Это узкий вопрос из связанного вопроса на многомерные интеграторы в Python. Необходима четверная интеграция с переменными пределами интегрирования, которую обеспечивает scipy. Однако scipy ограничен ошибками округления (при многократном или простом интегрировании), поскольку мы уменьшаем epsabs и epsrel. Эта проблема усугубляется при многомерном интегрировании, требующем постоянно уменьшающегося epsrel для внутренних интегралов. Может ли пользователь установить точность, используемую scipy? Mpmath не обеспечивает четверного интегрирования. Я упустил что-то, что позволило бы повысить производительность при интеграции scipy за счет требуемой точности?
Ниже приведено предупреждение и выдержки из отладки, отображаемые по ссылке scipy в строке 860 для кода, уже предоставленного в предыдущий вопрос.
"""
Warning (from warnings module):
File "C:\Users\makis\AppData\Local\Programs\Python\Python37-32\lib\site-packages\scipy\integrate\quadpack.py", line 860
**opt)
IntegrationWarning: The occurrence of roundoff error is detected, which prevents
the requested tolerance from being achieved. The error may be
underestimated.
"""
#For above, "degug: go to line 860" is displayed below:
# Author: Travis Oliphant 2001
# Author: Nathan Woods 2013 (nquad &c)
from __future__ import division, print_function, absolute_import
import sys
import warnings
from functools import partial
from . import _quadpack
import numpy
from numpy import Inf
__all__ = ['quad', 'dblquad', 'tplquad', 'nquad', 'quad_explain',
'IntegrationWarning']
error = _quadpack.error
class IntegrationWarning(UserWarning):
"""
Warning on issues during integration.
"""
pass
def quad_explain(output=sys.stdout):
"""
Print extra information about integrate.quad() parameters and returns.
Parameters
----------
output : instance with "write" method, optional
Information about `quad` is passed to ``output.write()``.
Default is ``sys.stdout``.
Returns
-------
None
"""
output.write(quad.__doc__)
def quad(func, a, b, args=(), full_output=0, epsabs=1.49e-8, epsrel=1.49e-8,
limit=50, points=None, weight=None, wvar=None, wopts=None, maxp1=50,
limlst=50):
"""
Compute a definite integral.
Integrate func from `a` to `b` (possibly infinite interval) using a
technique from the Fortran library QUADPACK.
Parameters
----------
func : {function, scipy.LowLevelCallable}
A Python function or method to integrate. If `func` takes many
arguments, it is integrated along the axis corresponding to the
first argument.
If the user desires improved integration performance, then `f` may
be a `scipy.LowLevelCallable` with one of the signatures::
double func(double x)
double func(double x, void *user_data)
double func(int n, double *xx)
double func(int n, double *xx, void *user_data)
The ``user_data`` is the data contained in the `scipy.LowLevelCallable`.
In the call forms with ``xx``, ``n`` is the length of the ``xx``
array which contains ``xx[0] == x`` and the rest of the items are
numbers contained in the ``args`` argument of quad.
In addition, certain ctypes call signatures are supported for
backward compatibility, but those should not be used in new code.
a : float
Lower limit of integration (use -numpy.inf for -infinity).
b : float
Upper limit of integration (use numpy.inf for +infinity).
args : tuple, optional
Extra arguments to pass to `func`.
full_output : int, optional
Non-zero to return a dictionary of integration information.
If non-zero, warning messages are also suppressed and the
message is appended to the output tuple.
Returns
-------
y : float
The integral of func from `a` to `b`.
abserr : float
An estimate of the absolute error in the result.
infodict : dict
A dictionary containing additional information.
Run scipy.integrate.quad_explain() for more information.
message
A convergence message.
explain
Appended only with 'cos' or 'sin' weighting and infinite
integration limits, it contains an explanation of the codes in
infodict['ierlst']
Other Parameters
----------------
epsabs : float or int, optional
Absolute error tolerance.
epsrel : float or int, optional
Relative error tolerance.
limit : float or int, optional
An upper bound on the number of subintervals used in the adaptive
algorithm.
points : (sequence of floats,ints), optional
A sequence of break points in the bounded integration interval
where local difficulties of the integrand may occur (e.g.,
singularities, discontinuities). The sequence does not have
to be sorted.
weight : float or int, optional
String indicating weighting function. Full explanation for this
and the remaining arguments can be found below.
wvar : optional
Variables for use with weighting functions.
wopts : optional
Optional input for reusing Chebyshev moments.
maxp1 : float or int, optional
An upper bound on the number of Chebyshev moments.
limlst : int, optional
Upper bound on the number of cycles (>=3) for use with a sinusoidal
weighting and an infinite end-point.
....
"""
if not isinstance(args, tuple):
args = (args,)
# check the limits of integration: \int_a^b, expect a < b
flip, a, b = b < a, min(a, b), max(a, b)
if weight is None:
retval = _quad(func, a, b, args, full_output, epsabs, epsrel, limit,
points)
else:
retval = _quad_weight(func, a, b, args, full_output, epsabs, epsrel,
limlst, limit, maxp1, weight, wvar, wopts)
if flip:
retval = (-retval[0],) + retval[1:]
ier = retval[-1]
if ier == 0:
return retval[:-1]
msgs = {80: "A Python error occurred possibly while calling the function.",
1: "The maximum number of subdivisions (%d) has been achieved.\n If increasing the limit yields no improvement it is advised to analyze \n the integrand in order to determine the difficulties. If the position of a \n local difficulty can be determined (singularity, discontinuity) one will \n probably gain from splitting up the interval and calling the integrator \n on the subranges. Perhaps a special-purpose integrator should be used." % limit,
2: "The occurrence of roundoff error is detected, which prevents \n the requested tolerance from being achieved. The error may be \n underestimated.",
3: "Extremely bad integrand behavior occurs at some points of the\n integration interval.",
4: "The algorithm does not converge. Roundoff error is detected\n in the extrapolation table. It is assumed that the requested tolerance\n cannot be achieved, and that the returned result (if full_output = 1) is \n the best which can be obtained.",
5: "The integral is probably divergent, or slowly convergent.",
6: "The input is invalid.",
7: "Abnormal termination of the routine. The estimates for result\n and error are less reliable. It is assumed that the requested accuracy\n has not been achieved.",
'unknown': "Unknown error."}
if weight in ['cos','sin'] and (b == Inf or a == -Inf):
msgs[1] = "The maximum number of cycles allowed has been achieved., e.e.\n of subintervals (a+(k-1)c, a+kc) where c = (2*int(abs(omega)+1))\n *pi/abs(omega), for k = 1, 2, ..., lst. One can allow more cycles by increasing the value of limlst. Look at info['ierlst'] with full_output=1."
msgs[4] = "The extrapolation table constructed for convergence acceleration\n of the series formed by the integral contributions over the cycles, \n does not converge to within the requested accuracy. Look at \n info['ierlst'] with full_output=1."
msgs[7] = "Bad integrand behavior occurs within one or more of the cycles.\n Location and type of the difficulty involved can be determined from \n the vector info['ierlist'] obtained with full_output=1."
explain = {1: "The maximum number of subdivisions (= limit) has been \n achieved on this cycle.",
2: "The occurrence of roundoff error is detected and prevents\n the tolerance imposed on this cycle from being achieved.",
3: "Extremely bad integrand behavior occurs at some points of\n this cycle.",
4: "The integral over this cycle does not converge (to within the required accuracy) due to roundoff in the extrapolation procedure invoked on this cycle. It is assumed that the result on this interval is the best which can be obtained.",
5: "The integral over this cycle is probably divergent or slowly convergent."}
try:
msg = msgs[ier]
except KeyError:
msg = msgs['unknown']
if ier in [1,2,3,4,5,7]:
if full_output:
if weight in ['cos', 'sin'] and (b == Inf or a == Inf):
return retval[:-1] + (msg, explain)
else:
return retval[:-1] + (msg,)
else:
warnings.warn(msg, IntegrationWarning, stacklevel=2)
return retval[:-1]
elif ier == 6: # Forensic decision tree when QUADPACK throws ier=6
if epsabs <= 0: # Small error tolerance - applies to all methods
if epsrel < max(50 * sys.float_info.epsilon, 5e-29):
msg = ("If 'errabs'<=0, 'epsrel' must be greater than both"
" 5e-29 and 50*(machine epsilon).")
elif weight in ['sin', 'cos'] and (abs(a) + abs(b) == Inf):
msg = ("Sine or cosine weighted intergals with infinite domain"
" must have 'epsabs'>0.")
elif weight is None:
if points is None: # QAGSE/QAGIE
msg = ("Invalid 'limit' argument. There must be"
" at least one subinterval")
else: # QAGPE
if not (min(a, b) <= min(points) <= max(points) <= max(a, b)):
msg = ("All break points in 'points' must lie within the"
" integration limits.")
elif len(points) >= limit:
msg = ("Number of break points ({:d})"
" must be less than subinterval"
" limit ({:d})").format(len(points), limit)
else:
if maxp1 < 1:
msg = "Chebyshev moment limit maxp1 must be >=1."
elif weight in ('cos', 'sin') and abs(a+b) == Inf: # QAWFE
msg = "Cycle limit limlst must be >=3."
elif weight.startswith('alg'): # QAWSE
if min(wvar) < -1:
msg = "wvar parameters (alpha, beta) must both be >= -1."
if b < a:
msg = "Integration limits a, b must satistfy a<b."
elif weight == 'cauchy' and wvar in (a, b):
msg = ("Parameter 'wvar' must not equal"
" integration limits 'a' or 'b'.")
raise ValueError(msg)
def _quad(func,a,b,args,full_output,epsabs,epsrel,limit,points):
infbounds = 0
if (b != Inf and a != -Inf):
pass # standard integration
elif (b == Inf and a != -Inf):
infbounds = 1
bound = a
elif (b == Inf and a == -Inf):
infbounds = 2
bound = 0 # ignored
elif (b != Inf and a == -Inf):
infbounds = -1
bound = b
else:
raise RuntimeError("Infinity comparisons don't work for you.")
if points is None:
if infbounds == 0:
return _quadpack._qagse(func,a,b,args,full_output,epsabs,epsrel,limit)
else:
return _quadpack._qagie(func,bound,infbounds,args,full_output,epsabs,epsrel,limit)
else:
if infbounds != 0:
raise ValueError("Infinity inputs cannot be used with break points.")
else:
#Duplicates force function evaluation at sinular points
the_points = numpy.unique(points)
the_points = the_points[a < the_points]
the_points = the_points[the_points < b]
the_points = numpy.concatenate((the_points, (0., 0.)))
return _quadpack._qagpe(func,a,b,the_points,args,full_output,epsabs,epsrel,limit)
def _quad_weight(func,a,b,args,full_output,epsabs,epsrel,limlst,limit,maxp1,weight,wvar,wopts):
if weight not in ['cos','sin','alg','alg-loga','alg-logb','alg-log','cauchy']:
raise ValueError("%s not a recognized weighting function." % weight)
strdict = {'cos':1,'sin':2,'alg':1,'alg-loga':2,'alg-logb':3,'alg-log':4}
if weight in ['cos','sin']:
integr = strdict[weight]
if (b != Inf and a != -Inf): # finite limits
if wopts is None: # no precomputed chebyshev moments
return _quadpack._qawoe(func, a, b, wvar, integr, args, full_output,
epsabs, epsrel, limit, maxp1,1)
else: # precomputed chebyshev moments
momcom = wopts[0]
chebcom = wopts[1]
return _quadpack._qawoe(func, a, b, wvar, integr, args, full_output,
epsabs, epsrel, limit, maxp1, 2, momcom, chebcom)
elif (b == Inf and a != -Inf):
return _quadpack._qawfe(func, a, wvar, integr, args, full_output,
epsabs,limlst,limit,maxp1)
elif (b != Inf and a == -Inf): # remap function and interval
if weight == 'cos':
def thefunc(x,*myargs):
y = -x
func = myargs[0]
myargs = (y,) + myargs[1:]
return func(*myargs)
else:
def thefunc(x,*myargs):
y = -x
func = myargs[0]
myargs = (y,) + myargs[1:]
return -func(*myargs)
args = (func,) + args
return _quadpack._qawfe(thefunc, -b, wvar, integr, args,
full_output, epsabs, limlst, limit, maxp1)
else:
raise ValueError("Cannot integrate with this weight from -Inf to +Inf.")
else:
if a in [-Inf,Inf] or b in [-Inf,Inf]:
raise ValueError("Cannot integrate with this weight over an infinite interval.")
if weight.startswith('alg'):
integr = strdict[weight]
return _quadpack._qawse(func, a, b, wvar, integr, args,
full_output, epsabs, epsrel, limit)
else: # weight == 'cauchy'
return _quadpack._qawce(func, a, b, wvar, args, full_output,
epsabs, epsrel, limit)
def dblquad(func, a, b, gfun, hfun, args=(), epsabs=1.49e-8, epsrel=1.49e-8):
"""
Compute a double integral.
Return the double (definite) integral of ``func(y, x)`` from ``x = a..b``
and ``y = gfun(x)..hfun(x)``.
Parameters
----------
func : callable
A Python function or method of at least two variables: y must be the
first argument and x the second argument.
a, b : float
The limits of integration in x: `a` < `b`
gfun : callable or float
The lower boundary curve in y which is a function taking a single
floating point argument (x) and returning a floating point result
or a float indicating a constant boundary curve.
hfun : callable or float
The upper boundary curve in y (same requirements as `gfun`).
args : sequence, optional
Extra arguments to pass to `func`.
epsabs : float, optional
Absolute tolerance passed directly to the inner 1-D quadrature
integration. Default is 1.49e-8.
epsrel : float, optional
Relative tolerance of the inner 1-D integrals. Default is 1.49e-8.
Returns
-------
y : float
The resultant integral.
abserr : float
An estimate of the error.
Parameters
----------
func : function
A Python function or method of at least three variables in the
order (z, y, x).
a, b : float
The limits of integration in x: `a` < `b`
gfun : function or float
The lower boundary curve in y which is a function taking a single
floating point argument (x) and returning a floating point result
or a float indicating a constant boundary curve.
hfun : function or float
The upper boundary curve in y (same requirements as `gfun`).
qfun : function or float
The lower boundary surface in z. It must be a function that takes
two floats in the order (x, y) and returns a float or a float
indicating a constant boundary surface.
rfun : function or float
The upper boundary surface in z. (Same requirements as `qfun`.)
args : tuple, optional
Extra arguments to pass to `func`.
epsabs : float, optional
Absolute tolerance passed directly to the innermost 1-D quadrature
integration. Default is 1.49e-8.
epsrel : float, optional
Relative tolerance of the innermost 1-D integrals. Default is 1.49e-8.
Returns
-------
y : float
The resultant integral.
abserr : float
An estimate of the error.
......
.......
def ranges0(*args):
return [qfun(args[1], args[0]) if callable(qfun) else qfun,
rfun(args[1], args[0]) if callable(rfun) else rfun]
def ranges1(*args):
return [gfun(args[0]) if callable(gfun) else gfun,
hfun(args[0]) if callable(hfun) else hfun]
ranges = [ranges0, ranges1, [a, b]]
return nquad(func, ranges, args=args,
opts={"epsabs": epsabs, "epsrel": epsrel})
def nquad(func, ranges, args=None, opts=None, full_output=False):
"""
Integration over multiple variables.
Wraps `quad` to enable integration over multiple variables.
Various options allow improved integration of discontinuous functions, as
well as the use of weighted integration, and generally finer control of the
integration process.
Parameters
----------
func : {callable, scipy.LowLevelCallable}
The function to be integrated. Has arguments of ``x0, ... xn``,
``t0, tm``, where integration is carried out over ``x0, ... xn``, which
must be floats. Function signature should be
``func(x0, x1, ..., xn, t0, t1, ..., tm)``. Integration is carried out
in order. That is, integration over ``x0`` is the innermost integral,
and ``xn`` is the outermost.
If the user desires improved integration performance, then `f` may
be a `scipy.LowLevelCallable` with one of the signatures::
double func(int n, double *xx)
double func(int n, double *xx, void *user_data)
where ``n`` is the number of extra parameters and args is an array
of doubles of the additional parameters, the ``xx`` array contains the
coordinates. The ``user_data`` is the data contained in the
`scipy.LowLevelCallable`.
ranges : iterable object
Each element of ranges may be either a sequence of 2 numbers, or else
a callable that returns such a sequence. ``ranges[0]`` corresponds to
integration over x0, and so on. If an element of ranges is a callable,
then it will be called with all of the integration arguments available,
as well as any parametric arguments. e.g. if
``func = f(x0, x1, x2, t0, t1)``, then ``ranges[0]`` may be defined as
either ``(a, b)`` or else as ``(a, b) = range0(x1, x2, t0, t1)``.
args : iterable object, optional
Additional arguments ``t0, ..., tn``, required by `func`, `ranges`, and
``opts``.
opts : iterable object or dict, optional
Options to be passed to `quad`. May be empty, a dict, or
a sequence of dicts or functions that return a dict. If empty, the
default options from scipy.integrate.quad are used. If a dict, the same
options are used for all levels of integraion. If a sequence, then each
element of the sequence corresponds to a particular integration. e.g.
opts[0] corresponds to integration over x0, and so on. If a callable,
the signature must be the same as for ``ranges``. The available
options together with their default values are:
- epsabs = 1.49e-08
- epsrel = 1.49e-08
- limit = 50
- points = None
- weight = None
- wvar = None
- wopts = None
For more information on these options, see `quad` and `quad_explain`.
full_output : bool, optional
Partial implementation of ``full_output`` from scipy.integrate.quad.
The number of integrand function evaluations ``neval`` can be obtained
by setting ``full_output=True`` when calling nquad.
Returns
-------
result : float
The result of the integration.
abserr : float
The maximum of the estimates of the absolute error in the various
integration results.
out_dict : dict, optional
A dict containing additional information on the integration.
See Also
--------
quad : 1-dimensional numerical integration
dblquad, tplquad : double and triple integrals
fixed_quad : fixed-order Gaussian quadrature
quadrature : adaptive Gaussian quadrature
........
"""
depth = len(ranges)
ranges = [rng if callable(rng) else _RangeFunc(rng) for rng in ranges]
if args is None:
args = ()
if opts is None:
opts = [dict([])] * depth
if isinstance(opts, dict):
opts = [_OptFunc(opts)] * depth
else:
opts = [opt if callable(opt) else _OptFunc(opt) for opt in opts]
return _NQuad(func, ranges, opts, full_output).integrate(*args)
class _RangeFunc(object):
def __init__(self, range_):
self.range_ = range_
def __call__(self, *args):
"""Return stored value.
*args needed because range_ can be float or func, and is called with
variable number of parameters.
"""
return self.range_
class _OptFunc(object):
def __init__(self, opt):
self.opt = opt
def __call__(self, *args):
"""Return stored dict."""
return self.opt
class _NQuad(object):
def __init__(self, func, ranges, opts, full_output):
self.abserr = 0
self.func = func
self.ranges = ranges
self.opts = opts
self.maxdepth = len(ranges)
self.full_output = full_output
if self.full_output:
self.out_dict = {'neval': 0}
def integrate(self, *args, **kwargs):
depth = kwargs.pop('depth', 0)
if kwargs:
raise ValueError('unexpected kwargs')
# Get the integration range and options for this depth.
ind = -(depth + 1)
fn_range = self.ranges[ind]
low, high = fn_range(*args)
fn_opt = self.opts[ind]
opt = dict(fn_opt(*args))
if 'points' in opt:
opt['points'] = [x for x in opt['points'] if low <= x <= high]
if depth + 1 == self.maxdepth:
f = self.func
else:
f = partial(self.integrate, depth=depth+1)
quad_r = quad(f, low, high, args=args, full_output=self.full_output,
**opt)
value = quad_r[0]
abserr = quad_r[1]
if self.full_output:
infodict = quad_r[2]
# The 'neval' parameter in full_output returns the total
# number of times the integrand function was evaluated.
# Therefore, only the innermost integration loop counts.
if depth + 1 == self.maxdepth:
self.out_dict['neval'] += infodict['neval']
self.abserr = max(self.abserr, abserr)
if depth > 0:
return value
else:
# Final result of n-D integration with error
if self.full_output:
return value, self.abserr, self.out_dict
else:
return value, self.abserr
