xfddlmZddlmZmZmZmZddlZddl Z ddl Z ddl m Z ddl mZddl mZmZddlmZgdZdgd Zdgd ZGd d eZerddlmZGddeZneZdhdZedZee_didZdjdZ dkdZ dZ!dldZ"dldZ#dZ$dldZ%dldZ&dmdZ'd Z(d!Z)d"Z* dnd#Z+dd$ddgd d%fdd&gd'd d(fd&d)gd*d+d,fd$d-gd.d/d0fdd1gd2d3d4fdd5gd6d7d8fd9d:gd;dd?gd@dAdBfdCdDgdEdFdGfddHgdIdJdKfdLdMgdNdOdPfddQgdRdSdTfdUdVgdWdXdYfd9dZgd[d\d]fd^Z,dod_Z-d`Z.e dadbdcgZ/d)dddddedfZ0y)p) annotations) TYPE_CHECKINGCallableAnycastN) namedtuple)roots_legendre)gammaln logsumexp) _rng_spawn) fixed_quad quadraturerombergromb trapezoidtrapzsimpssimpsoncumulative_trapezoidcumtrapz newton_cotesqmc_quadAccuracyWarningcttdrtj||||Stj||||S)a Integrate along the given axis using the composite trapezoidal rule. If `x` is provided, the integration happens in sequence along its elements - they are not sorted. Integrate `y` (`x`) along each 1d slice on the given axis, compute :math:`\int y(x) dx`. When `x` is specified, this integrates along the parametric curve, computing :math:`\int_t y(t) dt = \int_t y(t) \left.\frac{dx}{dt}\right|_{x=x(t)} dt`. Parameters ---------- y : array_like Input array to integrate. x : array_like, optional The sample points corresponding to the `y` values. If `x` is None, the sample points are assumed to be evenly spaced `dx` apart. The default is None. dx : scalar, optional The spacing between sample points when `x` is None. The default is 1. axis : int, optional The axis along which to integrate. Returns ------- trapezoid : float or ndarray Definite integral of `y` = n-dimensional array as approximated along a single axis by the trapezoidal rule. If `y` is a 1-dimensional array, then the result is a float. If `n` is greater than 1, then the result is an `n`-1 dimensional array. See Also -------- cumulative_trapezoid, simpson, romb Notes ----- Image [2]_ illustrates trapezoidal rule -- y-axis locations of points will be taken from `y` array, by default x-axis distances between points will be 1.0, alternatively they can be provided with `x` array or with `dx` scalar. Return value will be equal to combined area under the red lines. References ---------- .. [1] Wikipedia page: https://en.wikipedia.org/wiki/Trapezoidal_rule .. [2] Illustration image: https://en.wikipedia.org/wiki/File:Composite_trapezoidal_rule_illustration.png Examples -------- Use the trapezoidal rule on evenly spaced points: >>> import numpy as np >>> from scipy import integrate >>> integrate.trapezoid([1, 2, 3]) 4.0 The spacing between sample points can be selected by either the ``x`` or ``dx`` arguments: >>> integrate.trapezoid([1, 2, 3], x=[4, 6, 8]) 8.0 >>> integrate.trapezoid([1, 2, 3], dx=2) 8.0 Using a decreasing ``x`` corresponds to integrating in reverse: >>> integrate.trapezoid([1, 2, 3], x=[8, 6, 4]) -8.0 More generally ``x`` is used to integrate along a parametric curve. We can estimate the integral :math:`\int_0^1 x^2 = 1/3` using: >>> x = np.linspace(0, 1, num=50) >>> y = x**2 >>> integrate.trapezoid(y, x) 0.33340274885464394 Or estimate the area of a circle, noting we repeat the sample which closes the curve: >>> theta = np.linspace(0, 2 * np.pi, num=1000, endpoint=True) >>> integrate.trapezoid(np.cos(theta), x=np.sin(theta)) 3.141571941375841 ``trapezoid`` can be applied along a specified axis to do multiple computations in one call: >>> a = np.arange(6).reshape(2, 3) >>> a array([[0, 1, 2], [3, 4, 5]]) >>> integrate.trapezoid(a, axis=0) array([1.5, 2.5, 3.5]) >>> integrate.trapezoid(a, axis=1) array([2., 8.]) rxdxaxis)hasattrnprryrrrs =/usr/lib/python3/dist-packages/scipy/integrate/_quadrature.pyrrs9Pr;||Ar55xxQ2D11c t||||S)z}An alias of `trapezoid`. `trapz` is kept for backwards compatibility. For new code, prefer `trapezoid` instead. r)rr"s r$rrs Q! ..r%c eZdZy)rN)__name__ __module__ __qualname__r%r$rrsr%r)ProtocolceZdZUded<y)CacheAttributeszdict[int, tuple[Any, Any]]cacheN)r(r)r*__annotations__r+r%r$r.r.s))r%r.c"tt|SN)rr.)funcs r$cache_decoratorr4s  &&r%c|tjvrtj|St|tj|<tj|S)zX Cache roots_legendre results to speed up calls of the fixed_quad function. )_cached_roots_legendrer/r )ns r$r6r6sK  " ( ((%++A..&4Q&7  # ! ' ' **r%c,t|\}}tj|}tj|stj|r t d||z |dzzdz |z}||z dz tj |||g|zdzdfS)a Compute a definite integral using fixed-order Gaussian quadrature. Integrate `func` from `a` to `b` using Gaussian quadrature of order `n`. Parameters ---------- func : callable A Python function or method to integrate (must accept vector inputs). If integrating a vector-valued function, the returned array must have shape ``(..., len(x))``. a : float Lower limit of integration. b : float Upper limit of integration. args : tuple, optional Extra arguments to pass to function, if any. n : int, optional Order of quadrature integration. Default is 5. Returns ------- val : float Gaussian quadrature approximation to the integral none : None Statically returned value of None See Also -------- quad : adaptive quadrature using QUADPACK dblquad : double integrals tplquad : triple integrals romberg : adaptive Romberg quadrature quadrature : adaptive Gaussian quadrature romb : integrators for sampled data simpson : integrators for sampled data cumulative_trapezoid : cumulative integration for sampled data ode : ODE integrator odeint : ODE integrator Examples -------- >>> from scipy import integrate >>> import numpy as np >>> f = lambda x: x**8 >>> integrate.fixed_quad(f, 0.0, 1.0, n=4) (0.1110884353741496, None) >>> integrate.fixed_quad(f, 0.0, 1.0, n=5) (0.11111111111111102, None) >>> print(1/9.0) # analytical result 0.1111111111111111 >>> integrate.fixed_quad(np.cos, 0.0, np.pi/2, n=4) (0.9999999771971152, None) >>> integrate.fixed_quad(np.cos, 0.0, np.pi/2, n=5) (1.000000000039565, None) >>> np.sin(np.pi/2)-np.sin(0) # analytical result 1.0 z8Gaussian quadrature is only available for finite limits.@rrN)r6r!realisinf ValueErrorsum)r3abargsr7rwr#s r$r r s| "! $DAq  A xx{bhhqk*+ + 1qs C!A aC9rvvaQ.R8 8$ >>r%Fc*|rfd}|Sfd}|S)aoVectorize the call to a function. This is an internal utility function used by `romberg` and `quadrature` to create a vectorized version of a function. If `vec_func` is True, the function `func` is assumed to take vector arguments. Parameters ---------- func : callable User defined function. args : tuple, optional Extra arguments for the function. vec_func : bool, optional True if the function func takes vector arguments. Returns ------- vfunc : callable A function that will take a vector argument and return the result. c|gSr2r+)rrCr3s r$vfunczvectorize1..vfuncs>D> !r%cBtj|r |gStj|}|dg}t|}t |dt |}tj |f|}||d<td|D]}||g||<|S)NrdtyperIr:)r!isscalarasarraylengetattrtypeemptyrange)ry0r7rIoutputirCr3s r$rGzvectorize1..vfuncs{{1~A~~% 1 Aad"T"BAABb2EXXqd%0FF1I1a[ . 1--q  .Mr%r+)r3rCvec_funcrGs`` r$ vectorize1rVs 2 " L  Lr%r:c t|ts|f}t|||} tj} tj} t |dz|}t ||dzD]?} t| ||d| d} t| | z } | } | |ks| |t| zks<| | fStjd|| fzt| | fS)a Compute a definite integral using fixed-tolerance Gaussian quadrature. Integrate `func` from `a` to `b` using Gaussian quadrature with absolute tolerance `tol`. Parameters ---------- func : function A Python function or method to integrate. a : float Lower limit of integration. b : float Upper limit of integration. args : tuple, optional Extra arguments to pass to function. tol, rtol : float, optional Iteration stops when error between last two iterates is less than `tol` OR the relative change is less than `rtol`. maxiter : int, optional Maximum order of Gaussian quadrature. vec_func : bool, optional True or False if func handles arrays as arguments (is a "vector" function). Default is True. miniter : int, optional Minimum order of Gaussian quadrature. Returns ------- val : float Gaussian quadrature approximation (within tolerance) to integral. err : float Difference between last two estimates of the integral. See Also -------- romberg : adaptive Romberg quadrature fixed_quad : fixed-order Gaussian quadrature quad : adaptive quadrature using QUADPACK dblquad : double integrals tplquad : triple integrals romb : integrator for sampled data simpson : integrator for sampled data cumulative_trapezoid : cumulative integration for sampled data ode : ODE integrator odeint : ODE integrator Examples -------- >>> from scipy import integrate >>> import numpy as np >>> f = lambda x: x**8 >>> integrate.quadrature(f, 0.0, 1.0) (0.11111111111111106, 4.163336342344337e-17) >>> print(1/9.0) # analytical result 0.1111111111111111 >>> integrate.quadrature(np.cos, 0.0, np.pi/2) (0.9999999999999536, 3.9611425250996035e-11) >>> np.sin(np.pi/2)-np.sin(0) # analytical result 1.0 rUr:r+rz-maxiter (%d) exceeded. Latest difference = %e) isinstancetuplerVr!infmaxrQr abswarningswarnr)r3rArBrCtolrtolmaxiterrUminiterrGvalerrr7newvals r$rr#sB dE "w tTH 5E &&C &&C'!)W%G 7GAI & E1aQ/2&*o 9d3s8m+  8O   ;wn L   8Or%c8t|}|||<t|Sr2)listrZ)trTvaluels r$tuplesetrlxs QA AaD 8Or%c"t|||||S)zAn alias of `cumulative_trapezoid`. `cumtrapz` is kept for backwards compatibility. For new code, prefer `cumulative_trapezoid` instead. )rrrinitial)r)r#rrrrns r$rrs Q2D' JJr%ctj|}||}ntj|}|jdk(r>> from scipy import integrate >>> import numpy as np >>> import matplotlib.pyplot as plt >>> x = np.linspace(-2, 2, num=20) >>> y = x >>> y_int = integrate.cumulative_trapezoid(y, x, initial=0) >>> plt.plot(x, y_int, 'ro', x, y[0] + 0.5 * x**2, 'b-') >>> plt.show() Nr:r2If given, shape of x must be 1-D or the same as y.r<7If given, length of x along axis must be the same as y.r;z'`initial` parameter should be a scalar.rJ)r!rLndimdiffreshaperMshaper?rlslicecumsumrKrh concatenatefullrI) r#rrrrndrundslice1slice2ress r$rrsn 1 Ay  JJqM 66Q; AC!&&LEE$K % A \S\ )*+ +%A 774=AGGDMA- -*+ + QWWB uT{nR'uQ~ >F uT{nR'uT2 ?F ))A6QvY./#5D AC{{7#FG GSYYd nnbggeWCIIFL"&( Jr%c t|j}|d}d}tdf|z}t||t|||} t||t|dz|dz|} t||t|dz|dz|} |3t j || d|| zz|| z|} | |dz z} | St j ||} t||t|||}t||t|dz|dz|}| |jtd}| |jtd}||z}||z}t j||t j||dk7 }|d z || d t jd |t j||dk7 z z|| |t j||t j||dk7 zzz|| d |z zzz}t j ||} | S) Nrr:@r<@F)copyoutwhereg@r;?) rMrurvrlr!r@rsastypefloat true_divide zeros_like)r#startstoprrrr{step slice_allslice0r|r}resulthsl0sl1h0h1hsumhprodh0divh1tmps r$_basic_simpsonrs# QWWB } Dtr!I iuUD$'? @F iuU1Wd1fd'C DF iuU1Wd1fd'C DFy& C& M1AfI=DI"s(, M% GGAD !y$eT4(@Ay$eAgtAvt(DE sV]]5u] - sV]]5u] -BwR..RR]]2->bAgN3h!F)W46MM'4J6=l"DDEF)t')~~dE:<--:M>> from scipy import integrate >>> import numpy as np >>> x = np.arange(0, 10) >>> y = np.arange(0, 10) >>> integrate.simpson(y, x) 40.5 >>> y = np.power(x, 3) >>> integrate.simpson(y, x) 1640.5 >>> integrate.quad(lambda x: x**3, 0, 9)[0] 1640.25 >>> integrate.simpson(y, x, even='first') 1644.5 rNr:rprqzWThe 'even' keyword is deprecated as of SciPy 1.11.0 and will be removed in SciPy 1.13.0r stacklevelgr)avglastfirstrz>Parameter 'even' must be 'simpson', 'avg', 'last', or 'first'.r?r<rr)rr)rrrr;)r!rLrMrurtrZr?r^r_DeprecationWarningrvrlrasfarrayfloat64rssqueezerr)r#rrrrr{Nlast_dxfirst_dx returnshapeshapex saveshaperdrrr|r}slice3rhm2hm1diffsnumdenalphabetaetas r$rrsN 1 A QWWB  AGHK} JJqM qww<1 S2XF771:F4LIK %-(A \S\ )*+ + 774=A *+ +   & 1  1uz4[NR' 'tY : :-  6ir2Fir2F}F)ai/ 3=AfI& $9: :CD 9 #Aq!A#q"d;Fir2Fir2Fir2F RH%A}y$b"a0@Ay$b$0BC 27714#89ZZc 6ZZc 68adai-!ad(QqT/1Cqtad{#CNNMM#&Qh EA$!)cAaDj1Q4//Cad(C>>MM#&Qh Dadai-Cad(adQqTk*C..MM#&Qh C eAfIoQvY6QvYF FF # #ir2Fir2F}F)ai/ 3w;& !F) 34 4C#Aq!A#q"d;F ? "iq1Fiq1F}U6]+af .>> 3x<61V9!45 5C nQ1Q32t< >> from scipy import integrate >>> import numpy as np >>> x = np.arange(10, 14.25, 0.25) >>> y = np.arange(3, 12) >>> integrate.romb(y) 56.0 >>> y = np.sin(np.power(x, 2.5)) >>> integrate.romb(y) -0.742561336672229 >>> integrate.romb(y, show=True) Richardson Extrapolation Table for Romberg Integration ====================================================== -0.81576 4.63862 6.45674 -1.10581 -3.02062 -3.65245 -2.57379 -3.06311 -3.06595 -3.05664 -1.34093 -0.92997 -0.78776 -0.75160 -0.74256 ====================================================== -0.742561336672229 # may vary r:rz=Number of samples must be one plus a non-negative power of 2.NrrJr;)rrrr<rzE*** Printing table only supported for integrals of a single data set.r8z%%%d.%dfz6Richardson Extrapolation Table for Romberg Integration= )sepend r)r!rLrMrur?rvrlrrQr@rKprint TypeError IndexError)r#rrshowr{NsampsNintervr7kRrrslicem1rslice_RrrrrTjprevpreciswidthformstrtitles r$rr sz 1 A QWWB WWT]FQhG A A g+ a Q g+ G|45 5 At#I iq )Fy$+G"**Ru--A6QwZ',Q.AfIG!!E!D4 1ac] ! 7D%tT*BC  AaC8q7T)B'BBC1a& q!A# GAa1X;DQ!QqSz] 2ac A~FFAq!fI G S {{1V9% + , a Q!E6?2GLE %s5z)t >1Q3Z qs8A'Aq!fI-378  #E " # aV9!z*  z*  s$H."I.IIIIc"|dkr td|dk(rd||d||dzzS|dz }t|d|dz |z }|dd|zz}||tj|zz}tj||d}|S)aU Perform part of the trapezoidal rule to integrate a function. Assume that we had called difftrap with all lower powers-of-2 starting with 1. Calling difftrap only returns the summation of the new ordinates. It does _not_ multiply by the width of the trapezoids. This must be performed by the caller. 'function' is the function to evaluate (must accept vector arguments). 'interval' is a sequence with lower and upper limits of integration. 'numtraps' is the number of trapezoids to use (must be a power-of-2). rz#numtraps must be > 0 in difftrap().r:rrr<)r?rr!aranger@)functionintervalnumtrapsnumtosumrloxpointsss r$ _difftraprs1}>?? QHXa[)(8A;*??@@A: (1+hqk) *8 3qkC!G#q299X... FF8F#! ,r%c(d|z}||z|z |dz z S)z Compute the differences for the Romberg quadrature corrections. See Forman Acton's "Real Computing Made Real," p 143. rrr+)rBcrrs r$ _romberg_diffrs$ q&C !GaK#) $$r%cdx}}tdt|dtd|tdtddztt|D]Z}td d |z|d |dz d |zz fzdt|d zD]}td |||zdtd\tdtd|||dtdd t|d z zd zdy)NrzRomberg integration ofrrfromz %6s %9s %9s)StepsStepSizeResultsz%6d %9frr:r;z%9fzThe final result isafterzfunction evaluations.)rreprrQrM)rrresmatrTrs r$ _printresmatrs IA "DN< &( "I -: :; 3v;  i1a4(1+hqk"9BE!BCCMqs 3A %6!9Q<(c 2 3 b   "I 137 '1s6{1}%a')@Ar%c tj|stj|r tdt|||} d} ||g} ||z } t | | | } | | z}|gg}tj }|d}t d|dzD]}| dz} | t | | | z } | | z| z g}t |D]'}|jt|||||dz)||}||dz }|r|j|t||z }||ks||t|zkrn#|}tjd||fzt|r t| | ||S)a Romberg integration of a callable function or method. Returns the integral of `function` (a function of one variable) over the interval (`a`, `b`). If `show` is 1, the triangular array of the intermediate results will be printed. If `vec_func` is True (default is False), then `function` is assumed to support vector arguments. Parameters ---------- function : callable Function to be integrated. a : float Lower limit of integration. b : float Upper limit of integration. Returns ------- results : float Result of the integration. Other Parameters ---------------- args : tuple, optional Extra arguments to pass to function. Each element of `args` will be passed as a single argument to `func`. Default is to pass no extra arguments. tol, rtol : float, optional The desired absolute and relative tolerances. Defaults are 1.48e-8. show : bool, optional Whether to print the results. Default is False. divmax : int, optional Maximum order of extrapolation. Default is 10. vec_func : bool, optional Whether `func` handles arrays as arguments (i.e., whether it is a "vector" function). Default is False. See Also -------- fixed_quad : Fixed-order Gaussian quadrature. quad : Adaptive quadrature using QUADPACK. dblquad : Double integrals. tplquad : Triple integrals. romb : Integrators for sampled data. simpson : Integrators for sampled data. cumulative_trapezoid : Cumulative integration for sampled data. ode : ODE integrator. odeint : ODE integrator. References ---------- .. [1] 'Romberg's method' https://en.wikipedia.org/wiki/Romberg%27s_method Examples -------- Integrate a gaussian from 0 to 1 and compare to the error function. >>> from scipy import integrate >>> from scipy.special import erf >>> import numpy as np >>> gaussian = lambda x: 1/np.sqrt(np.pi) * np.exp(-x**2) >>> result = integrate.romberg(gaussian, 0, 1, show=True) Romberg integration of from [0, 1] :: Steps StepSize Results 1 1.000000 0.385872 2 0.500000 0.412631 0.421551 4 0.250000 0.419184 0.421368 0.421356 8 0.125000 0.420810 0.421352 0.421350 0.421350 16 0.062500 0.421215 0.421350 0.421350 0.421350 0.421350 32 0.031250 0.421317 0.421350 0.421350 0.421350 0.421350 0.421350 The final result is 0.421350396475 after 33 function evaluations. >>> print("%g %g" % (2*result, erf(1))) 0.842701 0.842701 z5Romberg integration only available for finite limits.rXr:rrz,divmax (%d) exceeded. Latest difference = %e)r!r>r?rVrr[rQappendrr]r^r_rr)rrArBrCr`rardivmaxrUrGr7rintrangeordsumrrrelast_rowrTrowr lastresults r$rrsj xx{bhhqk./ / x 9E A1vH1uH uh *F  FhZF &&CayH 1fQh  Q)E8Q//& 1$%q @A JJ}Xa[#a&!A#> ? @Qac]  MM# &:%& 9dS[00   :fc] J   UHf- Mr%r r)r:r:Zr)r:rrr:rP-) rrriii )K2rrrii@/))irrriixriC) + rrrri iri_7) `)iDrrrrii?# i^) ) }=8Krr r r r iiip) ><sB(:ihrrrrriii0 i0) I"!jmirrrrrrl& l7iR0P) @7@!!Nd7ipRr rrrrri$MY(r(r'r&r%r$r#r"l`:l@ Al@d@*)i`p`*oFg!f\a LRl@`r/r.r-r,r+r*r)lx=l7-)r:rrrr8rrrr  rrr!cv t|dz }|rtj|dz}n-tjtj|dk(rd}|rH|t vr@t |\}}}}}|tj|tz|z }|t||z fS|ddk7s|d|k7r td|t|z } d| zdz } tj|dz} | | ddtjfz} tjj| } tdD](}d| z| j| j| z } *d| ddddzz }| dddddfj||dz z}|dzdk(r|r||d zz }|dz}n ||dzz }|dz}|tj| |z|z }|dz}|tj |zt#|z }tj$|}|||zfS#t $r |}tj|dz}d}YwxYw) a Return weights and error coefficient for Newton-Cotes integration. Suppose we have (N+1) samples of f at the positions x_0, x_1, ..., x_N. Then an N-point Newton-Cotes formula for the integral between x_0 and x_N is: :math:`\int_{x_0}^{x_N} f(x)dx = \Delta x \sum_{i=0}^{N} a_i f(x_i) + B_N (\Delta x)^{N+2} f^{N+1} (\xi)` where :math:`\xi \in [x_0,x_N]` and :math:`\Delta x = \frac{x_N-x_0}{N}` is the average samples spacing. If the samples are equally-spaced and N is even, then the error term is :math:`B_N (\Delta x)^{N+3} f^{N+2}(\xi)`. Parameters ---------- rn : int The integer order for equally-spaced data or the relative positions of the samples with the first sample at 0 and the last at N, where N+1 is the length of `rn`. N is the order of the Newton-Cotes integration. equal : int, optional Set to 1 to enforce equally spaced data. Returns ------- an : ndarray 1-D array of weights to apply to the function at the provided sample positions. B : float Error coefficient. Notes ----- Normally, the Newton-Cotes rules are used on smaller integration regions and a composite rule is used to return the total integral. Examples -------- Compute the integral of sin(x) in [0, :math:`\pi`]: >>> from scipy.integrate import newton_cotes >>> import numpy as np >>> def f(x): ... return np.sin(x) >>> a = 0 >>> b = np.pi >>> exact = 2 >>> for N in [2, 4, 6, 8, 10]: ... x = np.linspace(a, b, N + 1) ... an, B = newton_cotes(N, 1) ... dx = (b - a) / N ... quad = dx * np.sum(an * f(x)) ... error = abs(quad - exact) ... print('{:2d} {:10.9f} {:.5e}'.format(N, quad, error)) ... 2 2.094395102 9.43951e-02 4 1.998570732 1.42927e-03 6 2.000017814 1.78136e-05 8 1.999999835 1.64725e-07 10 2.000000001 1.14677e-09 r:rJrrz1The sample positions must start at 0 and end at NrNr;r)rMr!rallrs Exception_builtincoeffsarrayrr?newaxislinalginvrQdotmathlogr exp)rnequalrnadavinbdbanyitinvecCCinvrTvecaiBNpowerp1facs r$rrxs;B  GAI 1Q3B VVBGGBK1$ %E  n$+A.BB "((2U+ +b 059R< 1 2! )* * eAhB R!B 99QqS>D d1bjj=!!A 99== D 1X.v --. cc1 C a1f  # !b& )B A  "X! "X! bffRY# #B qB  gbk )C ((3-C r#v:G   YYqs^sAH%H87H8c ttdsddlm}|t_ntj}t s d}t |t j|j}t j|j}t j||\}}|jd} ||zdz  t j||gj} t j"|} || k7r d }t |t j"|} || k7r d }t |||j$j'| }n-t)||j$j*s d}t ||j,|jdk7r d}t|t/|dd}|j0j3|}|dvr d}t || ||| | ||||f S#t$r} d}t|| d} ~ wwxYw#t$r-} d| d}tj |d fd } Yd} ~ Vd} ~ wwxYw)Nqmcr)statsz`func` must be callable.rz`func` must evaluate the integrand at points within the integration range; e.g. `func( (a + b) / 2)` must return the integrand at the centroid of the integration volume.zAException encountered when attempting vectorized call to `func`: z. For better performance, `func` should accept two-dimensional array `x` with shape `(len(a), n_points)` and return an array of the integrand value at each of the `n_points.rrc4tjd|S)Nr)rarr)r!apply_along_axis)rr3s r$rGz_qmc_quad_iv..vfunc s&&t"!< FTz*`log` must be boolean (`True` or `False`).)r rscipyrScallablerr! atleast_1drbroadcast_arraysrur4r?r6Tr^r_int64rRHaltonrY QMCEnginerzrN_qmccheck_random_state)r3rArBn_points n_estimatesqrngr<rSmessagedimerG n_points_intn_estimates_intrWrngs` r$ _qmc_quad_ivrks7 8U # D>,   aA aA  q! $DAq ''!*C) a!eq[ = RXXq!f    88H%L<2  hh{+Oo%5   |yy$ eii11 2L   vvH!!tZ.H ** ' ' 1C ->  1ac3 NNe ))!q( )  =S!,,  g!, = = =s0;G9 )H9 HHH I!"I  I QMCQuadResultintegralstandard_errori)rcrbrdr<c t||||||}|\ }}}}}}}} dd} dfd dfd dfd } tj||k(r=d} tj| dt |rtj dSddS||k} d | jd z}|| || c|| <|| <tj||z }||z }tj}t|j}tD]o}|j|}| jj|||j }||}| |||||<t#|dd ||i|j$}q||}| ||| }|r|dkr|tj&d zzn||z}t ||S)a Compute an integral in N-dimensions using Quasi-Monte Carlo quadrature. Parameters ---------- func : callable The integrand. Must accept a single argument ``x``, an array which specifies the point(s) at which to evaluate the scalar-valued integrand, and return the value(s) of the integrand. For efficiency, the function should be vectorized to accept an array of shape ``(d, n_points)``, where ``d`` is the number of variables (i.e. the dimensionality of the function domain) and `n_points` is the number of quadrature points, and return an array of shape ``(n_points,)``, the integrand at each quadrature point. a, b : array-like One-dimensional arrays specifying the lower and upper integration limits, respectively, of each of the ``d`` variables. n_estimates, n_points : int, optional `n_estimates` (default: 8) statistically independent QMC samples, each of `n_points` (default: 1024) points, will be generated by `qrng`. The total number of points at which the integrand `func` will be evaluated is ``n_points * n_estimates``. See Notes for details. qrng : `~scipy.stats.qmc.QMCEngine`, optional An instance of the QMCEngine from which to sample QMC points. The QMCEngine must be initialized to a number of dimensions ``d`` corresponding with the number of variables ``x1, ..., xd`` passed to `func`. The provided QMCEngine is used to produce the first integral estimate. If `n_estimates` is greater than one, additional QMCEngines are spawned from the first (with scrambling enabled, if it is an option.) If a QMCEngine is not provided, the default `scipy.stats.qmc.Halton` will be initialized with the number of dimensions determine from the length of `a`. log : boolean, default: False When set to True, `func` returns the log of the integrand, and the result object contains the log of the integral. Returns ------- result : object A result object with attributes: integral : float The estimate of the integral. standard_error : The error estimate. See Notes for interpretation. Notes ----- Values of the integrand at each of the `n_points` points of a QMC sample are used to produce an estimate of the integral. This estimate is drawn from a population of possible estimates of the integral, the value of which we obtain depends on the particular points at which the integral was evaluated. We perform this process `n_estimates` times, each time evaluating the integrand at different scrambled QMC points, effectively drawing i.i.d. random samples from the population of integral estimates. The sample mean :math:`m` of these integral estimates is an unbiased estimator of the true value of the integral, and the standard error of the mean :math:`s` of these estimates may be used to generate confidence intervals using the t distribution with ``n_estimates - 1`` degrees of freedom. Perhaps counter-intuitively, increasing `n_points` while keeping the total number of function evaluation points ``n_points * n_estimates`` fixed tends to reduce the actual error, whereas increasing `n_estimates` tends to decrease the error estimate. Examples -------- QMC quadrature is particularly useful for computing integrals in higher dimensions. An example integrand is the probability density function of a multivariate normal distribution. >>> import numpy as np >>> from scipy import stats >>> dim = 8 >>> mean = np.zeros(dim) >>> cov = np.eye(dim) >>> def func(x): ... # `multivariate_normal` expects the _last_ axis to correspond with ... # the dimensionality of the space, so `x` must be transposed ... return stats.multivariate_normal.pdf(x.T, mean, cov) To compute the integral over the unit hypercube: >>> from scipy.integrate import qmc_quad >>> a = np.zeros(dim) >>> b = np.ones(dim) >>> rng = np.random.default_rng() >>> qrng = stats.qmc.Halton(d=dim, seed=rng) >>> n_estimates = 8 >>> res = qmc_quad(func, a, b, n_estimates=n_estimates, qrng=qrng) >>> res.integral, res.standard_error (0.00018429555666024108, 1.0389431116001344e-07) A two-sided, 99% confidence interval for the integral may be estimated as: >>> t = stats.t(df=n_estimates-1, loc=res.integral, ... scale=res.standard_error) >>> t.interval(0.99) (0.0001839319802536469, 0.00018465913306683527) Indeed, the value reported by `scipy.stats.multivariate_normal` is within this range. >>> stats.multivariate_normal.cdf(b, mean, cov, lower_limit=a) 0.00018430867675187443 cx|r!t|tj|zStj||zSr2)r r!r<r@) integrandsdAr<s r$ sum_productzqmc_quad..sum_products0 Z(266":5 566*r/* *r%ct|r!t|tjz Stj|Sr2)r r!r<mean) estimatesr<rcs r$ruzqmc_quad..means. 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