a Re@sddlmZddlmZmZmZmZddlZddl Z ddl Z ddl m Z ddl mZddl mZmZddlmZgdZdd d Zdd dZGdddeZerddlmZGdddeZneZdddddZeddZee_dddZddd Zdd%d&Z d'd(Z!dd)d*Z"dd+d,Z#d-d.Z$dd/d0Z%dd1d2Z&dd3d4Z'd5d6Z(d7d8Z)d9d:Z*dd=d>Z+d$d?d$d$gd d@fd$dAgdBd dCfdAdDgdEdFdGfd?dHgdIdJdKfddLgdMdNdOfd$dPgdQdRdSfdTdUgdVdWdXfdYdZgd[d\d]fd^d_gd`dadbfddcgdddedffdgdhgdidjdkfd$dlgdmdndofdpdqgdrdsdtfdTdugdvdwdxfdyZ,ddzd{Z-d|d}Z.e d~ddgZ/dDddddddZ0dS)) annotations) TYPE_CHECKINGCallableAnycastN) namedtuple)roots_legendre)gammaln logsumexp) _rng_spawn) fixed_quad quadraturerombergromb trapezoidtrapzsimpssimpsoncumulative_trapezoidcumtrapz newton_cotesqmc_quadAccuracyWarning?cCs2ttdrtj||||dStj||||dSdS)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.]) rxdxaxisN)hasattrnprryrrrr#O/var/www/sunrise/env/lib/python3.9/site-packages/scipy/integrate/_quadrature.pyrsh rcCst||||dS)z}An alias of `trapezoid`. `trapz` is kept for backwards compatibility. For new code, prefer `trapezoid` instead. r)rr!r#r#r$rsrc@s eZdZdS)rN)__name__ __module__ __qualname__r#r#r#r$rsr)Protocolc@seZdZUded<dS)CacheAttributeszdict[int, tuple[Any, Any]]cacheN)r%r&r'__annotations__r#r#r#r$r)s r)r)funcreturncCs tt|SN)rr)r,r#r#r$cache_decoratorsr0cCs,|tjvrtj|St|tj|<tj|S)zX Cache roots_legendre results to speed up calls of the fixed_quad function. )_cached_roots_legendrer*r)nr#r#r$r1s  r1r#cCsxt|\}}t|}t|s*t|r2td|||dd|}||dtj|||g|RdddfS)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)r1r realisinf ValueErrorsum)r,abargsr2rwr"r#r#r$r s >  r Fcs&|rfdd}nfdd}|S)aoVectorize the call to a function. This is an internal utility function used by `romberg` and `quadrature` to create a vectorized version of a function. If `vec_func` is True, the function `func` is assumed to take vector arguments. Parameters ---------- func : callable User defined function. args : tuple, optional Extra arguments for the function. vec_func : bool, optional True if the function func takes vector arguments. Returns ------- vfunc : callable A function that will take a vector argument and return the result. cs|gRSr.r#rr=r,r#r$vfuncszvectorize1..vfunccst|r|gRSt|}|dgR}t|}t|dt|}tj|f|d}||d<td|D]}||gR||<qr|S)NrdtyperBr4)r isscalarasarraylengetattrtypeemptyrange)rZy0r2rBoutputir@r#r$rAs  r#)r,r=vec_funcrAr#r@r$ vectorize1s rN"\O>2Tr4c Cst|ts|f}t|||d} tj} tj} t|d|}t||dD]D} t| ||d| d} t| | } | } | |ks| |t| krFqqFt d|| ft | | 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 rMr4r#rz-maxiter (%d) exceeded. Latest difference = %e) isinstancetuplerNr infmaxrJr abswarningswarnr)r,r;r<r=tolrtolmaxiterrMZminiterrAvalerrr2Znewvalr#r#r$r #s"A   r cCst|}|||<t|Sr.)listrS)trLvaluelr#r#r$tuplesetxsrbcCst|||||dS)zAn alias of `cumulative_trapezoid`. `cumtrapz` is kept for backwards compatibility. For new code, prefer `cumulative_trapezoid` instead. )rrrinitial)r)r"rrrrcr#r#r$rsrc CsZt|}|dur|}nt|}|jdkrVt|}dg|j}d||<||}n,t|jt|jkrttdntj||d}|j||j|dkrtdt|j}tt df||t dd}tt df||t dd} tj ||||| d|d} |durVt |s$tdt | j}d||<tj tj||| jd | g|d} | S) a Cumulatively integrate y(x) using the composite trapezoidal rule. Parameters ---------- y : array_like Values to integrate. x : array_like, optional The coordinate to integrate along. If None (default), use spacing `dx` between consecutive elements in `y`. dx : float, optional Spacing between elements of `y`. Only used if `x` is None. axis : int, optional Specifies the axis to cumulate. Default is -1 (last axis). initial : scalar, optional If given, insert this value at the beginning of the returned result. Typically this value should be 0. Default is None, which means no value at ``x[0]`` is returned and `res` has one element less than `y` along the axis of integration. Returns ------- res : ndarray The result of cumulative integration of `y` along `axis`. If `initial` is None, the shape is such that the axis of integration has one less value than `y`. If `initial` is given, the shape is equal to that of `y`. See Also -------- numpy.cumsum, numpy.cumprod quad : adaptive quadrature using QUADPACK romberg : adaptive Romberg quadrature quadrature : adaptive Gaussian quadrature fixed_quad : fixed-order Gaussian quadrature dblquad : double integrals tplquad : triple integrals romb : integrators for sampled data ode : ODE integrators odeint : ODE integrators Examples -------- >>> 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() Nr4r2If given, shape of x must be 1-D or the same as y.r67If given, length of x along axis must be the same as y.r5z'`initial` parameter should be a scalar.rC)r rEndimdiffreshaperFshaper9rbsliceZcumsumrDr^Z concatenatefullrB) r"rrrrcdrindslice1slice2resr#r#r$rs67        "   rc Cst|j}|durd}d}tdf|}t||t|||} t||t|d|d|} t||t|d|d|} |durtj|| d|| || |d} | |d9} ntj||d} t||t|||}t||t|d|d|}| |jtdd}| |jtdd}||}||}tj ||t ||dkd }|d || d tj d |t ||dkd || |tj ||t ||dkd || d |}tj||d} | S) Nrr4@r6@F)copyoutwhereg@r5r) rFrirjrbr r:rgZastypefloat true_divide zeros_like)r"startstoprrrrmstep slice_allslice0rnroresulthZsl0Zsl1Zh0h1ZhsumZhprodZh0divh1tmpr#r#r$_basic_simpsonsH &   rcCst|||||dS)zyAn alias of `simpson`. `simps` is kept for backwards compatibility. For new code, prefer `simpson` instead. )rrreven)r)r"rrrrr#r#r$r srcCsft|}t|j}|j|}|}|}d} |durt|}t|jdkr|dg|} |jd| |<|j} d} |t| }nt|jt|jkrtd|j||krtd|durtjdt dd|ddkr>> from scipy import integrate >>> import numpy as np >>> x = np.arange(0, 10) >>> y = np.arange(0, 10) >>> integrate.simpson(y, x) 40.5 >>> y = np.power(x, 3) >>> integrate.simpson(y, x) 1640.5 >>> integrate.quad(lambda x: x**3, 0, 9)[0] 1640.25 >>> integrate.simpson(y, x, even='first') 1644.5 rNr4rdrezWThe 'even' keyword is deprecated as of SciPy 1.11.0 and will be removed in SciPy 1.13.0rq stacklevelgr)avglastfirstrz>Parameter 'even' must be 'simpson', 'avg', 'last', or 'first'.r?r6rurs)rr)rrrr5)r rErFrirhrSr9rWrXDeprecationWarningrjrbrZasfarrayfloat64rgZsqueezeryrz)r"rrrrrmNZlast_dxZfirst_dxZ returnshapeZshapexZ saveshaper\rr~rnroZslice3rZhm2Zhm1ZdiffsnumZdenalphabetaetar#r#r$rsg                $  (           rc Cst|}t|j}|j|}|d}d}d}||krH|dK}|d7}q.||krXtdi} tdf|} t| |d} t| |d} |tj|td} || || d| | d<| }|}}}td|dD]}|dL}t||t|||}|dL}d | |ddf| ||j |d | |df<td|dD]J}| ||df}||| |d|dfdd |>d| ||f<q4| d} q|r|t | dst d nz |d}Wnt t fyd }Yn0z |d}Wnt t fyd}Yn0d||f}d}t |dt|dddt|dD]8}t|dD]}t || ||fddqBt q2t dt|| ||fS)a Romberg integration using samples of a function. Parameters ---------- y : array_like A vector of ``2**k + 1`` equally-spaced samples of a function. dx : float, optional The sample spacing. Default is 1. axis : int, optional The axis along which to integrate. Default is -1 (last axis). show : bool, optional When `y` is a single 1-D array, then if this argument is True print the table showing Richardson extrapolation from the samples. Default is False. Returns ------- romb : ndarray The integrated result for `axis`. See Also -------- quad : adaptive quadrature using QUADPACK romberg : adaptive Romberg quadrature quadrature : adaptive Gaussian quadrature fixed_quad : fixed-order Gaussian quadrature dblquad : double integrals tplquad : triple integrals simpson : integrators for sampled data cumulative_trapezoid : cumulative integration for sampled data ode : ODE integrators odeint : ODE integrators Examples -------- >>> 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 r4rz=Number of samples must be one plus a non-negative power of 2.NrrCr5)rrrr6rqzE*** Printing table only supported for integrals of a single data set.r3z%%%d.%dfz6Richardson Extrapolation Table for Romberg Integration= )sepend r)r rErFrir9rjrbrxrJr:rDprint TypeError IndexError)r"rrshowrmZNsampsZNintervr2kRr~rZslicem1rZslice_Rr{r|r}rLjprevZpreciswidthZformstrtitler#r#r$r s`=       08        rcCs|dkrtdn||dkr6d||d||dS|d}t|d|d|}|dd|}||t|}tj||dd}|SdS)aU Perform part of the trapezoidal rule to integrate a function. Assume that we had called difftrap with all lower powers-of-2 starting with 1. Calling difftrap only returns the summation of the new ordinates. It does _not_ multiply by the width of the trapezoids. This must be performed by the caller. 'function' is the function to evaluate (must accept vector arguments). 'interval' is a sequence with lower and upper limits of integration. 'numtraps' is the number of trapezoids to use (must be a power-of-2). rz#numtraps must be > 0 in difftrap().r4rrqr6N)r9rxr aranger:)functionintervalZnumtrapsZnumtosumrZloxZpointssr#r#r$ _difftraps  rcCsd|}||||dS)z Compute the differences for the Romberg quadrature corrections. See Forman Acton's "Real Computing Made Real," p 143. rrrr#)r<crrr#r#r$ _romberg_diffsrcCsd}}tdt|ddtd|tdtddtt|D]b}td d ||d |dd |fddt|d D]}td |||ddqtdqDtdtd|||ddtdd t|d d ddS)NrzRomberg integration ofrrfromz %6s %9s %9s)ZStepsZStepSizeZResultsz%6d %9frqr4r5z%9fzThe final result isafterzfunction evaluations.)rreprrJrF)rrresmatrLrr#r#r$ _printresmats  , r`sbO> c  CsLt|st|rtdt|||d} d} ||g} ||} t| | | } | | }|gg}tj}|d}td|dD]}| d9} | t| | | 7} | | | g}t|D]"}|t|||||dq||}||d}|r||t ||}||ks||t |krq6|}qvt d||ft |rHt | | ||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.rQr4rrqz,divmax (%d) exceeded. Latest difference = %e)r r8r9rNrrTrJappendrrVrWrXrr)rr;r<r=rYrZrZdivmaxrMrAr2rZintrangeZordsumrrr]Zlast_rowrLrowrZ lastresultr#r#r$rs@U        rrq r)r4r4Zr)r4rrr4rP-) rrriii )KrPrPrrii@/))irrriixriC) + rrrri iri_7) `)iDrrrrii?# i^) ) }=8Krrrrriiip) ><sB(:ihrrrrriii0 i0) I"!jmirrrrrrl& l7iR0P) @7@!!Nd7ipRrrrrrri$MY(rrrrrrrl`:l@ Al@d@*)i`p`*oFg!f\a LRl@`rrrrrrrlx=l7-)r4rqrrr3rrrrrrrrcCsz>> 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 r4rCrrz1The sample positions must start at 0 and end at NrqNr5rs)rFr rallrg Exception_builtincoeffsarrayrxr9ZnewaxisZlinalginvrJdotmathlogr exp)ZrnequalrnadavinbdbZanyiZtiZnvecCZCinvrLZvecZaiBNpowerp1Zfacr#r#r$rxsFA        $    rc sttdsddlm}|t_ntj}ts8d}t|t|}t|}t ||\}}|j d} z||dWn2t y} zd}t || WYd} ~ n d} ~ 00zt ||gj} WnLt y} z2d| d}tj|d d fd d } WYd} ~ n d} ~ 00t|} || kr@d }t|t|} || kr`d}t||durx|j| }nt||jjsd}t||j|j dkrd}t |t|dd}|j|}|dvrd}t|| ||| | ||||f S)Nqmcr)statsz`func` must be callable.rqz`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.rrcstjd|dS)Nr)rZarr)r Zapply_along_axisr?r/r#r$rA sz_qmc_quad_iv..vfuncz`n_points` must be an integer.z!`n_estimates` must be an integer.z8`qrng` must be an instance of scipy.stats.qmc.QMCEngine.z`qrng` must be initialized with dimensionality equal to the number of variables in `a`, i.e., `qrng.random().shape[-1]` must equal `a.shape[0]`.rng_seed>FTz*`log` must be boolean (`True` or `False`).)rrZscipyr callablerr Z atleast_1drtbroadcast_arraysrirr9rTrWrXint64r ZHaltonrRZ QMCEnginerlrGZ_qmcZcheck_random_state)r,r;r<n_points n_estimatesqrngrr messageZdimerAZ n_points_intZn_estimates_intr rngr#r/r$ _qmc_quad_ivs^    "        r QMCQuadResultintegralstandard_errori)rrrrc st||||||}|\ }}}}}}}} ddd} dfdd dfdd dfd d } t||krd } tj| d dt|rtj nddS||k} d| jdd}|| || || <|| <t||}||}t }t |j }t D]V}| |}| j|||j}||}| |||||<t|fd||i|j}q||}| |||d}|r|dkr|tjdn||}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 FcSs(|rt|t|St||SdSr.)r r rr:) integrandsdArr#r#r$ sum_productszqmc_quad..sum_productcs$|rt|tSt|SdSr.)r r rmean) estimatesr)rr#r$rszqmc_quad..meanNrcs||p ||}|rjt||\}}t||tjdf}t|dd}tdtd|t|Stj||dSdS)N?rr6rrq)ddof)r rZvstackpir r7rstd)rmr!rtemprg)rrr#r$r#s  zqmc_quad..stdcsJ|p ||}|p ||d|d}|r8|dtS|tSdS)Nr4)r!rr)r rsqrt)rr$rrrrr#r#r$sems zqmc_quad..semz^A lower limit was equal to an upper limit, so the value of the integral is zero by definition.rqrrr6seed)r$rr )F)F)NrF)NNF)rr anyrWrXrrTr:prodzerosr rrJrandomr ZscalerrHZ _init_quadr")r,r;r<rrrrr=rr rr(rZi_swapsignArrZrngsrLsamplerrrrr#r'r$r2s6n         &r)Nrr)Nrr)r#r3)r#F)r#rOrOrPTr4)NrrN)NrrN)NrrN)NrrN)rrF)r#rrFrF)r)1 __future__rtypingrrrrnumpyr rrW collectionsrZ scipy.specialrr r Zscipy._lib._utilr __all__rrWarningrr(r)r0r1dictr*r rNr rbrrrrrrrrrrrrrrrr#r#r#r$s     p    G - U ]' x          # mJ