(phSSKrSSKrSSKJr SSKJr SSKr/SQr"SS\ 5r SSjr S r S r SS jrSS jrSS jr"SS5r"SS5r"SS5rg)N)partial) _quadpack)quaddblquadtplquadnquadIntegrationWarningc\rSrSrSrSrg)r z' Warning on issues during integration. N)__name__ __module__ __qualname____firstlineno____doc____static_attributes__r O/var/www/html/venv/lib/python3.13/site-packages/scipy/integrate/_quadpack_py.pyr r s  rr c^[U[5(dU4nX!:[X5[X5p!nU(aoU4SjnU4Sjn[ UXX4UXgXXXSS9n[ UXX4UXgXXXSS9nUSSUS--nUSSUS--nUU4nU(a0nUSS US 'USS US 'UU4-nU$U c[ TXX4XVUU5 nO0UbS n[ R"U[SS 9 [TXX4XVXXX5 nU(a US*4USS -nUSnUS:XaUS S$SSUS3SSSSSSSS. nU S;a@U[R:XdU[R*:XaSUS'SUS'SUS'S S!S"S#S$S%.nUUnUS';ahU(aBU S;a3U[R:XdU[R*:Xa US SUW4-$US SU4-$[ R"U[SS 9 US S$US(:XGa@US::aaU[S)[RR-S*5:aS+nGO U S,;a+[!U5[!U5-[R:XaS-nOU cgUcS.nO[X5[U5s=::a[U5s=::a[X5::dO S/nO[#U5U:aS0[#U5S1S2US1S33nOoU S:aS4nOfU S;a"[!X-5[R:XaS5nO>U R%S65(a[U 5S:aS7nX!:aS8nOU S9:Xa XU4;aS:n['U5e![a US&nGNf=f);u? 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) c*>T"U/UQ76R$N)imagxargsfuncs rimfuncquad..imfunc>D>&& &rc*>T"U/UQ76R$r)realrs rrefuncquad..refuncr rF) complex_funcry?rNr"rzfBreak points cannot be specified when using weighted integrand. Continuing, ignoring specified points.) stacklevelz0.z@Invalid 'limit' argument. There must be at least one subintervalzDAll break points in 'points' must lie within the integration limits.zNumber of break points (dz') must be less than subinterval limit ()z)Chebyshev moment limit maxp1 must be >=1.zCycle limit limlst must be >=3.algz1wvar parameters (alpha, beta) must both be >= -1.z*Integration limits a, b must satistfy aParameter 'wvar' must not equal integration limits 'a' or 'b'.) isinstancetupleminmaxr_quadwarningswarnr _quad_weightnpinfKeyErrorsys float_infoepsilonabslen startswith ValueError)rabr full_outputepsabsepsrellimitpointsweightwvarwoptsmaxp1limlstr%fliprr# re_retval im_retvalintegralerror_estimateretvalmsgexpmsgiermsgsexplains` rrrsp dE " "wA 3q9QD ' 't&U< t&U< Q<"Yq\/1"19Q<7>) F&qr]F6N&qr]F6Nvi'F  ~tQ4fe  ST7(NL'5(- )D0AK1< 9 Q = Q F Q 0Q$L+ ,3i m 'Q"&&[A"&&Lcr{c7^33cr{cV++ MM#1a @#2;   Q;B!7!77??:>)s1vA"&&/H0^~3A S[LCKL3q9L2C[E)5c&k!_ECCH)1NCqyA>)c!#h"&&.@7""5))t9r>MC5FC8#A9 S/g 9os(L--M?Mc Sn U[R:waU[R*:waOU[R:XaU[R*:waSn Un OgU[R:XaU[R*:XaSn Sn O9U[R:waU[R*:XaSn Un O [S5eUc:U S:Xa[R"XX#XEXg5$[R "UW XUXVU5$U S:wa [ S5e[R"U5n XU :n XU:n [R"U S45n [R"XX+X4XVU5 $)Nrrr&r(z(Infinity comparisons don't work for you.z1Infinity inputs cannot be used with break points.)rb) r@rA RuntimeErrorr_qagse_qagierIunique concatenate_qagpe) rrJrKrrLrMrNrOrP infboundsbound the_pointss rr<r<LsAI RVV bffW  rvv+!w,  rvv+!w,  rvv+!w, EFF ~ >##D1+VR R##D%+$*E; ; >PQ Q6*J# N3J#N3JX(>?J##DQD$*E; ;rc U S;a[U S35eSSSSSSS.n U S;GaXnU[R:wa\U[R*:waGU c[R"XX+XUXVXS5 $U S nU Sn[R"XX+XXEXhU SUU5$U[R:Xa/U[R*:wa[R "XXX4XWX5 $U[R:waEU[R*:Xa0U S :XaS nOS nU4U-n[R "UU*XUXEXxU 5 $[S 5eU[R*[R4;d%U[R*[R4;a Sn[U5eU R S5(aXn[R"XX+XXEXh5 $[R"XX+X4XVU5 $)N)r1r2r6alg-logaalg-logbalg-logr7z% not a recognized weighting function.rr&r*r+)r1r2r6rmrnror0rr1c.U*nUSnU4USS-nU"U6$Nrrr rmyargsyrs rthefunc_quad_weight..thefuncs.A!!9DTF12J.F=(rc0U*nUSnU4USS-nU"U6*$rqr rrs rrurvs0A!!9DTF12J.F &M>)rz4Cannot integrate with this weight from -Inf to +Inf.zD##GaRt$/O OST T "&&"&&! !QBFF7BFF*;%;TGW% %   U # #_F##DQf$/H H##DQd$*E; ;rc4^^UU4Sjn[XX//UXgS.S9$)u 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 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. **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. For each level of integration, ``qagse`` is used for finite limits or ``qagie`` is used if either limit (or both!) are infinite. 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. 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. 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. That is, :math:`\int^{x=2}_{x=0} \int^{y=1}_{y=0} x y^2 \,dy \,dx`. >>> import numpy as np >>> from scipy import integrate >>> f = lambda y, x: x*y**2 >>> integrate.dblquad(f, 0, 2, 0, 1) (0.6666666666666667, 7.401486830834377e-15) Calculate :math:`\int^{x=\pi/4}_{x=0} \int^{y=\cos(x)}_{y=\sin(x)} 1 \,dy \,dx`. >>> f = lambda y, x: 1 >>> integrate.dblquad(f, 0, np.pi/4, np.sin, np.cos) (0.41421356237309503, 1.1083280054755938e-14) Calculate :math:`\int^{x=1}_{x=0} \int^{y=2-x}_{y=x} a x y \,dy \,dx` for :math:`a=1, 3`. >>> f = lambda y, x, a: a*x*y >>> integrate.dblquad(f, 0, 1, lambda x: x, lambda x: 2-x, args=(1,)) (0.33333333333333337, 5.551115123125783e-15) >>> integrate.dblquad(f, 0, 1, lambda x: x, lambda x: 2-x, args=(3,)) (0.9999999999999999, 1.6653345369377348e-14) Compute the two-dimensional Gaussian Integral, which is the integral of the Gaussian function :math:`f(x,y) = e^{-(x^{2} + y^{2})}`, over :math:`(-\infty,+\infty)`. That is, compute the integral :math:`\iint^{+\infty}_{-\infty} e^{-(x^{2} + y^{2})} \,dy\,dx`. >>> f = lambda x, y: np.exp(-(x ** 2 + y ** 2)) >>> integrate.dblquad(f, -np.inf, np.inf, -np.inf, np.inf) (3.141592653589777, 2.5173086737433208e-08) cz>[T5(a T"US5OT[T5(a T"US5/$T/$Nrcallablergfunhfuns r temp_rangesdblquad..temp_ranges"B!)$T!W T!)$T!W ; ;59; ;rrMrNroptsr ) rrJrKrrrrMrNrs `` rrrs)x; aV,4"5 77rc J^^^^UU4Sjn UU4Sjn XX//n [X UXS.S9$)u7 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 fixed_quad : Fixed-order Gaussian quadrature dblquad : Double integrals nquad : N-dimensional integrals romb : Integrators for sampled data simpson : Integrators 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. **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. For each level of integration, ``qagse`` is used for finite limits or ``qagie`` is used, if either limit (or both!) are infinite. 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. 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. 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. That is, :math:`\int^{x=2}_{x=1} \int^{y=3}_{y=2} \int^{z=1}_{z=0} x y z \,dz \,dy \,dx`. >>> import numpy as np >>> from scipy import integrate >>> f = lambda z, y, x: x*y*z >>> integrate.tplquad(f, 1, 2, 2, 3, 0, 1) (1.8749999999999998, 3.3246447942574074e-14) Calculate :math:`\int^{x=1}_{x=0} \int^{y=1-2x}_{y=0} \int^{z=1-x-2y}_{z=0} x y z \,dz \,dy \,dx`. Note: `qfun`/`rfun` takes arguments in the order (x, y), even though ``f`` takes arguments in the order (z, y, x). >>> f = lambda z, y, x: x*y*z >>> integrate.tplquad(f, 0, 1, 0, lambda x: 1-2*x, 0, lambda x, y: 1-x-2*y) (0.05416666666666668, 2.1774196738157757e-14) Calculate :math:`\int^{x=1}_{x=0} \int^{y=1}_{y=0} \int^{z=1}_{z=0} a x y z \,dz \,dy \,dx` for :math:`a=1, 3`. >>> f = lambda z, y, x, a: a*x*y*z >>> integrate.tplquad(f, 0, 1, 0, 1, 0, 1, args=(1,)) (0.125, 5.527033708952211e-15) >>> integrate.tplquad(f, 0, 1, 0, 1, 0, 1, args=(3,)) (0.375, 1.6581101126856635e-14) Compute the three-dimensional Gaussian Integral, which is the integral of the Gaussian function :math:`f(x,y,z) = e^{-(x^{2} + y^{2} + z^{2})}`, over :math:`(-\infty,+\infty)`. That is, compute the integral :math:`\iiint^{+\infty}_{-\infty} e^{-(x^{2} + y^{2} + z^{2})} \,dz \,dy\,dx`. >>> f = lambda x, y, z: np.exp(-(x ** 2 + y ** 2 + z ** 2)) >>> integrate.tplquad(f, -np.inf, np.inf, -np.inf, np.inf, -np.inf, np.inf) (5.568327996830833, 4.4619078828029765e-08) c>[T5(aT"USUS5OT[T5(aT"USUS5/$T/$)Nrrr)rqfunrfuns rranges0tplquad..ranges0sT*24..T!Wd1g&d*24..T!Wd1g&D D>BD Drcz>[T5(a T"US5OT[T5(a T"US5/$T/$rrrs rranges1tplquad..ranges1rrrrr) rrJrKrrrrrrMrNrrrangess ```` rrr*s4LD; 'F D"5 77rc[U5nUVs/sH n[U5(aUO [U5PM" nnUcSnUc[/5/U-n[ U[5(a[ U5/U-nO-UVs/sH n[U5(aUO [ U5PM" nn[ XX45R"U6$s snfs snf)u# 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. Where ``t0, ... tm`` are extra arguments passed in args. 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 variables and args. The ``xx`` array contains the coordinates and extra arguments. ``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. ``ranges[0]`` 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 ``ranges[0]`` 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., ``opts[0]`` 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`. 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 Notes ----- For valid results, the integral must converge; behavior for divergent integrals is not guaranteed. **Details of QUADPACK level routines** `nquad` 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`. 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. Examples -------- Calculate .. math:: \int^{1}_{-0.15} \int^{0.8}_{0.13} \int^{1}_{-1} \int^{1}_{0} f(x_0, x_1, x_2, x_3) \,dx_0 \,dx_1 \,dx_2 \,dx_3 , where .. math:: f(x_0, x_1, x_2, x_3) = \begin{cases} x_0^2+x_1 x_2-x_3^3+ \sin{x_0}+1 & (x_0-0.2 x_3-0.5-0.25 x_1 > 0) \\ x_0^2+x_1 x_2-x_3^3+ \sin{x_0}+0 & (x_0-0.2 x_3-0.5-0.25 x_1 \leq 0) \end{cases} . >>> import numpy as np >>> 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) >>> 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}) Calculate .. math:: \int^{t_0+t_1+1}_{t_0+t_1-1} \int^{x_2+t_0^2 t_1^3+1}_{x_2+t_0^2 t_1^3-1} \int^{t_0 x_1+t_1 x_2+1}_{t_0 x_1+t_1 x_2-1} f(x_0,x_1, x_2,t_0,t_1) \,dx_0 \,dx_1 \,dx_2, where .. math:: f(x_0, x_1, x_2, t_0, t_1) = \begin{cases} x_0 x_2^2 + \sin{x_1}+2 & (x_0+t_1 x_1-t_0 > 0) \\ x_0 x_2^2 +\sin{x_1}+1 & (x_0+t_1 x_1-t_0 \leq 0) \end{cases} and :math:`(t_0, t_1) = (0, 1)` . >>> def func2(x0, x1, x2, t0, t1): ... return x0*x2**2 + np.sin(x1) + 1 + (1 if x0+t1*x1-t0>0 else 0) >>> def lim0(x1, x2, t0, t1): ... return [t0*x1 + t1*x2 - 1, t0*x1 + t1*x2 + 1] >>> def lim1(x2, t0, t1): ... return [x2 + t0**2*t1**3 - 1, x2 + t0**2*t1**3 + 1] >>> def lim2(t0, t1): ... return [t0 + t1 - 1, t0 + t1 + 1] >>> def opts0(x1, x2, t0, t1): ... return {'points' : [t0 - t1*x1]} >>> def opts1(x2, t0, t1): ... return {} >>> def opts2(t0, t1): ... return {} >>> integrate.nquad(func2, [lim0, lim1, lim2], args=(0,1), ... opts=[opts0, opts1, opts2]) (36.099919226771625, 1.8546948553373528e-07) r )rGr _RangeFuncdictr8_OptFunc_NQuad integrate)rrrrrLdepthrngopts rr r s\ KECI J6CXc]]c 376F J | |RzE!$%'CGH4Cx}}(3-74H $ 2 < rr3NNNNr3r3F)r rr)NNF)rCr= functoolsrrnumpyr@__all__ UserWarningr rr<r?rrr rrrr rrrs  G  ELJL!&up ;F4;nA7HAHP7fyDx  0*0*r