package scipy

Get an attribute of this module as a Py.Object.t. This is useful to pass a Python function to another function.

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 : float The limits of integration in x: `a` < `b` gfun : callable or float The lower boundary curve in y which is a function taking a single floating point argument (x) and returning a floating point result or a float indicating a constant boundary curve. hfun : callable or float 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. `dblquad`` tries to obtain an accuracy of ``abs(i-result) <= max(epsabs, epsrel*abs(i))`` where ``i`` = inner integral of ``func(y, x)`` from ``gfun(x)`` to ``hfun(x)``, and ``result`` is the numerical approximation. See `epsrel` below. epsrel : float, optional Relative tolerance of the inner 1-D integrals. Default is 1.49e-8. If ``epsabs <= 0``, `epsrel` must be greater than both 5e-29 and ``50 * (machine epsilon)``. See `epsabs` above.

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

Examples --------

Compute the double integral of ``x * y**2`` over the box ``x`` ranging from 0 to 2 and ``y`` ranging from 0 to 1.

>>> from scipy import integrate >>> f = lambda y, x: x*y**2 >>> integrate.dblquad(f, 0, 2, lambda x: 0, lambda x: 1) (0.6666666666666667, 7.401486830834377e-15)

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, scipy.LowLevelCallable 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.

If the user desires improved integration performance, then `f` may be a `scipy.LowLevelCallable` with one of the signatures::

double func(int n, double *xx) double func(int n, double *xx, void *user_data)

where ``n`` is the number of extra parameters and args is an array of doubles of the additional parameters, the ``xx`` array contains the coordinates. The ``user_data`` is the data contained in the `scipy.LowLevelCallable`. ranges : iterable object Each element of ranges may be either a sequence of 2 numbers, or else a callable that returns such a sequence. ``ranges0`` 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, as well as any parametric arguments. e.g., if ``func = f(x0, x1, x2, t0, t1)``, then ``ranges0`` may be defined as either ``(a, b)`` or else as ``(a, b) = range0(x1, x2, t0, t1)``. args : iterable object, optional Additional arguments ``t0, ..., tn``, required by `func`, `ranges`, and ``opts``. 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.quad are 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., opts0 corresponds to integration over x0, and so on. If a callable, the signature must be the same as for ``ranges``. 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

For more information on these options, see `quad` and `quad_explain`.

full_output : bool, optional Partial implementation of ``full_output`` from scipy.integrate.quad. The number of integrand function evaluations ``neval`` can be obtained by setting ``full_output=True`` when calling nquad.

Returns ------- result : float The result of the integration. abserr : float The maximum of the estimates of the absolute error in the various integration results. out_dict : dict, optional A dict containing additional information on the integration.

See Also -------- quad : 1-D 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,{},{},{}, full_output=True) (1.5267454070738633, 2.9437360001402324e-14, 'neval': 388962)

>>> 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)

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 ``xx0 == 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. 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 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 ``EI[1], ..., EI[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 ``(pts1, pts2)`` where ``pts0`` and ``pts2`` are adjacent elements of ``infodict'pts'``, then (aa,bb) has level l if ``|bb-aa| = |pts2-pts1| * 2**(-l)``. 'ndin' A rank-1 integer array of length P+2. After the first integration over the intervals (pts1, pts2), 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.

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

Calculate :math:`\int^1_0 x^2 + y^2 dx` with ctypes, holding y parameter as 1::

testlib.c => double func(int n, double argsn) 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)

Print extra information about integrate.quad() parameters and returns.

Parameters ---------- output : instance with 'write' method, optional Information about `quad` is passed to ``output.write()``. Default is ``sys.stdout``.

Returns ------- None

Examples -------- We can show detailed information of the `integrate.quad` function in stdout:

>>> from scipy.integrate import quad_explain >>> quad_explain()

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 : float The limits of integration in x: `a` < `b` gfun : function or float The lower boundary curve in y which is a function taking a single floating point argument (x) and returning a floating point result or a float indicating a constant boundary curve. hfun : function or float The upper boundary curve in y (same requirements as `gfun`). qfun : function or float The lower boundary surface in z. It must be a function that takes two floats in the order (x, y) and returns a float or a float indicating a constant boundary surface. rfun : function or float The upper boundary surface in z. (Same requirements as `qfun`.) args : tuple, optional 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

Examples --------

Compute the triple integral of ``x * y * z``, over ``x`` ranging from 1 to 2, ``y`` ranging from 2 to 3, ``z`` ranging from 0 to 1.

>>> from scipy import integrate >>> f = lambda z, y, x: x*y*z >>> integrate.tplquad(f, 1, 2, lambda x: 2, lambda x: 3, ... lambda x, y: 0, lambda x, y: 1) (1.8750000000000002, 3.324644794257407e-14)