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# 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):
pass
def quad_explain(output=sys.stdout):
"""
Print extra information about integrate.quad() parameters and returns.
Parameters
----------
output : instance with "write" method
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
A Python function or method to integrate. If `func` takes many
arguments, it is integrated along the axis corresponding to the
first argument.
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.
See Also
--------
dblquad : double integral
tplquad : triple integral
nquad : n-dimensional integrals (uses `quad` recursively)
fixed_quad : fixed-order Gaussian quadrature
quadrature : adaptive Gaussian quadrature
odeint : ODE integrator
ode : ODE integrator
simps : integrator for sampled data
romb : integrator for sampled data
scipy.special : for coefficients and roots of orthogonal polynomials
Notes
-----
**Extra information for quad() inputs and outputs**
If full_output is non-zero, then the third output argument
(infodict) is a dictionary with entries as tabulated below. For
infinite limits, the range is transformed to (0,1) and the
optional outputs are given with respect to this transformed range.
Let M be the input argument limit and let K be infodict['last'].
The entries are:
'neval'
The number of function evaluations.
'last'
The number, K, of subintervals produced in the subdivision process.
'alist'
A rank-1 array of length M, the first K elements of which are the
left end points of the subintervals in the partition of the
integration range.
'blist'
A rank-1 array of length M, the first K elements of which are the
right end points of the subintervals.
'rlist'
A rank-1 array of length M, the first K elements of which are the
integral approximations on the subintervals.
'elist'
A rank-1 array of length M, the first K elements of which are the
moduli of the absolute error estimates on the subintervals.
'iord'
A rank-1 integer array of length M, the first L elements of
which are pointers to the error estimates over the subintervals
with L=K if K<=M/2+2 or L=M+1-K otherwise. Let I be the sequence
infodict['iord'] and let E be the sequence infodict['elist'].
Then E[I[1]], ..., E[I[L]] forms a decreasing sequence.
If the input argument points is provided (i.e. it is not None),
the following additional outputs are placed in the output
dictionary. Assume the points sequence is of length P.
'pts'
A rank-1 array of length P+2 containing the integration limits
and the break points of the intervals in ascending order.
This is an array giving the subintervals over which integration
will occur.
'level'
A rank-1 integer array of length M (=limit), containing the
subdivision levels of the subintervals, i.e., if (aa,bb) is a
subinterval of (pts[1], pts[2]) where pts[0] and pts[2] are
adjacent elements of infodict['pts'], then (aa,bb) has level l if
|bb-aa|=|pts[2]-pts[1]| * 2**(-l).
'ndin'
A rank-1 integer array of length P+2. After the first integration
over the intervals (pts[1], pts[2]), the error estimates over some
of the intervals may have been increased artificially in order to
put their subdivision forward. This array has ones in slots
corresponding to the subintervals for which this happens.
**Weighting the integrand**
The input variables, *weight* and *wvar*, are used to weight the
integrand by a select list of functions. Different integration
methods are used to compute the integral with these weighting
functions. The possible values of weight and the corresponding
weighting functions are.
========== =================================== =====================
``weight`` Weight function used ``wvar``
========== =================================== =====================
'cos' cos(w*x) wvar = w
'sin' sin(w*x) wvar = w
'alg' g(x) = ((x-a)**alpha)*((b-x)**beta) wvar = (alpha, beta)
'alg-loga' g(x)*log(x-a) wvar = (alpha, beta)
'alg-logb' g(x)*log(b-x) wvar = (alpha, beta)
'alg-log' g(x)*log(x-a)*log(b-x) wvar = (alpha, beta)
'cauchy' 1/(x-c) wvar = c
========== =================================== =====================
wvar holds the parameter w, (alpha, beta), or c depending on the weight
selected. In these expressions, a and b are the integration limits.
For the 'cos' and 'sin' weighting, additional inputs and outputs are
available.
For finite integration limits, the integration is performed using a
Clenshaw-Curtis method which uses Chebyshev moments. For repeated
calculations, these moments are saved in the output dictionary:
'momcom'
The maximum level of Chebyshev moments that have been computed,
i.e., if M_c is infodict['momcom'] then the moments have been
computed for intervals of length |b-a|* 2**(-l), l=0,1,...,M_c.
'nnlog'
A rank-1 integer array of length M(=limit), containing the
subdivision levels of the subintervals, i.e., an element of this
array is equal to l if the corresponding subinterval is
|b-a|* 2**(-l).
'chebmo'
A rank-2 array of shape (25, maxp1) containing the computed
Chebyshev moments. These can be passed on to an integration
over the same interval by passing this array as the second
element of the sequence wopts and passing infodict['momcom'] as
the first element.
If one of the integration limits is infinite, then a Fourier integral is
computed (assuming w neq 0). If full_output is 1 and a numerical error
is encountered, besides the error message attached to the output tuple,
a dictionary is also appended to the output tuple which translates the
error codes in the array info['ierlst'] to English messages. The output
information dictionary contains the following entries instead of 'last',
'alist', 'blist', 'rlist', and 'elist':
'lst'
The number of subintervals needed for the integration (call it K_f).
'rslst'
A rank-1 array of length M_f=limlst, whose first K_f elements
contain the integral contribution over the interval (a+(k-1)c,
a+kc) where c = (2*floor(|w|) + 1) * pi / |w| and k=1,2,...,K_f.
'erlst'
A rank-1 array of length M_f containing the error estimate
corresponding to the interval in the same position in
infodict['rslist'].
'ierlst'
A rank-1 integer array of length M_f containing an error flag
corresponding to the interval in the same position in
infodict['rslist']. See the explanation dictionary (last entry
in the output tuple) for the meaning of the codes.
Examples
--------
Calculate :math:`\\int^4_0 x^2 dx` and compare with an analytic result
>>> from scipy import integrate
>>> x2 = lambda x: x**2
>>> integrate.quad(x2, 0, 4)
(21.333333333333332, 2.3684757858670003e-13)
>>> print(4**3 / 3.) # analytical result
21.3333333333
Calculate :math:`\\int^\\infty_0 e^{-x} dx`
>>> invexp = lambda x: np.exp(-x)
>>> integrate.quad(invexp, 0, np.inf)
(1.0, 5.842605999138044e-11)
>>> f = lambda x,a : a*x
>>> y, err = integrate.quad(f, 0, 1, args=(1,))
>>> y
0.5
>>> y, err = integrate.quad(f, 0, 1, args=(3,))
>>> y
1.5
"""
if not isinstance(args, tuple):
args = (args,)
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)
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)
return retval[:-1]
else:
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:
nl = len(points)
the_points = numpy.zeros((nl+2,), float)
the_points[:nl] = points
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[:3] == '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 _infunc(x,func,gfun,hfun,more_args):
a = gfun(x)
b = hfun(x)
myargs = (x,) + more_args
return quad(func,a,b,args=myargs)[0]
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) : tuple
The limits of integration in x: `a` < `b`
gfun : callable
The lower boundary curve in y which is a function taking a single
floating point argument (x) and returning a floating point result: a
lambda function can be useful here.
hfun : callable
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
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.
See also
--------
quad : single integral
tplquad : triple integral
nquad : N-dimensional integrals
fixed_quad : fixed-order Gaussian quadrature
quadrature : adaptive Gaussian quadrature
odeint : ODE integrator
ode : ODE integrator
simps : integrator for sampled data
romb : integrator for sampled data
scipy.special : for coefficients and roots of orthogonal polynomials
"""
return quad(_infunc,a,b,(func,gfun,hfun,args),epsabs=epsabs,epsrel=epsrel)
def _infunc2(y,x,func,qfun,rfun,more_args):
a2 = qfun(x,y)
b2 = rfun(x,y)
myargs = (y,x) + more_args
return quad(func,a2,b2,args=myargs)[0]
def tplquad(func, a, b, gfun, hfun, qfun, rfun, args=(), epsabs=1.49e-8,
epsrel=1.49e-8):
"""
Compute a triple (definite) integral.
Return the triple integral of ``func(z, y, x)`` from ``x = a..b``,
``y = gfun(x)..hfun(x)``, and ``z = qfun(x,y)..rfun(x,y)``.
Parameters
----------
func : function
A Python function or method of at least three variables in the
order (z, y, x).
(a,b) : tuple
The limits of integration in x: `a` < `b`
gfun : function
The lower boundary curve in y which is a function taking a single
floating point argument (x) and returning a floating point result:
a lambda function can be useful here.
hfun : function
The upper boundary curve in y (same requirements as `gfun`).
qfun : function
The lower boundary surface in z. It must be a function that takes
two floats in the order (x, y) and returns a float.
rfun : function
The upper boundary surface in z. (Same requirements as `qfun`.)
args : Arguments
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.
See Also
--------
quad: Adaptive quadrature using QUADPACK
quadrature: Adaptive Gaussian quadrature
fixed_quad: Fixed-order Gaussian quadrature
dblquad: Double integrals
nquad : N-dimensional integrals
romb: Integrators for sampled data
simps: Integrators for sampled data
ode: ODE integrators
odeint: ODE integrators
scipy.special: For coefficients and roots of orthogonal polynomials
"""
return dblquad(_infunc2,a,b,gfun,hfun,(func,qfun,rfun,args),epsabs=epsabs,epsrel=epsrel)
def nquad(func, ranges, args=None, opts=None):
"""
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
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.
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.
e.g. if ``func = f(x0, x1, x2)``, then ``ranges[0]`` may be defined as
either ``(a, b)`` or else as ``(a, b) = range0(x1, x2)``.
args : iterable object, optional
Additional arguments ``t0, ..., tn``, required by `func`.
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.quadare 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. 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
The ``full_output`` option from `quad` is unavailable, due to the
complexity of handling the large amount of data such an option would
return for this kind of nested integration. For more information on
these options, see `quad` and `quad_explain`.
Returns
-------
result : float
The result of the integration.
abserr : float
The maximum of the estimates of the absolute error in the various
integration results.
See Also
--------
quad : 1-dimensional numerical integration
dblquad, tplquad : double and triple integrals
fixed_quad : fixed-order Gaussian quadrature
quadrature : adaptive Gaussian quadrature
Examples
--------
>>> from scipy import integrate
>>> func = lambda x0,x1,x2,x3 : x0**2 + x1*x2 - x3**3 + np.sin(x0) + (
... 1 if (x0-.2*x3-.5-.25*x1>0) else 0)
>>> points = [[lambda (x1,x2,x3) : 0.2*x3 + 0.5 + 0.25*x1], [], [], []]
>>> def opts0(*args, **kwargs):
... return {'points':[0.2*args[2] + 0.5 + 0.25*args[0]]}
>>> integrate.nquad(func, [[0,1], [-1,1], [.13,.8], [-.15,1]],
... opts=[opts0,{},{},{}])
(1.5267454070738633, 2.9437360001402324e-14)
>>> scale = .1
>>> def func2(x0, x1, x2, x3, t0, t1):
... return x0*x1*x3**2 + np.sin(x2) + 1 + (1 if x0+t1*x1-t0>0 else 0)
>>> def lim0(x1, x2, x3, t0, t1):
... return [scale * (x1**2 + x2 + np.cos(x3)*t0*t1 + 1) - 1,
... scale * (x1**2 + x2 + np.cos(x3)*t0*t1 + 1) + 1]
>>> def lim1(x2, x3, t0, t1):
... return [scale * (t0*x2 + t1*x3) - 1,
... scale * (t0*x2 + t1*x3) + 1]
>>> def lim2(x3, t0, t1):
... return [scale * (x3 + t0**2*t1**3) - 1,
... scale * (x3 + t0**2*t1**3) + 1]
>>> def lim3(t0, t1):
... return [scale * (t0+t1) - 1, scale * (t0+t1) + 1]
>>> def opts0(x1, x2, x3, t0, t1):
... return {'points' : [t0 - t1*x1]}
>>> def opts1(x2, x3, t0, t1):
... return {}
>>> def opts2(x3, t0, t1):
... return {}
>>> def opts3(t0, t1):
... return {}
>>> integrate.nquad(func2, [lim0, lim1, lim2, lim3], args=(0,0),
opts=[opts0, opts1, opts2, opts3])
(25.066666666666666, 2.7829590483937256e-13)
"""
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 = [opts] * depth
else:
opts = [opt if callable(opt) else _OptFunc(opt) for opt in opts]
return _NQuad(func, ranges, opts).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):
self.abserr = 0
self.func = func
self.ranges = ranges
self.opts = opts
self.maxdepth = len(ranges)
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)
value, abserr = quad(f, low, high, args=args, **opt)
self.abserr = max(self.abserr, abserr)
if depth > 0:
return value
else:
# Final result of n-D integration with error
return value, self.abserr
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