projektAI/venv/Lib/site-packages/scipy/integrate/_quad_vec.py

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2021-06-06 22:13:05 +02:00
import sys
import copy
import heapq
import collections
import functools
import numpy as np
from scipy._lib._util import MapWrapper
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(LRUDict, self).__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(object):
"""
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(object):
"""
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(object):
def __init__(self, **kwargs):
self.__keys = kwargs.keys()
self.__dict__.update(**kwargs)
def __repr__(self):
return "_Bunch({})".format(", ".join("{}={}".format(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):
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.
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.
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 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)
# 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("invalid integration bounds a={}, b={}".format(a, 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("unknown quadrature {!r}".format(quadrature)) 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 dint, derr, dround_err, subint, dneval in mapwrapper(_subdivide_interval, to_process):
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