657 lines
21 KiB
Python
657 lines
21 KiB
Python
|
import sys
|
||
|
|
import copy
|
||
|
|
import heapq
|
||
|
|
import collections
|
||
|
|
import functools
|
||
|
|
|
||
|
|
import numpy as np
|
||
|
|
|
||
|
|
from scipy._lib._util import MapWrapper, _FunctionWrapper
|
||
|
|
|
||
|
|
|
||
|
|
class LRUDict(collections.OrderedDict):
|
||
|
|
def __init__(self, max_size):
|
||
|
|
self.__max_size = max_size
|
||
|
|
|
||
|
|
def __setitem__(self, key, value):
|
||
|
|
existing_key = (key in self)
|
||
|
|
super().__setitem__(key, value)
|
||
|
|
if existing_key:
|
||
|
|
self.move_to_end(key)
|
||
|
|
elif len(self) > self.__max_size:
|
||
|
|
self.popitem(last=False)
|
||
|
|
|
||
|
|
def update(self, other):
|
||
|
|
# Not needed below
|
||
|
|
raise NotImplementedError()
|
||
|
|
|
||
|
|
|
||
|
|
class SemiInfiniteFunc:
|
||
|
|
"""
|
||
|
|
Argument transform from (start, +-oo) to (0, 1)
|
||
|
|
"""
|
||
|
|
def __init__(self, func, start, infty):
|
||
|
|
self._func = func
|
||
|
|
self._start = start
|
||
|
|
self._sgn = -1 if infty < 0 else 1
|
||
|
|
|
||
|
|
# Overflow threshold for the 1/t**2 factor
|
||
|
|
self._tmin = sys.float_info.min**0.5
|
||
|
|
|
||
|
|
def get_t(self, x):
|
||
|
|
z = self._sgn * (x - self._start) + 1
|
||
|
|
if z == 0:
|
||
|
|
# Can happen only if point not in range
|
||
|
|
return np.inf
|
||
|
|
return 1 / z
|
||
|
|
|
||
|
|
def __call__(self, t):
|
||
|
|
if t < self._tmin:
|
||
|
|
return 0.0
|
||
|
|
else:
|
||
|
|
x = self._start + self._sgn * (1 - t) / t
|
||
|
|
f = self._func(x)
|
||
|
|
return self._sgn * (f / t) / t
|
||
|
|
|
||
|
|
|
||
|
|
class DoubleInfiniteFunc:
|
||
|
|
"""
|
||
|
|
Argument transform from (-oo, oo) to (-1, 1)
|
||
|
|
"""
|
||
|
|
def __init__(self, func):
|
||
|
|
self._func = func
|
||
|
|
|
||
|
|
# Overflow threshold for the 1/t**2 factor
|
||
|
|
self._tmin = sys.float_info.min**0.5
|
||
|
|
|
||
|
|
def get_t(self, x):
|
||
|
|
s = -1 if x < 0 else 1
|
||
|
|
return s / (abs(x) + 1)
|
||
|
|
|
||
|
|
def __call__(self, t):
|
||
|
|
if abs(t) < self._tmin:
|
||
|
|
return 0.0
|
||
|
|
else:
|
||
|
|
x = (1 - abs(t)) / t
|
||
|
|
f = self._func(x)
|
||
|
|
return (f / t) / t
|
||
|
|
|
||
|
|
|
||
|
|
def _max_norm(x):
|
||
|
|
return np.amax(abs(x))
|
||
|
|
|
||
|
|
|
||
|
|
def _get_sizeof(obj):
|
||
|
|
try:
|
||
|
|
return sys.getsizeof(obj)
|
||
|
|
except TypeError:
|
||
|
|
# occurs on pypy
|
||
|
|
if hasattr(obj, '__sizeof__'):
|
||
|
|
return int(obj.__sizeof__())
|
||
|
|
return 64
|
||
|
|
|
||
|
|
|
||
|
|
class _Bunch:
|
||
|
|
def __init__(self, **kwargs):
|
||
|
|
self.__keys = kwargs.keys()
|
||
|
|
self.__dict__.update(**kwargs)
|
||
|
|
|
||
|
|
def __repr__(self):
|
||
|
|
return "_Bunch({})".format(", ".join(f"{k}={repr(self.__dict__[k])}"
|
||
|
|
for k in self.__keys))
|
||
|
|
|
||
|
|
|
||
|
|
def quad_vec(f, a, b, epsabs=1e-200, epsrel=1e-8, norm='2', cache_size=100e6,
|
||
|
|
limit=10000, workers=1, points=None, quadrature=None, full_output=False,
|
||
|
|
*, args=()):
|
||
|
|
r"""Adaptive integration of a vector-valued function.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
f : callable
|
||
|
|
Vector-valued function f(x) to integrate.
|
||
|
|
a : float
|
||
|
|
Initial point.
|
||
|
|
b : float
|
||
|
|
Final point.
|
||
|
|
epsabs : float, optional
|
||
|
|
Absolute tolerance.
|
||
|
|
epsrel : float, optional
|
||
|
|
Relative tolerance.
|
||
|
|
norm : {'max', '2'}, optional
|
||
|
|
Vector norm to use for error estimation.
|
||
|
|
cache_size : int, optional
|
||
|
|
Number of bytes to use for memoization.
|
||
|
|
limit : float or int, optional
|
||
|
|
An upper bound on the number of subintervals used in the adaptive
|
||
|
|
algorithm.
|
||
|
|
workers : int or map-like callable, optional
|
||
|
|
If `workers` is an integer, part of the computation is done in
|
||
|
|
parallel subdivided to this many tasks (using
|
||
|
|
:class:`python:multiprocessing.pool.Pool`).
|
||
|
|
Supply `-1` to use all cores available to the Process.
|
||
|
|
Alternatively, supply a map-like callable, such as
|
||
|
|
:meth:`python:multiprocessing.pool.Pool.map` for evaluating the
|
||
|
|
population in parallel.
|
||
|
|
This evaluation is carried out as ``workers(func, iterable)``.
|
||
|
|
points : list, optional
|
||
|
|
List of additional breakpoints.
|
||
|
|
quadrature : {'gk21', 'gk15', 'trapezoid'}, optional
|
||
|
|
Quadrature rule to use on subintervals.
|
||
|
|
Options: 'gk21' (Gauss-Kronrod 21-point rule),
|
||
|
|
'gk15' (Gauss-Kronrod 15-point rule),
|
||
|
|
'trapezoid' (composite trapezoid rule).
|
||
|
|
Default: 'gk21' for finite intervals and 'gk15' for (semi-)infinite
|
||
|
|
full_output : bool, optional
|
||
|
|
Return an additional ``info`` dictionary.
|
||
|
|
args : tuple, optional
|
||
|
|
Extra arguments to pass to function, if any.
|
||
|
|
|
||
|
|
.. versionadded:: 1.8.0
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
res : {float, array-like}
|
||
|
|
Estimate for the result
|
||
|
|
err : float
|
||
|
|
Error estimate for the result in the given norm
|
||
|
|
info : dict
|
||
|
|
Returned only when ``full_output=True``.
|
||
|
|
Info dictionary. Is an object with the attributes:
|
||
|
|
|
||
|
|
success : bool
|
||
|
|
Whether integration reached target precision.
|
||
|
|
status : int
|
||
|
|
Indicator for convergence, success (0),
|
||
|
|
failure (1), and failure due to rounding error (2).
|
||
|
|
neval : int
|
||
|
|
Number of function evaluations.
|
||
|
|
intervals : ndarray, shape (num_intervals, 2)
|
||
|
|
Start and end points of subdivision intervals.
|
||
|
|
integrals : ndarray, shape (num_intervals, ...)
|
||
|
|
Integral for each interval.
|
||
|
|
Note that at most ``cache_size`` values are recorded,
|
||
|
|
and the array may contains *nan* for missing items.
|
||
|
|
errors : ndarray, shape (num_intervals,)
|
||
|
|
Estimated integration error for each interval.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
The algorithm mainly follows the implementation of QUADPACK's
|
||
|
|
DQAG* algorithms, implementing global error control and adaptive
|
||
|
|
subdivision.
|
||
|
|
|
||
|
|
The algorithm here has some differences to the QUADPACK approach:
|
||
|
|
|
||
|
|
Instead of subdividing one interval at a time, the algorithm
|
||
|
|
subdivides N intervals with largest errors at once. This enables
|
||
|
|
(partial) parallelization of the integration.
|
||
|
|
|
||
|
|
The logic of subdividing "next largest" intervals first is then
|
||
|
|
not implemented, and we rely on the above extension to avoid
|
||
|
|
concentrating on "small" intervals only.
|
||
|
|
|
||
|
|
The Wynn epsilon table extrapolation is not used (QUADPACK uses it
|
||
|
|
for infinite intervals). This is because the algorithm here is
|
||
|
|
supposed to work on vector-valued functions, in an user-specified
|
||
|
|
norm, and the extension of the epsilon algorithm to this case does
|
||
|
|
not appear to be widely agreed. For max-norm, using elementwise
|
||
|
|
Wynn epsilon could be possible, but we do not do this here with
|
||
|
|
the hope that the epsilon extrapolation is mainly useful in
|
||
|
|
special cases.
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
[1] R. Piessens, E. de Doncker, QUADPACK (1983).
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
We can compute integrations of a vector-valued function:
|
||
|
|
|
||
|
|
>>> from scipy.integrate import quad_vec
|
||
|
|
>>> import numpy as np
|
||
|
|
>>> import matplotlib.pyplot as plt
|
||
|
|
>>> alpha = np.linspace(0.0, 2.0, num=30)
|
||
|
|
>>> f = lambda x: x**alpha
|
||
|
|
>>> x0, x1 = 0, 2
|
||
|
|
>>> y, err = quad_vec(f, x0, x1)
|
||
|
|
>>> plt.plot(alpha, y)
|
||
|
|
>>> plt.xlabel(r"$\alpha$")
|
||
|
|
>>> plt.ylabel(r"$\int_{0}^{2} x^\alpha dx$")
|
||
|
|
>>> plt.show()
|
||
|
|
|
||
|
|
"""
|
||
|
|
a = float(a)
|
||
|
|
b = float(b)
|
||
|
|
|
||
|
|
if args:
|
||
|
|
if not isinstance(args, tuple):
|
||
|
|
args = (args,)
|
||
|
|
|
||
|
|
# create a wrapped function to allow the use of map and Pool.map
|
||
|
|
f = _FunctionWrapper(f, args)
|
||
|
|
|
||
|
|
# Use simple transformations to deal with integrals over infinite
|
||
|
|
# intervals.
|
||
|
|
kwargs = dict(epsabs=epsabs,
|
||
|
|
epsrel=epsrel,
|
||
|
|
norm=norm,
|
||
|
|
cache_size=cache_size,
|
||
|
|
limit=limit,
|
||
|
|
workers=workers,
|
||
|
|
points=points,
|
||
|
|
quadrature='gk15' if quadrature is None else quadrature,
|
||
|
|
full_output=full_output)
|
||
|
|
if np.isfinite(a) and np.isinf(b):
|
||
|
|
f2 = SemiInfiniteFunc(f, start=a, infty=b)
|
||
|
|
if points is not None:
|
||
|
|
kwargs['points'] = tuple(f2.get_t(xp) for xp in points)
|
||
|
|
return quad_vec(f2, 0, 1, **kwargs)
|
||
|
|
elif np.isfinite(b) and np.isinf(a):
|
||
|
|
f2 = SemiInfiniteFunc(f, start=b, infty=a)
|
||
|
|
if points is not None:
|
||
|
|
kwargs['points'] = tuple(f2.get_t(xp) for xp in points)
|
||
|
|
res = quad_vec(f2, 0, 1, **kwargs)
|
||
|
|
return (-res[0],) + res[1:]
|
||
|
|
elif np.isinf(a) and np.isinf(b):
|
||
|
|
sgn = -1 if b < a else 1
|
||
|
|
|
||
|
|
# NB. explicitly split integral at t=0, which separates
|
||
|
|
# the positive and negative sides
|
||
|
|
f2 = DoubleInfiniteFunc(f)
|
||
|
|
if points is not None:
|
||
|
|
kwargs['points'] = (0,) + tuple(f2.get_t(xp) for xp in points)
|
||
|
|
else:
|
||
|
|
kwargs['points'] = (0,)
|
||
|
|
|
||
|
|
if a != b:
|
||
|
|
res = quad_vec(f2, -1, 1, **kwargs)
|
||
|
|
else:
|
||
|
|
res = quad_vec(f2, 1, 1, **kwargs)
|
||
|
|
|
||
|
|
return (res[0]*sgn,) + res[1:]
|
||
|
|
elif not (np.isfinite(a) and np.isfinite(b)):
|
||
|
|
raise ValueError(f"invalid integration bounds a={a}, b={b}")
|
||
|
|
|
||
|
|
norm_funcs = {
|
||
|
|
None: _max_norm,
|
||
|
|
'max': _max_norm,
|
||
|
|
'2': np.linalg.norm
|
||
|
|
}
|
||
|
|
if callable(norm):
|
||
|
|
norm_func = norm
|
||
|
|
else:
|
||
|
|
norm_func = norm_funcs[norm]
|
||
|
|
|
||
|
|
parallel_count = 128
|
||
|
|
min_intervals = 2
|
||
|
|
|
||
|
|
try:
|
||
|
|
_quadrature = {None: _quadrature_gk21,
|
||
|
|
'gk21': _quadrature_gk21,
|
||
|
|
'gk15': _quadrature_gk15,
|
||
|
|
'trapz': _quadrature_trapezoid, # alias for backcompat
|
||
|
|
'trapezoid': _quadrature_trapezoid}[quadrature]
|
||
|
|
except KeyError as e:
|
||
|
|
raise ValueError(f"unknown quadrature {quadrature!r}") from e
|
||
|
|
|
||
|
|
# Initial interval set
|
||
|
|
if points is None:
|
||
|
|
initial_intervals = [(a, b)]
|
||
|
|
else:
|
||
|
|
prev = a
|
||
|
|
initial_intervals = []
|
||
|
|
for p in sorted(points):
|
||
|
|
p = float(p)
|
||
|
|
if not (a < p < b) or p == prev:
|
||
|
|
continue
|
||
|
|
initial_intervals.append((prev, p))
|
||
|
|
prev = p
|
||
|
|
initial_intervals.append((prev, b))
|
||
|
|
|
||
|
|
global_integral = None
|
||
|
|
global_error = None
|
||
|
|
rounding_error = None
|
||
|
|
interval_cache = None
|
||
|
|
intervals = []
|
||
|
|
neval = 0
|
||
|
|
|
||
|
|
for x1, x2 in initial_intervals:
|
||
|
|
ig, err, rnd = _quadrature(x1, x2, f, norm_func)
|
||
|
|
neval += _quadrature.num_eval
|
||
|
|
|
||
|
|
if global_integral is None:
|
||
|
|
if isinstance(ig, (float, complex)):
|
||
|
|
# Specialize for scalars
|
||
|
|
if norm_func in (_max_norm, np.linalg.norm):
|
||
|
|
norm_func = abs
|
||
|
|
|
||
|
|
global_integral = ig
|
||
|
|
global_error = float(err)
|
||
|
|
rounding_error = float(rnd)
|
||
|
|
|
||
|
|
cache_count = cache_size // _get_sizeof(ig)
|
||
|
|
interval_cache = LRUDict(cache_count)
|
||
|
|
else:
|
||
|
|
global_integral += ig
|
||
|
|
global_error += err
|
||
|
|
rounding_error += rnd
|
||
|
|
|
||
|
|
interval_cache[(x1, x2)] = copy.copy(ig)
|
||
|
|
intervals.append((-err, x1, x2))
|
||
|
|
|
||
|
|
heapq.heapify(intervals)
|
||
|
|
|
||
|
|
CONVERGED = 0
|
||
|
|
NOT_CONVERGED = 1
|
||
|
|
ROUNDING_ERROR = 2
|
||
|
|
NOT_A_NUMBER = 3
|
||
|
|
|
||
|
|
status_msg = {
|
||
|
|
CONVERGED: "Target precision reached.",
|
||
|
|
NOT_CONVERGED: "Target precision not reached.",
|
||
|
|
ROUNDING_ERROR: "Target precision could not be reached due to rounding error.",
|
||
|
|
NOT_A_NUMBER: "Non-finite values encountered."
|
||
|
|
}
|
||
|
|
|
||
|
|
# Process intervals
|
||
|
|
with MapWrapper(workers) as mapwrapper:
|
||
|
|
ier = NOT_CONVERGED
|
||
|
|
|
||
|
|
while intervals and len(intervals) < limit:
|
||
|
|
# Select intervals with largest errors for subdivision
|
||
|
|
tol = max(epsabs, epsrel*norm_func(global_integral))
|
||
|
|
|
||
|
|
to_process = []
|
||
|
|
err_sum = 0
|
||
|
|
|
||
|
|
for j in range(parallel_count):
|
||
|
|
if not intervals:
|
||
|
|
break
|
||
|
|
|
||
|
|
if j > 0 and err_sum > global_error - tol/8:
|
||
|
|
# avoid unnecessary parallel splitting
|
||
|
|
break
|
||
|
|
|
||
|
|
interval = heapq.heappop(intervals)
|
||
|
|
|
||
|
|
neg_old_err, a, b = interval
|
||
|
|
old_int = interval_cache.pop((a, b), None)
|
||
|
|
to_process.append(
|
||
|
|
((-neg_old_err, a, b, old_int), f, norm_func, _quadrature)
|
||
|
|
)
|
||
|
|
err_sum += -neg_old_err
|
||
|
|
|
||
|
|
# Subdivide intervals
|
||
|
|
for parts in mapwrapper(_subdivide_interval, to_process):
|
||
|
|
dint, derr, dround_err, subint, dneval = parts
|
||
|
|
neval += dneval
|
||
|
|
global_integral += dint
|
||
|
|
global_error += derr
|
||
|
|
rounding_error += dround_err
|
||
|
|
for x in subint:
|
||
|
|
x1, x2, ig, err = x
|
||
|
|
interval_cache[(x1, x2)] = ig
|
||
|
|
heapq.heappush(intervals, (-err, x1, x2))
|
||
|
|
|
||
|
|
# Termination check
|
||
|
|
if len(intervals) >= min_intervals:
|
||
|
|
tol = max(epsabs, epsrel*norm_func(global_integral))
|
||
|
|
if global_error < tol/8:
|
||
|
|
ier = CONVERGED
|
||
|
|
break
|
||
|
|
if global_error < rounding_error:
|
||
|
|
ier = ROUNDING_ERROR
|
||
|
|
break
|
||
|
|
|
||
|
|
if not (np.isfinite(global_error) and np.isfinite(rounding_error)):
|
||
|
|
ier = NOT_A_NUMBER
|
||
|
|
break
|
||
|
|
|
||
|
|
res = global_integral
|
||
|
|
err = global_error + rounding_error
|
||
|
|
|
||
|
|
if full_output:
|
||
|
|
res_arr = np.asarray(res)
|
||
|
|
dummy = np.full(res_arr.shape, np.nan, dtype=res_arr.dtype)
|
||
|
|
integrals = np.array([interval_cache.get((z[1], z[2]), dummy)
|
||
|
|
for z in intervals], dtype=res_arr.dtype)
|
||
|
|
errors = np.array([-z[0] for z in intervals])
|
||
|
|
intervals = np.array([[z[1], z[2]] for z in intervals])
|
||
|
|
|
||
|
|
info = _Bunch(neval=neval,
|
||
|
|
success=(ier == CONVERGED),
|
||
|
|
status=ier,
|
||
|
|
message=status_msg[ier],
|
||
|
|
intervals=intervals,
|
||
|
|
integrals=integrals,
|
||
|
|
errors=errors)
|
||
|
|
return (res, err, info)
|
||
|
|
else:
|
||
|
|
return (res, err)
|
||
|
|
|
||
|
|
|
||
|
|
def _subdivide_interval(args):
|
||
|
|
interval, f, norm_func, _quadrature = args
|
||
|
|
old_err, a, b, old_int = interval
|
||
|
|
|
||
|
|
c = 0.5 * (a + b)
|
||
|
|
|
||
|
|
# Left-hand side
|
||
|
|
if getattr(_quadrature, 'cache_size', 0) > 0:
|
||
|
|
f = functools.lru_cache(_quadrature.cache_size)(f)
|
||
|
|
|
||
|
|
s1, err1, round1 = _quadrature(a, c, f, norm_func)
|
||
|
|
dneval = _quadrature.num_eval
|
||
|
|
s2, err2, round2 = _quadrature(c, b, f, norm_func)
|
||
|
|
dneval += _quadrature.num_eval
|
||
|
|
if old_int is None:
|
||
|
|
old_int, _, _ = _quadrature(a, b, f, norm_func)
|
||
|
|
dneval += _quadrature.num_eval
|
||
|
|
|
||
|
|
if getattr(_quadrature, 'cache_size', 0) > 0:
|
||
|
|
dneval = f.cache_info().misses
|
||
|
|
|
||
|
|
dint = s1 + s2 - old_int
|
||
|
|
derr = err1 + err2 - old_err
|
||
|
|
dround_err = round1 + round2
|
||
|
|
|
||
|
|
subintervals = ((a, c, s1, err1), (c, b, s2, err2))
|
||
|
|
return dint, derr, dround_err, subintervals, dneval
|
||
|
|
|
||
|
|
|
||
|
|
def _quadrature_trapezoid(x1, x2, f, norm_func):
|
||
|
|
"""
|
||
|
|
Composite trapezoid quadrature
|
||
|
|
"""
|
||
|
|
x3 = 0.5*(x1 + x2)
|
||
|
|
f1 = f(x1)
|
||
|
|
f2 = f(x2)
|
||
|
|
f3 = f(x3)
|
||
|
|
|
||
|
|
s2 = 0.25 * (x2 - x1) * (f1 + 2*f3 + f2)
|
||
|
|
|
||
|
|
round_err = 0.25 * abs(x2 - x1) * (float(norm_func(f1))
|
||
|
|
+ 2*float(norm_func(f3))
|
||
|
|
+ float(norm_func(f2))) * 2e-16
|
||
|
|
|
||
|
|
s1 = 0.5 * (x2 - x1) * (f1 + f2)
|
||
|
|
err = 1/3 * float(norm_func(s1 - s2))
|
||
|
|
return s2, err, round_err
|
||
|
|
|
||
|
|
|
||
|
|
_quadrature_trapezoid.cache_size = 3 * 3
|
||
|
|
_quadrature_trapezoid.num_eval = 3
|
||
|
|
|
||
|
|
|
||
|
|
def _quadrature_gk(a, b, f, norm_func, x, w, v):
|
||
|
|
"""
|
||
|
|
Generic Gauss-Kronrod quadrature
|
||
|
|
"""
|
||
|
|
|
||
|
|
fv = [0.0]*len(x)
|
||
|
|
|
||
|
|
c = 0.5 * (a + b)
|
||
|
|
h = 0.5 * (b - a)
|
||
|
|
|
||
|
|
# Gauss-Kronrod
|
||
|
|
s_k = 0.0
|
||
|
|
s_k_abs = 0.0
|
||
|
|
for i in range(len(x)):
|
||
|
|
ff = f(c + h*x[i])
|
||
|
|
fv[i] = ff
|
||
|
|
|
||
|
|
vv = v[i]
|
||
|
|
|
||
|
|
# \int f(x)
|
||
|
|
s_k += vv * ff
|
||
|
|
# \int |f(x)|
|
||
|
|
s_k_abs += vv * abs(ff)
|
||
|
|
|
||
|
|
# Gauss
|
||
|
|
s_g = 0.0
|
||
|
|
for i in range(len(w)):
|
||
|
|
s_g += w[i] * fv[2*i + 1]
|
||
|
|
|
||
|
|
# Quadrature of abs-deviation from average
|
||
|
|
s_k_dabs = 0.0
|
||
|
|
y0 = s_k / 2.0
|
||
|
|
for i in range(len(x)):
|
||
|
|
# \int |f(x) - y0|
|
||
|
|
s_k_dabs += v[i] * abs(fv[i] - y0)
|
||
|
|
|
||
|
|
# Use similar error estimation as quadpack
|
||
|
|
err = float(norm_func((s_k - s_g) * h))
|
||
|
|
dabs = float(norm_func(s_k_dabs * h))
|
||
|
|
if dabs != 0 and err != 0:
|
||
|
|
err = dabs * min(1.0, (200 * err / dabs)**1.5)
|
||
|
|
|
||
|
|
eps = sys.float_info.epsilon
|
||
|
|
round_err = float(norm_func(50 * eps * h * s_k_abs))
|
||
|
|
|
||
|
|
if round_err > sys.float_info.min:
|
||
|
|
err = max(err, round_err)
|
||
|
|
|
||
|
|
return h * s_k, err, round_err
|
||
|
|
|
||
|
|
|
||
|
|
def _quadrature_gk21(a, b, f, norm_func):
|
||
|
|
"""
|
||
|
|
Gauss-Kronrod 21 quadrature with error estimate
|
||
|
|
"""
|
||
|
|
# Gauss-Kronrod points
|
||
|
|
x = (0.995657163025808080735527280689003,
|
||
|
|
0.973906528517171720077964012084452,
|
||
|
|
0.930157491355708226001207180059508,
|
||
|
|
0.865063366688984510732096688423493,
|
||
|
|
0.780817726586416897063717578345042,
|
||
|
|
0.679409568299024406234327365114874,
|
||
|
|
0.562757134668604683339000099272694,
|
||
|
|
0.433395394129247190799265943165784,
|
||
|
|
0.294392862701460198131126603103866,
|
||
|
|
0.148874338981631210884826001129720,
|
||
|
|
0,
|
||
|
|
-0.148874338981631210884826001129720,
|
||
|
|
-0.294392862701460198131126603103866,
|
||
|
|
-0.433395394129247190799265943165784,
|
||
|
|
-0.562757134668604683339000099272694,
|
||
|
|
-0.679409568299024406234327365114874,
|
||
|
|
-0.780817726586416897063717578345042,
|
||
|
|
-0.865063366688984510732096688423493,
|
||
|
|
-0.930157491355708226001207180059508,
|
||
|
|
-0.973906528517171720077964012084452,
|
||
|
|
-0.995657163025808080735527280689003)
|
||
|
|
|
||
|
|
# 10-point weights
|
||
|
|
w = (0.066671344308688137593568809893332,
|
||
|
|
0.149451349150580593145776339657697,
|
||
|
|
0.219086362515982043995534934228163,
|
||
|
|
0.269266719309996355091226921569469,
|
||
|
|
0.295524224714752870173892994651338,
|
||
|
|
0.295524224714752870173892994651338,
|
||
|
|
0.269266719309996355091226921569469,
|
||
|
|
0.219086362515982043995534934228163,
|
||
|
|
0.149451349150580593145776339657697,
|
||
|
|
0.066671344308688137593568809893332)
|
||
|
|
|
||
|
|
# 21-point weights
|
||
|
|
v = (0.011694638867371874278064396062192,
|
||
|
|
0.032558162307964727478818972459390,
|
||
|
|
0.054755896574351996031381300244580,
|
||
|
|
0.075039674810919952767043140916190,
|
||
|
|
0.093125454583697605535065465083366,
|
||
|
|
0.109387158802297641899210590325805,
|
||
|
|
0.123491976262065851077958109831074,
|
||
|
|
0.134709217311473325928054001771707,
|
||
|
|
0.142775938577060080797094273138717,
|
||
|
|
0.147739104901338491374841515972068,
|
||
|
|
0.149445554002916905664936468389821,
|
||
|
|
0.147739104901338491374841515972068,
|
||
|
|
0.142775938577060080797094273138717,
|
||
|
|
0.134709217311473325928054001771707,
|
||
|
|
0.123491976262065851077958109831074,
|
||
|
|
0.109387158802297641899210590325805,
|
||
|
|
0.093125454583697605535065465083366,
|
||
|
|
0.075039674810919952767043140916190,
|
||
|
|
0.054755896574351996031381300244580,
|
||
|
|
0.032558162307964727478818972459390,
|
||
|
|
0.011694638867371874278064396062192)
|
||
|
|
|
||
|
|
return _quadrature_gk(a, b, f, norm_func, x, w, v)
|
||
|
|
|
||
|
|
|
||
|
|
_quadrature_gk21.num_eval = 21
|
||
|
|
|
||
|
|
|
||
|
|
def _quadrature_gk15(a, b, f, norm_func):
|
||
|
|
"""
|
||
|
|
Gauss-Kronrod 15 quadrature with error estimate
|
||
|
|
"""
|
||
|
|
# Gauss-Kronrod points
|
||
|
|
x = (0.991455371120812639206854697526329,
|
||
|
|
0.949107912342758524526189684047851,
|
||
|
|
0.864864423359769072789712788640926,
|
||
|
|
0.741531185599394439863864773280788,
|
||
|
|
0.586087235467691130294144838258730,
|
||
|
|
0.405845151377397166906606412076961,
|
||
|
|
0.207784955007898467600689403773245,
|
||
|
|
0.000000000000000000000000000000000,
|
||
|
|
-0.207784955007898467600689403773245,
|
||
|
|
-0.405845151377397166906606412076961,
|
||
|
|
-0.586087235467691130294144838258730,
|
||
|
|
-0.741531185599394439863864773280788,
|
||
|
|
-0.864864423359769072789712788640926,
|
||
|
|
-0.949107912342758524526189684047851,
|
||
|
|
-0.991455371120812639206854697526329)
|
||
|
|
|
||
|
|
# 7-point weights
|
||
|
|
w = (0.129484966168869693270611432679082,
|
||
|
|
0.279705391489276667901467771423780,
|
||
|
|
0.381830050505118944950369775488975,
|
||
|
|
0.417959183673469387755102040816327,
|
||
|
|
0.381830050505118944950369775488975,
|
||
|
|
0.279705391489276667901467771423780,
|
||
|
|
0.129484966168869693270611432679082)
|
||
|
|
|
||
|
|
# 15-point weights
|
||
|
|
v = (0.022935322010529224963732008058970,
|
||
|
|
0.063092092629978553290700663189204,
|
||
|
|
0.104790010322250183839876322541518,
|
||
|
|
0.140653259715525918745189590510238,
|
||
|
|
0.169004726639267902826583426598550,
|
||
|
|
0.190350578064785409913256402421014,
|
||
|
|
0.204432940075298892414161999234649,
|
||
|
|
0.209482141084727828012999174891714,
|
||
|
|
0.204432940075298892414161999234649,
|
||
|
|
0.190350578064785409913256402421014,
|
||
|
|
0.169004726639267902826583426598550,
|
||
|
|
0.140653259715525918745189590510238,
|
||
|
|
0.104790010322250183839876322541518,
|
||
|
|
0.063092092629978553290700663189204,
|
||
|
|
0.022935322010529224963732008058970)
|
||
|
|
|
||
|
|
return _quadrature_gk(a, b, f, norm_func, x, w, v)
|
||
|
|
|
||
|
|
|
||
|
|
_quadrature_gk15.num_eval = 15
|