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def quad(func, a, b, args=(), full_output=0, epsabs=1.49e-08, epsrel=1.49e-08, limit=50, points=None, weight=None, wvar=None, wopts=None, maxp1=50, limlst=50, complex_func=False)
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.
complex_func : bool, optional
Indicate if the function's (`func`) return type is real
(``complex_func=False``: default) or complex (``complex_func=True``).
In both cases, the function's argument is real.
If full_output is also non-zero, the `infodict`, `message`, and
`explain` for the real and complex components are returned in
a dictionary with keys "real output" and "imag output".
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.
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. Default is 1.49e-8. `quad` tries to obtain
an accuracy of ``abs(i-result) <= max(epsabs, epsrel*abs(i))``
where ``i`` = integral of `func` from `a` to `b`, and ``result`` is the
numerical approximation. See `epsrel` below.
epsrel : float or int, optional
Relative error tolerance. Default is 1.49e-8.
If ``epsabs <= 0``, `epsrel` must be greater than both 5e-29
and ``50 * (machine epsilon)``. See `epsabs` above.
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. Note that this option cannot be used in conjunction
with ``weight``.
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
simpson : integrator for sampled data
romb : integrator for sampled data
scipy.special : for coefficients and roots of orthogonal polynomials
Notes
-----
For valid results, the integral must converge; behavior for divergent
integrals is not guaranteed.
**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, and these do not support specifying break points. 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.
**Details of QUADPACK level routines**
`quad` calls routines from the FORTRAN library QUADPACK. This section
provides details on the conditions for each routine to be called and a
short description of each routine. The routine called depends on
`weight`, `points` and the integration limits `a` and `b`.
================ ============== ========== =====================
QUADPACK routine `weight` `points` infinite bounds
================ ============== ========== =====================
qagse None No No
qagie None No Yes
qagpe None Yes No
qawoe 'sin', 'cos' No No
qawfe 'sin', 'cos' No either `a` or `b`
qawse 'alg*' No No
qawce 'cauchy' No No
================ ============== ========== =====================
The following provides a short description from [1]_ for each
routine.
qagse
is an integrator based on globally adaptive interval
subdivision in connection with extrapolation, which will
eliminate the effects of integrand singularities of
several types.
qagie
handles integration over infinite intervals. The infinite range is
mapped onto a finite interval and subsequently the same strategy as
in ``QAGS`` is applied.
qagpe
serves the same purposes as QAGS, but also allows the
user to provide explicit information about the location
and type of trouble-spots i.e. the abscissae of internal
singularities, discontinuities and other difficulties of
the integrand function.
qawoe
is an integrator for the evaluation of
:math:`\int^b_a \cos(\omega x)f(x)dx` or
:math:`\int^b_a \sin(\omega x)f(x)dx`
over a finite interval [a,b], where :math:`\omega` and :math:`f`
are specified by the user. The rule evaluation component is based
on the modified Clenshaw-Curtis technique
An adaptive subdivision scheme is used in connection
with an extrapolation procedure, which is a modification
of that in ``QAGS`` and allows the algorithm to deal with
singularities in :math:`f(x)`.
qawfe
calculates the Fourier transform
:math:`\int^\infty_a \cos(\omega x)f(x)dx` or
:math:`\int^\infty_a \sin(\omega x)f(x)dx`
for user-provided :math:`\omega` and :math:`f`. The procedure of
``QAWO`` is applied on successive finite intervals, and convergence
acceleration by means of the :math:`\varepsilon`-algorithm is applied
to the series of integral approximations.
qawse
approximate :math:`\int^b_a w(x)f(x)dx`, with :math:`a < b` where
:math:`w(x) = (x-a)^{\alpha}(b-x)^{\beta}v(x)` with
:math:`\alpha,\beta > -1`, where :math:`v(x)` may be one of the
following functions: :math:`1`, :math:`\log(x-a)`, :math:`\log(b-x)`,
:math:`\log(x-a)\log(b-x)`.
The user specifies :math:`\alpha`, :math:`\beta` and the type of the
function :math:`v`. A globally adaptive subdivision strategy is
applied, with modified Clenshaw-Curtis integration on those
subintervals which contain `a` or `b`.
qawce
compute :math:`\int^b_a f(x) / (x-c)dx` where the integral must be
interpreted as a Cauchy principal value integral, for user specified
:math:`c` and :math:`f`. The strategy is globally adaptive. Modified
Clenshaw-Curtis integration is used on those intervals containing the
point :math:`x = c`.
**Integration of Complex Function of a Real Variable**
A complex valued function, :math:`f`, of a real variable can be written as
:math:`f = g + ih`. Similarly, the integral of :math:`f` can be
written as
.. math::
\int_a^b f(x) dx = \int_a^b g(x) dx + i\int_a^b h(x) dx
assuming that the integrals of :math:`g` and :math:`h` exist
over the interval :math:`[a,b]` [2]_. Therefore, ``quad`` integrates
complex-valued functions by integrating the real and imaginary components
separately.
References
----------
.. [1] Piessens, Robert; de Doncker-Kapenga, Elise;
Überhuber, Christoph W.; Kahaner, David (1983).
QUADPACK: A subroutine package for automatic integration.
Springer-Verlag.
ISBN 978-3-540-12553-2.
.. [2] McCullough, Thomas; Phillips, Keith (1973).
Foundations of Analysis in the Complex Plane.
Holt Rinehart Winston.
ISBN 0-03-086370-8
Examples
--------
Calculate :math:`\int^4_0 x^2 dx` and compare with an analytic result
>>> from scipy import integrate
>>> import numpy as np
>>> 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)
Calculate :math:`\int^1_0 a x \,dx` for :math:`a = 1, 3`
>>> 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
Calculate :math:`\int^1_0 x^2 + y^2 dx` with ctypes, holding
y parameter as 1::
testlib.c =>
double func(int n, double args[n]){
return args[0]*args[0] + args[1]*args[1];}
compile to library testlib.*
::
from scipy import integrate
import ctypes
lib = ctypes.CDLL('/home/.../testlib.*') #use absolute path
lib.func.restype = ctypes.c_double
lib.func.argtypes = (ctypes.c_int,ctypes.c_double)
integrate.quad(lib.func,0,1,(1))
#(1.3333333333333333, 1.4802973661668752e-14)
print((1.0**3/3.0 + 1.0) - (0.0**3/3.0 + 0.0)) #Analytic result
# 1.3333333333333333
Be aware that pulse shapes and other sharp features as compared to the
size of the integration interval may not be integrated correctly using
this method. A simplified example of this limitation is integrating a
y-axis reflected step function with many zero values within the integrals
bounds.
>>> y = lambda x: 1 if x<=0 else 0
>>> integrate.quad(y, -1, 1)
(1.0, 1.1102230246251565e-14)
>>> integrate.quad(y, -1, 100)
(1.0000000002199108, 1.0189464580163188e-08)
>>> integrate.quad(y, -1, 10000)
(0.0, 0.0)
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