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