scipy.integrate._quadpack_py:quad

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 weighted integrals with 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. The integration is performed using a 21-point Gauss-Kronrod quadrature within each subinterval.

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 ${\int }_a^b\cos (\omega x)f(x)dx$ or ${\int }_a^b\sin (\omega x)f(x)dx$ over a finite interval [a,b], where $\omega$ and $f$ 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 $f(x)$.

qawfe

calculates the Fourier transform ${\int }_a^{\infty }\cos (\omega x)f(x)dx$ or ${\int }_a^{\infty }\sin (\omega x)f(x)dx$ for user-provided $\omega$ and $f$. The procedure of QAWO is applied on successive finite intervals, and convergence acceleration by means of the $\epsilon$-algorithm is applied to the series of integral approximations.

qawse

approximate ${\int }_a^{b}w(x)f(x)dx$, with $a<b$ where $w(x)=(x-a{)}^{\alpha }(b-x{)}^{\beta }v(x)$ with $\alpha ,\beta >-1$, where $v(x)$ may be one of the following functions: $1$, $\log (x-a)$, $\log (b-x)$, $\log (x-a)\log (b-x)$.

The user specifies $\alpha$, $\beta$ and the type of the function $v$. A globally adaptive subdivision strategy is applied, with modified Clenshaw-Curtis integration on those subintervals which contain a or b.

qawce

compute ${\int }_a^{b}f(x)/(x-c)dx$ where the integral must be interpreted as a Cauchy principal value integral, for user specified $c$ and $f$. The strategy is globally adaptive. Modified Clenshaw-Curtis integration is used on those intervals containing the point $x=c$.

Integration of Complex Function of a Real Variable

A complex valued function, $f$, of a real variable can be written as $f=g+ih$. Similarly, the integral of $f$ can be written as

assuming that the integrals of $g$ and $h$ exist over the interval $[a,b]$ ^[2]. Therefore, quad integrates complex-valued functions by integrating the real and imaginary components separately.

Array API Standard Support

quad has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

====================  ====================  ====================
Library               CPU                   GPU
====================  ====================  ====================
NumPy                 ✅                     n/a                 
CuPy                  n/a                   ⛔                   
PyTorch               ⛔                     ⛔                   
JAX                   ⛔                     ⛔                   
Dask                  ⛔                     n/a                 
====================  ====================  ====================

See dev-arrayapi for more information.