scipy.integrate.quad¶

scipy.integrate.
quad
(func, a, b, args=(), full_output=0, epsabs=1.49e08, epsrel=1.49e08, limit=50, points=None, weight=None, wvar=None, wopts=None, maxp1=50, limlst=50)[source]¶ 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 thescipy.LowLevelCallable
. In the call forms withxx
,n
is the length of thexx
array which containsxx[0] == x
and the rest of the items are numbers contained in theargs
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
Nonzero to return a dictionary of integration information. If nonzero, 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 endpoint.
See also
dblquad
 double integral
tplquad
 triple integral
nquad
 ndimensional integrals (uses
quad
recursively) fixed_quad
 fixedorder 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 nonzero, 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 rank1 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 rank1 array of length M, the first K elements of which are the right end points of the subintervals.
 ‘rlist’
 A rank1 array of length M, the first K elements of which are the integral approximations on the subintervals.
 ‘elist’
 A rank1 array of length M, the first K elements of which are the moduli of the absolute error estimates on the subintervals.
 ‘iord’
 A rank1 integer array of length M, the first L elements of
which are pointers to the error estimates over the subintervals
with
L=K
ifK<=M/2+2
orL=M+1K
otherwise. Let I be the sequenceinfodict['iord']
and let E be the sequenceinfodict['elist']
. ThenE[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 rank1 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 rank1 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])
wherepts[0]
andpts[2]
are adjacent elements ofinfodict['pts']
, then (aa,bb) has level l ifbbaa = pts[2]pts[1] * 2**(l)
.  ‘ndin’
 A rank1 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) = ((xa)**alpha)*((bx)**beta) wvar = (alpha, beta) ‘algloga’ g(x)*log(xa) wvar = (alpha, beta) ‘alglogb’ g(x)*log(bx) wvar = (alpha, beta) ‘alglog’ g(x)*log(xa)*log(bx) wvar = (alpha, beta) ‘cauchy’ 1/(xc) 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 ClenshawCurtis 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
isinfodict['momcom']
then the moments have been computed for intervals of lengthba * 2**(l)
,l=0,1,...,M_c
.  ‘nnlog’
 A rank1 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
ba* 2**(l)
.  ‘chebmo’
 A rank2 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 rank1 array of length M_f=limlst, whose first
K_f
elements contain the integral contribution over the interval(a+(k1)c, a+kc)
wherec = (2*floor(w) + 1) * pi / w
andk=1,2,...,K_f
.  ‘erlst’
 A rank1 array of length
M_f
containing the error estimate corresponding to the interval in the same position ininfodict['rslist']
.  ‘ierlst’
 A rank1 integer array of length
M_f
containing an error flag corresponding to the interval in the same position ininfodict['rslist']
. See the explanation dictionary (last entry in the output tuple) for the meaning of the codes.
Examples
Calculate \(\int^4_0 x^2 dx\) and compare with an analytic result
>>> from scipy import integrate >>> x2 = lambda x: x**2 >>> integrate.quad(x2, 0, 4) (21.333333333333332, 2.3684757858670003e13) >>> print(4**3 / 3.) # analytical result 21.3333333333
Calculate \(\int^\infty_0 e^{x} dx\)
>>> invexp = lambda x: np.exp(x) >>> integrate.quad(invexp, 0, np.inf) (1.0, 5.842605999138044e11)
>>> 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 \(\int^1_0 x^2 + y^2 dx\) 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, 1.4802973661668752e14) print((1.0**3/3.0 + 1.0)  (0.0**3/3.0 + 0.0)) #Analytic result # 1.3333333333333333