scipy.integrate.quad

scipy.integrate.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)

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.

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 \(\int^4_0 x^2 dx\) 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 \(\int^\infty_0 e^{-x} dx\)

>>> 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

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, 1.4802973661668752e-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)