path.vert[0]
# XXX still imperfect in the case of a high-precision starting point with
# relatively large radius... (do we care?)
main = Step(z0, z0.simple_approx(ctx=ctx))
path_mat = step_transition_matrix(dop, main, eps1, ctx=ctx)
if z0.keep_value():
res.append(point_dict(z0, identity_matrix(ZZ, dop.order())))
for step in path:
main, dev = step.chain_simple(main.end, ctx=ctx)
main_mat = step_transition_matrix(dop, main, eps1, ctx=ctx)
path_mat = main_mat*path_mat
if dev is not None:
dev_mat = path_mat
for sub in dev:
sub_mat = step_transition_matrix(dop, sub, eps1, ctx=ctx)
dev_mat = sub_mat*dev_mat
res.append(point_dict(step.end, dev_mat))
cm = sage.structure.element.get_coercion_model()
real = (rings.RIF.has_coerce_map_from(dop.base_ring().base_ring())
and all(v.is_real() for v in path.vert))
OutputIntervals = cm.common_parent(
utilities.ball_field(eps, real),
*[rec["value"].base_ring() for rec in res])
for rec in res:
rec["value"] = rec["value"].change_ring(OutputIntervals)
return res
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 __init__(self, point, dop=None, singular=None, **kwds):
"""
INPUT:
- ``singular``: can be set to True to force this point to be considered
a singular point, even if this cannot be checked (e.g. because we only
have an enclosure)
TESTS::
sage: from ore_algebra import *
sage: from ore_algebra.analytic.path import Point
sage: Dops, x, Dx = DifferentialOperators()
sage: [Point(z, Dx)
....: for z in [1, 1/2, 1+I, QQbar(I), RIF(1/3), CIF(1/3), pi,
....: RDF(1), CDF(I), 0.5r, 0.5jr, 10r, QQbar(1), AA(1/3)]]
[1, 1/2, I + 1, I, [0.333333333333333...], [0.333333333333333...],
3.141592653589794?, ~1.0000, ~1.0000*I, ~0.50000, ~0.50000*I, 10,
1, 1/3]
sage: Point(sqrt(2), Dx).iv()
[1.414...]
sage: Point(RBF(0), (x-1)*x*Dx, singular=True).dist_to_sing()
1.000000000000000
"""
SageObject.__init__(self)
from sage.rings.complex_double import ComplexDoubleField_class
from sage.rings.complex_field import ComplexField_class
from sage.rings.complex_interval_field import ComplexIntervalField_class
from sage.rings.real_double import RealDoubleField_class
from sage.rings.real_mpfi import RealIntervalField_class
from sage.rings.real_mpfr import RealField_class
point = sage.structure.coerce.py_scalar_to_element(point)
try:
parent = point.parent()
except AttributeError:
raise TypeError("unexpected value for point: " + repr(point))
if isinstance(point, Point):
self.value = point.value
elif isinstance(parent, (RealBallField, ComplexBallField)):
self.value = point
elif isinstance(parent, number_field_base.NumberField):
_, hom = good_number_field(point.parent())
self.value = hom(point)
elif QQ.has_coerce_map_from(parent):
self.value = QQ.coerce(point)
elif QQbar.has_coerce_map_from(parent):
alg = QQbar.coerce(point)
NF, val, hom = alg.as_number_field_element()
if NF is QQ:
self.value = QQ.coerce(val) # parent may be ZZ
else:
embNF = number_field.NumberField(NF.polynomial(),
NF.variable_name(),
embedding=hom(NF.gen()))
self.value = val.polynomial()(embNF.gen())
elif isinstance(
parent,
(RealField_class, RealDoubleField_class, RealIntervalField_class)):
self.value = RealBallField(point.prec())(point)
elif isinstance(parent, (ComplexField_class, ComplexDoubleField_class,
ComplexIntervalField_class)):
self.value = ComplexBallField(point.prec())(point)
elif parent is sage.symbolic.ring.SR:
try:
return self.__init__(point.pyobject(), dop)
except TypeError:
pass
try:
return self.__init__(QQbar(point), dop)
except (TypeError, ValueError, NotImplementedError):
pass
try:
self.value = RLF(point)
except (TypeError, ValueError):
self.value = CLF(point)
else:
try:
self.value = RLF.coerce(point)
except TypeError:
self.value = CLF.coerce(point)
parent = self.value.parent()
assert (isinstance(
parent,
(number_field_base.NumberField, RealBallField, ComplexBallField))
or parent is RLF or parent is CLF)
if dop is None: # TBI
if isinstance(point, Point):
self.dop = point.dop
else:
self.dop = DifferentialOperator(dop.numerator())
self._force_singular = bool(singular)
self.options = kwds